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MATHEMATICS 3ºESO Alumno/a:__________________________

Curso:______

I.E.S. “MIGUEL DE CERVANTES”

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UNIT 1: FRACTIONS AND DECIMALS….on page 3

UNIT 2: POWERS AND ROOTS ……………..on page 20

UNIT 3: ALGEBRAIC EXPRESSIONS……..on page 30

UNIT 4: EQUATIONS…………………………………on page 44

UNIT 5: SYSTEMS OF

LINEAR EQUATIONS………………………………….on page 69

UNIT 6: PROPORTIONALITY…………………..on page 86

UNIT 7: PROGRESSIONS………………………….on page 106

UNIT 8 and 9: PLANE SHAPES

AND SOLIDS………………………………………………..on page 122

UNIT 11: FUNCTIONS………………………………on page 146

UNIT 12: LINEAR FUNCTIONS…………….on page 159

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UNIT 1

FRACTIONS AND DECIMALS

1. FRACTIONS

Exercise 1:

4

5

Exercise 2:

Exercise 3:

6

Exercise 4

Exercise 5:

7

Exercise 6:

Order of operations with fractions:

Remember: 1. Brackets.

2. Powers and roots.

3. Divisions and multiplications in the order they

appear.

4. Additions and subtractions in the order they

appear.

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Exercise 7:

Exercise 8:

Calculate, giving your answers in their simplest form.

a)

b)

c)

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10

Exercise 9:

In a group of 160 students,

were female. How many students were female?

Exercise 10:

Three – eighths of a cake that weighed 1.2 kg has been eaten. How much

does the remaining cake weigh?

Exercise 11:

This list gives the numbers of trees in a small wood:

beech: 32, oak: 74, elm: 2, ash: 15, chestnut: 15, yew: 1.

List each type as a fraction of the number of trees in the wood.

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Exercise 12:

Of the people invited to the party,

could not come because of illness and

could not come because of transport problems. What fraction of those invited

could not come?

Exercise 13:

Sadie has already driven

of the distance between college and home. She

wants to split the remaining distance into 5 equal parts. What fraction of the

whole journey is each part?

Exercise 14:

John eats

of a bar of chocolate. Linda eats

of what remains. What

fraction of the bar of chocolate have they eaten between them?

Exercise 15:

Complete this magic square. All the rows, columns and diagonals must add the

same.

1

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Exercise 16:

In an orchestra three fifths of the musicians are men and there are 16

women. How many musicians are there in the orchestra?

Exercise 17:

Three quarters of a kilo of cheese costs €9.75. How much does one kilo of

cheese cost?

Exercise 18:

On a tree plantation, 3 out of every 20 trees have been cut down. If 840

trees have been cut down, how many are left?

Exercise 19:

A company advertises some job vacancies and then hires 18 new employees.

This is three fifths of all of the applicants. How many people applied for

work?

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Exercise 20:

A shelf in a supermarket holds 80 one-quarter litre bottles and 44 one and a

half litre bottles. How many litres of water are there on the shelf?

Exercise 21:

A lorry’s tank contains 225 litres of diesel, and the gauge says the tank is ¾

full. How many litres can the tank hold?

Exercise 22:

Alberto moves forward 5/6 of metre with each step. How many steps must he

take to complete a 9 kilometre walk?

Exercise 23:

In a bicycle race, cyclist A has covered 4/5 of the total route and has 21 km

left before the finish line. How many kilometers are left before cyclist B

reaches the final line, if he has covered 6/7 of the route?

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Exercise 24:

John is planning for his holiday. He calculates that if he spends a third of his

saving on a plane ticket and a quarter on a hotel, he will still have €570 left.

How much does he have saved?

Exercise 25:

A farmer sells two-fifths of his corn harvest to an animal feed factory, and

a third to a neighbouring farmer. If he still has 40 tonnes left, how many

tones did he harvest?

2. DECIMAL NUMBERS

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Converting fractions to decimals

Converting decimals to fractions

. To convert a terminating decimal to a fraction:

1. As the numerator, we use the number without the decimal point.

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2. As the denominator, we use 1 followed by as many zeroes as digits as the

decimal part has.

3. Simplify the fraction.

. To convert a recurring decimal to a fraction:

a) When the repeating digits are after the decimal point (pure recurring decimal)

1. As the numerator, we use the number without the decimal point minus the

whole number part.

2. As the denominator, we use as many nines as digits as the period has.

3. Simplify the fraction.

b) When the repeating digits aren´t after the decimal point, the number has some

digits between the decimal point and the period (mixed recurring decimal)

1. As the numerator, we use the number without the decimal point minus

all digits before the repeating numbers.

2. As the denominator, we use as many nines as digits as the period has

followed by as many zeroes as digits are between the decimal point and the

repeating numbers.

Exercise 26:

Convert these decimals to fractions. Give your answers in their simplest form.

a) 0.32 b) 4.5 c) 5.3333… d) 8.35555…

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3. RATIONAL AND IRRATIONAL NUMBERS

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Exercise 27:

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UNIT 2 POWERS AND ROOTS

1. POWERS

Properties of powers

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Exercise 1:

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Exercise 2:

Exercise 3:

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Exercise 4:

Express with a positive index power:

b)

c)

d)

2. ROOTS

Square roots

Cube roots

Nth roots

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Exercise 5:

3. RADICALS

When irrational numbers are written in a form using nth roots, they are called

radicals and they give the value exactly. Radicals are exact answers, but their

decimal equivalents are not.

Rules for operating radicals

There are rules you can use to manipulate and simplify radicals. Here is a

summary of those you need to know. When you use them, look for factors

that are square numbers!

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Exercise 6:

Simplify each of the following expressions, leaving your answer in radical form:

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Exercise 7:

Simplify each of the following expressions, leaving your answer in radical form:

Exercise 8:

Convert the following fractions so that they share a common denominator, then

simplify them:

a)

b)

c)

d)

e)

f)

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4. STANDARD INDEX FORM FOR LARGE AND

SMALL NUMBERS

Operations with numbers in standard form

.Addition and subtraction

1) First we express the two amounts using the same power with a base

of ten. It´s better if you use the largest index.

2) Add or subtract the significant digits and keep the power.

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3) If the new significant digit has only a digit before the decimal

point, you have finished. If it has more than one or if it is the zero

you must move the decimal point until you only have one number before

the decimal point.

Example

.Multiplication and division

1) First multiply or divide the significant digits.

2) Multiply or divide the powers using their properties.

3) The same rule 3 as adding and subtracting in standard form.

Example

Exercise 9:

Write in standard form:

a) 493,000,000 b) 315,000,000,000

c) 0.0004464 d) 12.00056

e) 253 f) 2567.23

Exercise 10:

Write, with all their digits, the following numbers written in standard form:

a) 2.51 ·106 b) 9.32 ·10-8 c) 3,76 ·1012

index↑

decimal

point←

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Exercise 11:

Solve these operations using the standard form:

a) 7.77 ·109 – 6.5 ·107 b) 0.05 ·102 + 1.3 ·103

c) 3.73 ·10-2 + 5.1 ·102 d) (3.4 ·103) · (2.52 ·10-2)

e) (7.5 ·107) : (3·103) f) (8.06 ·109) · (6.5 ·107)

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UNIT 3

Algebraic expressionS

1. ALGEBRAIC EXPRESSIONS

Algebra is a branch of mathematics in which symbols, usually letters of the

alphabet, represent numbers or members of a specified set and are used to

represent quantities so that we can use letters for the arithmetical

operations such as +, −, ×, ÷ and the power.

What do you do when you want to refer to a number that you do not know?

Suppose you wanted to refer to the number of buildings in your town, but

haven't counted them yet. You could say 'blank' number of buildings, or

perhaps '?' number of buildings.

In mathematics, letters are often used to represent numbers that we don’t

know - so you could say 'x' number of buildings, or 'q' number of buildings.

An algebraic expression has numbers and letters linked by operations. The

letters are called variables. Every addend is called term.

The expression 5n + 3s has two terms: 5n and 3s.

You can simplify an algebraic expression by collecting like terms.

Like terms have exactly the same letters. ( 3x2 and −5x2 are like terms).

Exercise 1:

Simplify these expressions:

a) 4x 2y 2x 3y b) 7p 3q 5q p c) 5c 2b 2c 3b

Exercise 2:

In one month, Dan sends x texts.

a) Alice sends 4 times as many texts as Dan. How many texts does Alice

send?

b) Kris sends 8 more texts than Alice. How many texts does Kris send?

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Exercise 3:

In a pizza takeaway

•a medium pizza has 6 slices of tomato

•a large pizza has 10 slices of tomato

How many slices of tomato are needed for c medium pizzas and d large

pizzas?

2. MONOMIALS

A monomial is an algebraic expression containing one term which may be a

number, a variable or a product of numbers and variables, with no negative or

fractional exponents. (“Mono” implies one and the ending “nomial” is Greek for

part)

For example:

are monomials, but

aren’t monomials

The number is called coefficient and the variables are called literal part.

If the literal part of a monomial has only one letter, then the degree is the

exponent of the letter.

If the literal part of a monomial has more than one letter, then the degree

is the sum of the exponents of every letter

In the example the degree is 2+1=3

Exercise 4:

Complete the following table:

MONOMIAL VARIABLES COEFFICIENT LITERAL PART DEGREE

-3x2y3

7x3yz

5

32x3

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Addition and subtraction of monomials

You can add or subtract monomials only if they have the same literal part

(they are also called like terms). In this case, you sum or subtract the

coefficients and leave the same literal part. If monomials aren’t like, the

addition sign must be left in the expression

Look at these examples:

4xy + 3xy = 7xy ; 4x2-5x-x2+x = 3x2-4x

Exercise 5:

Collect like terms to simplify each expression:

a)

b) 5y 3x 2y 4x

c) 2x 35x 7x 1

d)

Multiplication of monomials

To multiply monomials you multiply the coefficients and the variables,

following the rules for working with powers. The result is always a monomial.

Exercise 6:

Multiply the following monomials:

a) b) c)

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Exercise 7:

Multiply and simplify the following expressions:

a) b)

c) – d)

e) f)

Division of monomials

To divide monomials you divide the coefficients and the variables, following

the rules for working with powers. The result not always is a monomial.

The result not always is a monomial. For example isn´t a

monomial.

Exercise 8:

a) b) c) d)

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3. POLINOMIALS

A polynomial is an algebraic sum of monomials. (Poly implies many)

For example: x2 + 2x , 3x2 + x3 + 5x + 6 , 4x − 6y + 8

If there are two monomials, it is called a binomial, for example 2 x + 2x

If there are three monomials, it is called a trinomial, for example 4x − 6y +

8

The degree of a polynomial is the degree of the highest-degree term that it

contains.

Examples: is a fourth-degree binomial; is a second-degree

trinomial

Polynomials are usually written this way, with the terms written in ‘decreasing’

order; that is, with the highest exponent first, the next highest next, and so

forth, until you get down to the constant term.

Evaluating a polynomial is the same as calculating its number value at a given

value of the variable. For instance:

Evaluate at

is the numerical value of the polynomial

Always remember to be careful with the minus signs!

Exercise 9:

Adding and subtracting polynomials

You can add or subtract monomials only if they have the same literal part,

that is, if they are like terms. In this case, you sum or subtract the

coefficients and leave the same literal part.

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Example1:

Example 2:

Exercise 10:

Multiplying monomials and polynomials

• Monomial x polynomial: distribute the monomial through the brackets.

Multiply all the terms inside the bracket by the term outside is called

expanding the bracket.

Polynomial x polynomial. In this case, we expand double brackets, that is,

each term in the first bracket multiplies each term in the second bracket.

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Exercise 11:

Exercise 12:

Calculate and then simplify your result:

a) b)

c) d)

e) f)

g) h)

i) j)

k) l)

m) n)

o)

p)

q)

r)

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Exercise 13:

A small box contains 12 chocolates. Sam buys y small boxes of chocolates.

a) Write an expression for the number of chocolates Sam buys.

A large box contains 20 chocolates. Sam buys 2 more of the large boxes than the

small ones.

b) Write an expression for the number of the large boxes of chocolates he

buys.

c) Find, in terms of y, the total number of chocolates in the large boxes that

Sam buys.

d) Find, in terms of y, the total number of chocolates Sam buys. Give your

answer in its simplest form.

Exercise 14:

Jake is n years old.

Jake’s sister is 4 years older than Jake.

Jake’s mother is 3 times older than his sister.

Jake’s father is 4 times older than Jake.

Jake’s uncle is 2 years younger than Jake’s father.

Jake’s grandmother is twice as old as Jake’s uncle.

a) Copy the table and write each person’s age in terms of n.

Jake Sister Mother Father Uncle Grandmother

n

b) Find, in terms of n, how much older Jake’s grandmother is than his mother. Give

your answer in its simplest form.

Exercise 15:

Write an expression involving brackets for the volume of this cuboid. Expand the

brackets and simplify your expression.

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Dividing polynomials

1) Write the polynomials in decreasing

order.

2) Find the first term of the quotient,

dividing the first term of the dividend

by the first term of the divisor.

3) Multiply the result times all the

terms of the divisor and subtract from

the dividend.

4) Repeat the process until the degree

of the remainder is less than the

degree of the divisor.

Exercise 16:

Calculate the following divisions:

a) b)

c) d)

e) f)

g) h)

i) j)

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4. FACTORISING Factorising is the reverse process of multiplying out

a bracket (expanding). The factorised expression

has a polynomial inside a bracket, and a term

outside.

This term outside must be a common term (a number

or a letter). It means that the number or the

letter (s) can be found in every term of the expression.

The trick is to see what can be factored out of every term in the expression.

To factorise an expression, look for a common factor for all the terms. If

you remove the brackets in the final expression, you obtain the original one.

Just don't make the mistake of thinking that "factoring" means "dividing off

and making disappear".

Remember: when the term to be factored out coincides with one of the

addends, the unit always remains:

Exercise 17:

Exercise 18:

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Exercise 19:

5. THREE ALGEBRAIC IDENTITIES.

The square of a sum

The rule in words:

‘The square of the sum of two terms is the sum of their squares plus two times

the product of the terms’.

Be careful here: ‘the square of the sum’ and ‘the sum of the squares’ sound very

similar, but are different; the square of the sum is and the sum of squares

is , and for example

The square of a difference

The rule in words:

‘The square of the difference of two terms is the sum of their squares minus two

times the product of the terms’.

The product of a sum and a difference

The rule in words:

‘The product of a sum and a difference of two terms is the difference of their

squares’.

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Exercise 20: Expand and simplify:

a) b)

c) d)

e) f)

g) h)

k) l)

m) n)

o) p)

q) r)

s) t)

u) v)

w) x)

y) z)

Exercise 21: Convert the following concepts into algebraic expressions containing a single

unknown or two unknowns.

a) A tenth of a number.

b) A number plus half that number.

c) The price of a pair of trousers which has been reduced by 20%.

d) The age that a girl was 5 years ago.

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e) Two-thirds of a number, minus 1.

f) The time it takes a cyclist to cover x kilometers if he is travelling at 20 km/h.

g) The perimeter of a rectangle with a base of 10 m and a height of x.

h) The sum of a number and another number which is 10 units larger.

i) The product of a number, x, and the number that comes after it.

j) The area of a triangle with a base of 5 m and a height of x.

k) The square of a number, x, minus double that number.

l) The area of a rectangle whose base is 8 cm greater than its height.

m) The amount I pay for two albums with the same original price, but one of which

is on sale for 15% off and the other for 10% off.

n) The number of animal legs on a farm with x hens and y rabbits.

o) The result of adding 5/9 of a number to another number.

p) The quotient of two numbers minus 5 units.

q) The square of the sum of two numbers.

r) 35% of a number plus another number which is 12% greater.

s) The cost of a mix of two types of paint, one of which costs €8/kg and the other

€10/kg.

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UNIT 4

EQUATIONS

KEYWORDS Variable Quadratic equation Unknown Quadratic formula Solution Discriminant Roots Linear equation Zeros

1. EQUATIONS, SOLUTION OF AN

EQUATION

An equation is an expression of equivalence containing at least one letter

(unknown) whose value we want to determine.

Examples: 2x + 3 = 15 , x2 -16 = 0 , x3 = 81 ,2x-5 = x+1

The solution of an equation is the set of values which, when substituted for

unknowns, make the equation a true statement.

Example: There are two values that make true the following equation:

X2 − 16 = 0 Solution: {x = −4, x = 4}

Solving an equation means finding its solution (or solutions) or determining that

it has no solution.

There are different types of equations. For example:

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Exercise 1:

Degree of an equation The degree of an equation that has not more than one variable in each term

is the exponent of the highest power to which that variable is raised in the

equation.

The equation 3x − 17 = 0 is a first-degree equation, since x is raised only to the

first power.

An example of a second-degree equation is 5x2 − 2x + 1 = 0.

An example of a third-degree equation is 4x3 − 7x2 = 0.

The equation 3x − 2y = 5 is of the first degree in two variables, x and y.

When more than one variable appears in a term, as in xy = 5 , it is necessary

to add the exponents of the variables within a term to get the degree of the

equation.

Since 1 + 1 = 2, the equation xy = 5 is of the second degree.

2. FIRST-DEGREE EQUATIONS

A first-degree equation is called a linear equation. The highest exponent of a

linear equation is 1. The standard form for a linear equation is:

ax + b = c; a, b and c can have any value, except that a can’t be zero.

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Examples:

a) 3x + 2 = -10. The solution of this equation is x = -4.

b) 4x -6 = 4 (x + 3) ↔ 4x - 6 = 4x + 12 ↔ 0x = 18 ↔ 0 = 18 False equation.

This equation has no solution.

c) 4x - 6 = 4 (x - 2) + 2↔ 4x - 6 = 4x - 6↔ 0x = 0 This equation is true for any

value of x. Any number is solution of this equation. It has infinite solutions.

Steps for solving a first degree equation in one variable

1. If there are brackets remove them (expand)

2. If there are denominators remove them (multiplying both sides by the

LCM of the denominators or writing fractions with the same denominator)

3. Transpose the like terms (move to one side the x-terms and the numbers

to the other side)

4. Combine like terms

5. Isolate the unknown (move the coefficient of the x to the other side)

6. Check the solution.

For example:

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Exercise 2:

Solutions: a) 5; b) ∞ sol. ; c) no sol. ; d) 5; e) 13; f) 1/2 ; g) 1/2 ; h) -5

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Exercise 3:

Solve the following equations:

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50

p)

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Solutions: a)- 2; b) 4 ; c) 7 ; d) 1 ; e) ¾; f) 0; g) 2/5; h) 3 ; i) 1 ; j) 10; k)

7 ; l) 1 ; m) 5; n) ½ ; o) 1/3 ; p) 0; q) 1; r) -3; s) -3; t) -1/8 ; u) 8 ;

v) -4; w) 25; x) -2/3; y) -7/2; z) 0

Exercise 4:

Solve the following equations, some of them have no solution, some of them have infinite

solutions:

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+1

Solutions: a) 7/5 ; b) 7 ; c)No sol. ; d) No sol.; e)∞ sol. ; f)2 ; g) 4 ; h) -

5; i) ∞ sol. j)6 ; k)No sol; l)No sol; m)-3/2 ;n)-3; o)∞ sol ; p)∞ sol; q)0 ;

r) ∞ sol s)-9/2 , t)No sol ; u)No sol ; v)∞ sol ;w)5

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3. QUADRATIC EQUATIONS

A second-degree equation is called a quadratic equation. The highest

exponent of a quadratic equation is 2. The standard form for a quadratic

equation is: ax2+ bx + c = 0; where a, b and c can have any value, except

that a can’t be zero.

Exercise 5:

How to solve quadratic equations?

Quadratic equations can be solved using a special formula called the quadratic

formula:

The ± means you need to do a plus AND a minus, and so there are normally

TWO solutions!

The answers it gives are the solutions to the quadratic equation, and are

often called roots, or sometimes zeros.

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Examples:

The previous examples show that the different types of solutions of the

second-degree equations depend on the value of 2 b − 4ac.

b2 − 4ac is called the discriminant.

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• When b2 − 4ac is positive, you will get two different solutions.

• When b2 − 4ac is zero, you get one double solution.

• When b2 − 4ac is negative, you get two complex solutions, we say that the

equation don’t have real solutions.

Sometimes you can find “incomplete” quadratic equations, for example:

16x2 − 25 = 0 , 7x2 + 11x = 0

You can solve these equations in an easy way, without using the quadratic

formula:

Exercise 6:

Solutions: a) 2,3; b) -1/3 double; c) No sol.; d) 1/2, -3; e) No sol; f) 2, -2;

g) 0, 37/11 ; h) 3, -3 ; i) No sol.

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Exercise 7:

Solve the following second-degree equations:

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Solutions: a) 1, -4; b) 2, 5; c) 1, 1/2 ;d) -1, -7; e) 1, -2/3 ;f)1, 3/2 ;

g) ½ , ¼ ; h)-3, 2/3 ; i) ½ double; j)-7 double; k)-2/3 double l)8, -8;

m)0,5/2; n) 3, -3; o) No sol.; p) ½, -1/2 ; q) 0, -4;r) 5, -5; s) 0, -4/3 ;

t)-2; u) 3, -1; v) 0, -2 w) 3, -3; x) no sol.; y)-1, -4; z)0, 4

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Exercise 8:

Solve the following equations:

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Solucitions: a) -1, 3; b)±5 ; c) 2, -7/3 ; d)1, -15/2 ; e)0, -4 ; f)No sol.;

g) -1/3 double; h) No sol.; i) 5, -1/2 ; j)4, 1; k) 0, 6; l) ± ; m) No

sol.; n)0, -5; o) 2, -3/2; p)1, -3/4

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Exercise 9:

Solutions: a) ±5 ; b) 0, -5/7 ; c) 10 ; d) -2/3 e) 0 ; f)No sol

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4. SOLVING PROBLEMS USING

EQUATIONS To solve word problems we must create an equation that expresses the

relationship between known quantities and the quantities we want to find. It

is helpful to follow the following steps:

1. Read the problem carefully and get a clear idea of what is known and what

is unknown.

2. Use an algebraic language to express the relationship between the

quantities mentioned in the problem.

3. Solve the equation.

4. Check the solution using the problem (not the equation)

5. Express the solution as an answer of the problem using a sentence and

paying attention with the unit of measure.

Exercise 10:

If we add 6 units to a number we obtain the same amount as if we subtract

two from double that number. What is the number?

Exercise 11:

The fence along a school`s rectangular playground measures 2,100 m and its

length is double its width. What are the playground’s dimensions?

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Exercise 12:

Carmen spends €122 on one pair of trousers for John and one pair for Helen.

If the trousers for Helen cost €16 more than the trousers for John, how

much did each pair of trousers cost?

Exercise 13:

The difference between four fifths of a number and two thirds of that same

number is 10 units. What is the number?

Exercise 14:

A triangle’s shortest side is 3 cm less than the second shortest side, and the

second shortest side is 5 cm less than the longest side. If the perimeter of

the triangle is 32 cm, how long is each side?

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Exercise 15:

Tom is 12 years old and his mother is 38. How many years must pass in order

for Tom’s age to be half his mother’s age?

Exercise 16:

Anne’s age is double the age of her daughter. Ten years ago her age was

triple her daughter’s age. How old are they?

Exercise 17:

There are 4 years difference between Sarah’s age and Helen’s age. In 10

years the sum of their ages will be 50. Calculate each of their ages.

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Exercise 18:

A newsagent sells pens for €3 and markers for €5. One week the newsagent

sells 15 pens more than markets, and takes in a total of €125. How many

markets and pens are sold that week?

Exercise 19:

I paid a bill for €410 with €50 and €20 notes. I used three more €20 notes

than €50 notes. How many notes of each amount did I spend?

Exercise 20:

I bought two shirts that had the same original price, but one of them had a

discount of 20%. I ended up paying €36 in total. What was the final price of

each shirt?

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Exercise 21:

A nail sinks 2/5 of its length into a wall after one hit with a hammer. With

the second hit it sinks 1.4 cm, and 1 cm (the rest of the nail) is left

exposed. Find the length of the nail.

Exercise 22:

Two identical tanks are full of water. A quarter of the liters of the first one

is removed, and then 30 liters more is removed. Five sixths of the liters of

the second tank is removed and then 19 liters is added. After this, the tanks

contain the same amount of water. What is the capacity of each tank?

Exercise 23:

A trader mixes 10 kg of rice which costs €2.5/kg with another, inferior type

of rice which costs @1.8/kg. How many kilos of cheaper rice does he have to

add for the mix to end up costing €2/kg?

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Exercise 24:

Six kilograms of coffee that costs €9/kg is mixed with 10 kg of another

cheaper type of coffee. The result is a mix that costs €8/kg. What is the

price of the cheaper coffee?

Exercise 25:

Find a number which addes to its squaare equals 72 (there are two solutions).

Exercise 26:

Calculate the lengths of the sides of the right-angled triangle:

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Exercise 27:

If we subtract double a specific number from its square, we obtain 35. Find

the number.

Exercise 28:

Find two consecutive numbers whose squares added together equal 145.

Exercise 29:

Find three consecutive numbers which, when the smallest number is multiply

by the middle number, equal the largest number plus 34.

Exercise 30:

One of the catheti of a right-angled triangle is 3 cm longer than the other.

If the area of the triangle is 54 cm2, how long are the catheti.

Exercise 31:

The length of a rectangle is 2 cm longer than the width and the area is 143

cm2, how long are its sides?

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Exercise 32:

The length of a rectangle exceeds the width by 2 cm and the diagonal is 10

cm long, find the width of the rectangle.

Exercise 33:

Two numbers which differ 3, have a product of 88. Find them.

Exercise 34:

The product of two consecutive odd numbers is 143. Find the numbers.

Exercise 35:

A stone is thrown in the air. After t seconds its height, h, above sea level is

given by the formula h = 80 + 3t- 5t2, Find the value of t when the stone

falls into the sea.

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UNIT 5

Systems of linear

EQUATIONS

1. SYSTEMS OF LINEAR

EQUATIONS

A system of linear equations (or linear system) is a collection of linear

equations involving the same set of variables.

For example:

Solving a system of equations means finding the values of the variables that

make all the equations true at the same time.

The solution of this system is x = 3 and y = 1.

Methods to solve systems of linear equations

1) The graphical method

2) The substitution method

3) The algebraic equation method

4) The elimination method

2. THE GRAPHICAL METHOD When you are solving systems, you are, graphically, finding intersection of lines.

(Remember that the graph of a linear equation, ax + by = c , is a straight line, and

its points are the solutions of the equation).

The graphical method consists of graphing every equation in the system and then

using the graph to find the coordinates of the point(s) where the graphs

intersect. The point of intersection is the solution

Example:

You have to write a table of value for every equation:

70

;

Plot these points in a grid:

Graph both equations very precisely. If you don’t graph neatly, your point of

intersection will be way off.

The solution is x = 1 and y = 2.

Substitute these values into both equations to check the solution.

Number of solutions of two-variable systems

For two-variable systems, there are three possible types of solutions:

Case 1: The two lines cross at exactly one point. This point is the only solution to

the system. The two straight-lines are secant in this point. These systems are

called determinate compatible systems.

Example:

Solution: the point (1 , 1)

x 0 3

Y 3 0 x 0 -1

y 1 0

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Case 2: Since parallel lines never cross, the system has no solution. These

systems are called incompatible systems.

Example:

Solution: the system has no solution, the lines are parallel

Case 3: When the two lines are the same line, any point of the line is solution to

the system, the system has infinite solutions. These systems are called

indeterminate compatible systems.

Example:

Solution: Infinite solutions. The lines are the same line.

Exercise 1:

Solutions: a) x=3 , y=2; b) ∞sol. c) No sol.

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3. THE SUBSTITUTION METHOD

To solve a system of equations using the substitution method, first isolate an

unknown in one of the equations and then insert its value into the other equation.

Remember: Always substitute into the other equation and always use

parentheses!!!

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For example:

a) We isolate the “x” in the first equation: (*)

b) We substitute x with 3+y in the second equation:

c) We solve the new first degree equation:

d) We plug the value into (*) :

e) The solution is:

f) We check the solution:

Exercise 2:

Solve using the substitution method:

a)

b)

c)

d)

e)

f)

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g)

h)

i)

Solutions:

; i) x=1; y=1

4. THE EQUALIZATION METHOD

The equalization or algebraic equation method consists of isolating the same

unknown in each and equating the result obtained.

For example:

a) We isolate the “ ” in both equations:

b) We equate the two expressions:

c) We solve the equation:

d) We insert this value into any of the expressions of :

e) Solutions:

f) We check the solution:

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Exercise 3:

Solve using the algebraic equation method:

a)

b)

c)

d)

e)

f)

g)

h)

i)

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Solutions:

5. THE ELIMINATION METHOD

The elimination method consists of multiplying one or both equations by an

appropriated number in such a way that when they are added together one of the

unknown disappears. Then you can use back-substitution to solve for the other

variable.

When using elimination, eliminate one variable at a time. It is also important to

write down “instructions” that indicate how you are manipulating the equations

going from step to step.

For example:

a) We multiply the first equation by -4:

We need that an unknown has the same coefficient with different sign!!!

b) We add the two equations together: -17 and solve:

c) To solve the other unknown, we can plug this value into one of the original

equations or we can repeat the process in order to eliminate the y.

The second way: We multiply the first equations by 3 and the second one by -5:

;

d) Solution:

e) We check the solution:

Exercise 4:

Solve using the elimination method:

a)

b)

c)

77

d)

e)

f)

g)

h)

i)

78

Solutions: a) x=3; y=2; b) x=3; y=4; c) x=1; y=-1; d) x=-1; y=-2; e) x=-2;

y=3; f) x=0; y=4; g) x=7; y=-3; h) x=2; y=1/2; i) x=2; y=-2

Exercise 5:

Solutions: a)x = 5/22; y= 40/11 ; b) x=-1; y= 2 ; c) x=-12; y =15/2; x=-5

y= 4

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Exercise 6:

Say which of the following systems are incompatible and which are

indeterminate:

a)

b)

c)

d)

e)

f) –

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Exercise 7:

Find the value of two numbers if their sum is 12 and their difference is 4.

Exercise 8:

The difference of two numbers is 3. Their sum is 13. Find the numbers.

Exercise 9:

The school that Stefan goes to is selling tickets to a choral

performance. On the first day of ticket sales, the school

sold 3 senior citizen tickets and 1 child ticket for a total of

$38. The school took in $52 on the second day by selling 3

senior citizen tickets and 2 child tickets. Find the price of a

senior citizen ticket and the price of a child ticket.

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Exercise 10:

The sum of the digits of a certain two-digit number is 7. Reversing its digits

increases the number by 9. What is the number?

Exercise 11:

A hotel has double and single rooms. In total there are 90 rooms and 165

beds. How many rooms are there of each type?

Exercise 12:

I paid a bill of €145 using €10 and €5 notes. I handed over a total of 17

notes. How many of each note did I use?

Exercise 13:

Find two numbers that add up to 160 and that are separated by a difference

of 6.

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Exercise 14:

The difference between two numbers is 12 units. Half of the smaller number

is equal to one-fifth of the larger number. What are the numbers?

Exercise 15:

Three years ago. Laura’s age was half of Ana’s age, and in seven years their

ages will add up to 50. How old are they?

Exercise 16:

Two years ago, Carlos’ age was triple the age of his son Luis, but in twelve

years his age will only be double that of Luis. Calculate their ages.

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Exercise 17:

On a test with 30 questions, students receive 2 points for each correct

answer and lose 1 point for each incorrect answer. A boy who answered all

the questions got a score of 24 points. How many correct answers and how

many incorrect answers did he get?

Exercise 18:

We mix two types of flour, one that costs €0.75/kg and another that costs

€1.15/kg, and obtain 50 kg of a mix which costs €1/kg. How much of each

type of flour is used in the mix?

Exercise 19:

We mix two types of coffee, one that costs €5/kg and another that costs

€7.5/kg. We obtain 30 kg of a mix which costs €6/kg. How much coffee of

each type does the mix contain?

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Exercise 20:

You want to mix two types of olive oil that cost €5/liter and €3/liter to

obtain 25 liters of a mix that will cost €3.80/liter. How many liters of each

type of olive oil do you need to mix?

Exercise 21:

In a language school, 430 students studied either English or French last year.

This year, students studying English have increased 18 % and students

studying French have increased 15 %. There are now 502 students studying

the two languages. Calculate how many English students and how many French

students there were last year.

Exercise 22:

I changed a lot of 20 cent coins for €1 coins, and now I have 12 coins less

than I did before I changed them. How many 20 cent coins did I have?

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Exercise 23:

A boat traveled 210 miles downstream and back. The trip downstream took

10 hours. The trip back took 70 hours. What is the speed of the boat in still

water? What is the speed of the current?

Exercise 24:

The senior classes at High School A and High School B planned separate trips

to New York City. The senior class at High School A rented and filled 1 van

and 6 buses with 372 students. High School B rented and filled 4 vans and 12

buses with 780 students. Each van and each bus carried the same number of

students. How many students can a van carry? How many students can a bus

carry?

86

UNIT 6

PROPORTIONALITY

1. RATIO AND PROPORTION

A ratio is a way of comparing amounts of something. It shows how much bigger one

thing is than another. We write it as “a to b”, a:b or

For example:

a) Use 1 measure screen wash to 10 measures water

b) Use 1 shovel of cement to 3 shovels of sand

c) Use 3 parts blue paint to 1 part white paint

Ratio is the number of parts to a mix. The paint mix is 4 parts, with 3 parts blue

paint and 1 part white paint.

For example, the ratio of screen wash to water is 1:10. This means for every 1

measure of screen wash there are 10 measures of water.

Mixing paint in the ratio 3:1 means 3 parts blue paint to 1 part white paint

A proportion is the equality of two ratios. We write a proportion as two equal

fractions.

For example, if two bottles of water contain three liters of water, four bottles

contain six liters of water. We write:

2. DIRECT PROPORTINALITY

Two quantities are in direct proportion when they increase or decrease in the

same ratio. For example you could increase something by doubling it or decrease

it by halving.

If we look at the example of mixing paint the ratio is 3 pots of blue paint to 1 pot

of white paint, or 3:1.

But this amount of paint will only decorate two walls of a room. If you wanted to

decorate the whole room, four walls, what amount of each colour do you need?

You have to double the amount of paint and increase it in the same ratio.

If we double the amount of blue paint we need 6 pots.

If we double the amount of white paint we need 2 pots.

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The amount of blue and white paint we need increase in direct proportion to each

other. Look at the table to see how as you use more blue paint you need more

white paint:

Pots of blue paint 3 6 9 12

Pots of white paint 1 2 3 4

Understanding proportion can help in making all kinds of calculations. It helps you

work out the value or amount of quantities either bigger or smaller than the one

about which you have information.

Constant of direct proportionality

If we calculate the quotient between pots of blue paint and pots of white paint, we

get equal ratios:

=…., the constant of proportionality is always 3.

You can solve this kind of problems using two different ways:

Example 1:

If you know the cost of 3 packets of batteries is £6.00, can you work out

the cost of 5 packets?

THE UNITARY METHOD

To solve this problem we need to know the cost of 1 packet.

If three packets cost £6.00, then you divide £6.00 by 3 to find the price of 1

packet. (6 ÷ 3 = 2)

Now you know that they cost £2.00 each, to work out the cost of 5 packets you

multiply £2.00 by 5. (2 x 5 = 10)

So, 5 packets of batteries cost £10.00

THE DIRECT RULE OF THREE

Packets of batteries 3 5

£ 6.00 x

5 packets of batteries cost £10.00

Example 2:

You've invited friends for a pizza supper. You already have the toppings, so

just need to make the pizza base. Looking in the recipe book you notice that

the quantities given in the recipe are for 2 people and you need to cook for 5!

Pizza base - to serve 2 people:

100 g flour ; 60 ml water ; 4 g yeast; 20 ml milk ; a pinch of salt

88

THE UNITARY METHOD

The trick here is to divide all the amounts by 2 to give you the quantities for 1

serving.

Then multiply the amounts by the number stated in the question, 5.

For 1 serving, divide by 2:

100 g ÷ 2 = 50 g 60 ml ÷ 2 = 30 ml 4 g ÷ 2 = 2 g 20 ml ÷ 2 = 10 ml

For 5 servings, multiply by 5:

50 g x 5 = 250 g flour; 3 0 ml x 5 = 150 ml water 2 g x 5 = 10 g yeast

10 ml x 5 = 50 ml milk

The pinch of salt is up to you!

Example 3

Yesterday I paid 12 euro for five hours of parking… How much will I pay

today if my car has been in the same car park for three and a half hours?

THE DIRECT RULE OF THREE

Hours 5 3.5

Euro 12 x

I will pay €8.40

3. INVERSE PROPORTINALITY

Two quantities are inversely proportional if one of them varies in the inverse

ratio as the other. If we multiply one of the quantities by a factor, we divide the

other quantity by the same factor.

For example: we want to put 300 kg of potatoes in bags. The size of the bags is

inversely proportional to the number of bags we need.

Weight of the bags (kg) 1 2 3 4

Number of bags 300 150 100 75

If the bags are bigger, we need fewer bags.

Constant of inverse proportionality The weight of the bags is inversely proportional to the number of bags, if we

calculate the product of weight and number of bags we always get the same

constant:

the constant of proportionality is always

300.

You can solve this kind of problems using two different ways:

Example 1

89

After a sports event, four cleaners take two and a half hours to clean up a

sports center. How long would three cleaners take to do the same job?

THE UNITARY METHOD

To solve this problem we need to know how many hours 1 person needs to do the

job.

If 4 cleaners take 2.5 hours, then you multiply 2.5 h by 4 to find the number of

hours which one cleaner needs. (4·2.5 = 10 h)

Now to work out the hours for 3 cleaners you divide 10 h by 3.

(10:3 = 3.3 h = 3h 20 min)

So, 3 cleaners take 3h and 20 min to do the same job.

THE INVERSE RULE OF THREE

Cleaners 4 3

Hours 2.5 x

Example 2

Four taps fill a water tank in 10 hours. How long do eight taps need to fill

the same tank?

THE INVERSE RULE OF THREE

Taps 4 8

Hours 10 x

HINT to solve the problems: The first step is to think if the quantities

are direct or inverse proportional

Exercise 1:

Marta needed 75 g of sugar to make a 600 g cake. How many grams will she

need to bake a 1 kg cake?

90

Exercise 2:

This morning one euro traded at 1.325 dollar. How many euro would you get

for 280 dollar?

Exercise 3:

Four people who live in the same building must pay a water bill of €360. The

bill is to be split proportionately between them, taking into account the area

of each of their flats. If the flats measure 80 m2, 100 m2, 120 m2 and 150

m2, how much must be paid for each of them?

Exercise 4:

A 200 g piece of cheese cost €2.80.How much will another piece of the same

cheese cost if it weighs 325 grams?

Exercise 5:

At a second-hand car dealer’s, a three-year-old car is sold for €12,000- If

its price varies inversely with its age, how much will it be sold when it is 10

years old?

91

Exercise 6:

Travelling at an average speed of 60 km/h, a lorry took 1 h 18 min to get

from town A to town B. How many minutes less will it take on the trip back if

it travels at an average speed of 65 km/h?

Exercise 7:

With its employees working 8 hours per day, a factory produces enough to fill

an order in 15 days. How long would it have taken if each worker had been

two hours more every day?

Exercise 8:

I bought an 18 carat gold chain that weighs 30 grams. How many grams of

pure gold does it contain? (Note: “18 carat gold” means that 18 out of 24

parts are pure gold and “pure gold” is 24 carats)

Exercise 9:

A jeweler makes a broach, mixing 15 grams of gold with 5 grams of tin.

What is the caratage (gold purity) of the alloy? (Note: “caratage” –the ratio

of gold to other metals in an alloy- is measured in carats. 1 carat means that

1 out of 24 parts are pure gold)

92

Exercise 10:

With two kilograms of oranges we can make 1.2 liters of juice. How much

juice can we make with:

a) 9 kg of oranges? b) 20 kg of oranges?

Exercise 11:

If three computers cost €1800, how much would five computers cost?

Exercise 12:

Three elephants eat 75 kg of food. How many kilograms of food eat:

a) Six elephants? b) 26 elephants?

93

4. COMPOUND PROPORTION PROBLEMS Proportionality is “compound” if there more than two magnitudes

Here are some examples of problems. You can solve this kind of problems using

two different ways:

Example 1:

Working 8 hours per day, a glass factory makes 6,000 bottles in 3 days. How

long would it take to make 10,000 bottles working 9 hours per day?

THE UNITARY METHOD

To make 6,000 bottles working 8h/day it takes 3 days

To make 6,000 bottles working 1h/day it takes 3·8 days

To make 1 bottles working 1h/day it takes

days

To make 10,000 bottles working 1h/day it takes

days

To make 10,000 bottles working 9h/day it takes

days

Working 9 hours per day, it would take 4 days to make 10,000 bottles.

THE COMPOUND RULE OF THREE

BOTTLES h/days DAYS

6,000 8 3

10,000 9 x

You always have to write the variable in the last magnitude. Then you write the

rule of three, if the proportionality is inverse, you invert the fraction.

In this problem, the number of bottles and the numbers of days are direct

proportional but, the number of hours per day is inverse proportional to the

number of days

94

Example 2:

Six people eat three days in a restaurant, the total cost is €216. How much

would it cost four people to eat for two days?

PEOPLE DAYS COST

6 3 216

4 2 X

The magnitudes “people” and “days” are direct proportional to the “cost”

€

Exercise 13:

Four cows produce 800 liters of milk in 5 days. How long will take 8 cows to

produce 2,000 liters?

Exercise 14:

Four people travel through the desert for 20 days, they drink 400 liters of

water. How much water would they need if they were:

a) 7 people travelling for 35 days?

b) 20 people travelling for 20 days?

c) 2 people travelling for 165 days?

95

Exercise 15:

56 workers build 120 cars in 30 days. How many cars would:

a) 5 workers build in 60 days?

b) 2 workers build in 2 days?

c) 1 worker builds in 1 day?

Exercise 16:

20 cows consume 200 kilos of feed per week. How many kilos of feed do 35

cows consume in one month?

Exercise 17:

Working 8 hours per day a textile factory makes 15,000 pairs of socks in 12

days. How many pairs of socks will it produce over the next ten days if it

doubles its working hours?

Exercise 18:

How many days does the factory (from the previous question) still working

double its normal hours, need to make 20,000 socks?

96

5. DIRECTLY PROPORTIONAL

DISTRIBUTIONS

To distribute an amount, N , in parts that are directly proportional to other

amounts: a, b, c

1st . We divide the total amount by the sum of the other amounts:

2nd. We multiply this value (k) by the number of units in each amount.

Example

Distribute €102 in directly proportional parts to 3, 2, and 1.

The parts are €51, €34 and €17

To check, the new amounts must add N (51+34+17=102)

Exercise 19:

A company distributes its benefits, €80,000, proportionally amongst its three

owners, considering the shares they own. Paul owns 8 shares, Ann owns 12

shares and Steven 20 shares. Calculate the benefit for each owner.

Exercise 20:

A father distributes €99 amongst his three sons in directly proportional parts

to their ages: 8, 12 and 15 years old.

97

6. INVERSELY PROPORTIONAL

DISTRIBUTIONS To distribute an amount, N , in parts that are inversely proportional to other

amounts: a, b, c

1st . We divide the total amount by the sum of the inverse of the other amounts:

2nd. We divide this value (k) by the number of units in each amount.

Example

Distribute €3900 in parts inversely proportional to 6,4 and 8.

We calculate the sum of the inverses:

We calculate :

Now: The first quantity is:

; the second quantity is:

and the third quantity is:

To check, the new amounts must add N (1200+1800+900=3900)

Exercise 21:

The parents of 3 sisters decide to distribute their total weekly pay, €30, in

inverse proportion to the days they were late to school last month. Nancy

was late 1 day, Sarah was late 2 days and Mary was late 3 days. Calculate

how much each receives.

98

Exercise 22:

I want to distribute €620 amongst three nephews, in inversely proportional

part to their ages: 1, 3 and 7 years. How much money do I have to give

them?

7. PERCENTAGES

To find a certain percentage of an amount, you multiply the amount by that

percentage and divide by one hundred.(You can use decimals)

Exercise 23:

Calculate the answers in your head:

a) A town has 800 inhabitants. Ten percent of them have never seen the sea.

How many of the inhabitants have never seen the sea?

b) A cake weighed 800 grams, but 75% of it has been eaten. How much does

the remaining cake weight?

c) My class has 28 students. 25% of them got an A in Maths. How many

didn´t get an A?

d) There are a million cars in a country. Ninety percent of them are 5 years

old or less. How many cars are over 5 years old?

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Exercise 24:

A farmer harvests 25,000 kilos of corn and sells 85% to an animal feed

factory. How many kilos did the feed factory buy?

Exercise 25:

A certain country has a population of eight million, and a study shows that

2.8% of the population is diabetic. How many diabetics does the country

have?

Exercise 26:

A restaurant manager buys 8 kilos of beef loin and orders that 60% of it be

frozen. How many kilos of beef will be frozen?

Exercise 27:

During a tournament a basketball player took 275 shots and made 68% of

them were baskets. How many baskets did he make?

100

Exercise 28:

Calculate the answers in your head:

a) There are 44 cars in a car park. 11 of them

are red. What percentage of the cars is red?

b) Julian earns €40 cleaning the neighbor`s garden. He saves thirty and

spends the rest. What percentage does he spend?

c) A flock has 27 sheep and 3 goats. What percentage of the flock is goats?

Exercise 29:

A bag contains 6 white beads and 9 black beads. What percentage of each

colour does the bag contain?

Exercise 30:

In a class of 30 students, 12 have signed up for dance lessons. What

percentage of the class is going to dance lessons?

Exercise 31:

The executive jobs in a company are divided up between 2 men and 18 women.

What percentage of the company’s executives are women?

101

Exercise 32:

A 1 kilo cake contains 150 grams of sugar. What percentage of the cake is

sugar?

Exercise 33:

The statics of two players in a basketball tournament are as follows:

Player A ↔ 85 shots ↔ 68 baskets

Player B ↔ 94 shots ↔ 72 baskets

What percentage of the shots taken by each player was successful? Which of

two players is more reliable?

Exercise 34:

Calculate the answers in your head:

a) 25 % of the students in my class got A in Physical Education. Seven

students got A. How many students are in the class?

b) Thirty rooms in a hotel are occupied. This is 75 % of all the rooms in the

hotel. How many rooms are there in total?

c) A shepherd sold 8 sheep, which made up 10 % of his flock. How many

sheep were in the flock?

d) A carton of eggs fell on the floor and 6 of them broke. This was 20 % of

all the eggs in the carton. How many eggs did the carton contain?

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Exercise 35:

A garage employs 27 women. This is 12 % of its total staff. How many

employees does the company have in total?

Exercise 36:

In a football team, 12 players have missed practice. This is 40 % of all the

players. How many players are on the team?

Exercise 37:

I went to the cinema and it cost 6 euro, which is 80 % of my weekly

allowance. What is my weekly allowance?

Exercise 38:

Carmen lost 12 kilos, which is 15 % of what she weighed one year ago. How

much did she weigh one year ago?

Exercise 39:

Fifty- two of a hospital’s beds are occupied. This is 13 % of the total

number of beds. How many beds does the hospital have?

103

Percentage increase

If we increase a quantity by a percentage, then we add that percentage of the

quantity to original.

For example: If €40 is increased by 25%, the results 40 € + 40€ · 0.25= 40 € + 10

€ = 50 €. This is the same as multiplying 40 by 1 + 0.25 ↔ 40 · 1.25 =50 €

To increase a p% is the same to calculate (100 + p) % of the amount.

Percentage decrease

Finding the percentage decrease is similar to finding the percentage increase, but

we subtract the percentage.

For example: If €40 is decreased by 25%, the results 40 € - 40€ · 0.25= 40 € - 10

€ = 30 €. This is the same as multiplying 40 by 1 - 0.25 ↔ 40 · 0.75 =30 €

To decrease a p% is the same to calculate (100 - p) % of the amount.

Consecutive percentages increase and decrease

To calculate consecutive percentages, we multiply the percentages.

For example: if we increase 40€ by 25%, then we decrease by 25% and after we

increase by 10%, the result is 40 € · (1.25 · 0.75 · 1.1) = 40€ · 1.03125 = 41.25 €

Exercise 40:

Calculate the answers in your head:

a) A roll of wire is 80 meters long. How much wire will be left if we cut off

25 % of the roll?

b) A loaf of bread that used to cost one euro has gone up in price by 20 %.

How much does it cost now?

c) A shop reduces the prices by 10 %. How much would I pay for a jumper

that used cost 40 euro?

d) Last year there were 600 students at my school. This year the number of

students enrolled has risen by 5 %. How many students are there this year?

104

Exercise 41:

A town has 6,500 inhabitants, but the population is expected to rise by 20 %

over the next five years. How big is the population expected to be in five

years?

Exercise 42:

The price of a bus ticket used to be €2, but today it goes up 5 %. How much

will a ticket cost from now on?

Exercise 43:

At the beginning of spring a reservoir contains 2,800 cubic decimeter of

water, but recent rainfall has increased its water by 40 %. How much water

does the reservoir contain now?

105

Exercise 44:

A coat used to cost €265. In the sales the price went down 20 %. What is

the price of the coat after the discount?

Exercise 45:

In a given population, 2480 people last year had the flu. This year the

number is 30 % lower. How many people had the flu this year?

Exercise 46:

A company with 1,675 employees cuts its staff by 8 %. How many employees

does it have after the cut?

106

UNIT 7

PROGRESSIONS

1. SEQUENCES AND THE Nth TERM

A sequence is a set of ordered numbers that follow a pattern, for example:

• 5, 9, 13, 17, 21, … are the first five terms of a sequence that goes up in 4s.

• 3, 6, 12, 24, 48, … are the first five terms of a sequence that doubles.

• 1, 4, 9, 16, 25, … is the sequence of square numbers.

• 1, 8, 27, 64, 125, … are the cube numbers

Example: Write the next two terms in each sequence.

a) 4, 5, 7, 10, 14, 19, …

The terms increase by 1, then 2, then 3. So the next two terms are 19 + 6 = 25

and 25 + 7 =32.

b) 0.6, 0.7, 0.8, 0.9, …

The terms increase by 0.1. So the next two terms are 0.9 + 0.1 = 1.0 and 1.0 + 0.1 =

1.1.

Exercise 1:

Copy each sequence and add the next two terms.

a) 4, 9, 14, 19, 24, ___ , ___ b) 100, 93, 86, 79, 72, ___ , ___

c) 54, 27, 13.5, 6.75, ___ , ___ d) 1, 1, 2, 3, 5, 8, ___ , ___

Exercise 2:

Write the first five terms of each of these well-known number patterns.

a) Multiples of 3

b) Powers of 2

c) Prime numbers

d) Square numbers over 100

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Exercise 3:

The triangular numbers form a sequence.

a) Copy the table and use the diagrams to complete it.

Diagrams 1 2 3 4 5

Number of

dots

b) Write the first 10 triangular numbers.

c) Why do you think the square numbers (1, 4, 9, 16, 25, …) got their name?

The general term

A sequence will have a rule that gives you a way to find the value of each term.

Example: the sequence 3, 5, 7, 9, … starts at 3 and jumps 2 every time:

But the rule should be a formula!

Saying “starts at 3 and jumps 2 every time” doesn’t tell us how to calculate the:

• 10th term,

• 100th term, or

• nth term, (where n could be any term number we want).

We want a formula with “n” in it (where n can be any term number).

So, what would the Rule for the sequence 3, 5, 7, 9, … be?

Firstly, we can see the sequence goes up 2 every time, so we can guess that the

Rule will be something like “2×n ” (where “n” is the term number). Let’s test it

out:

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n Term Test Rule

1 3 2n = 2 x 1 = 2

2 5 2n = 2 x 2 = 4

3 7 2n = 2 x 3 = 6

That nearly worked … but that Rule is too low by 1 every time, so let us try

changing it to:

So instead of saying “starts at 3 and jumps 2 every time” we write the rule as

the Rule for 3, 5, 7, 9, … is: 2n+1

Now, for example, we can calculate the 100th term: 2 x 100 + 1 = 201

Notation:

To make it easier to write down rules, we often use this special style:

an represents the general term (general rule or nth term) of the sequence, the

Nth term of a sequence is the expression that represents the value of any of its

terms.

So to mention the “5th term” you just write: a5

Example: an =3n+2

To find a5 , the 5th term, put n = 5 in the rule.

Exercise 4:

Exercise 5:

Write three more terms for the following sequences and write the Nth terms

for each sequences:

a) 20, 40, 60, 80, 100,

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b) 4, 8, 12, 16, 20,

c) 5, 9, 13, 17, 21,

d) 2, 4, 8, 16, 32,

e) 3, 5, 9, 17, 33,

Exercise 6:

Exercise 7:

Write the first three terms and the 10th term in the following sequences:

a)

b)

c)

d)

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2. ARITHMETIC SEQUENCES

An arithmetic sequence is a sequence in which each successive term is obtained by

adding a constant number to the previous term, then the difference between

consecutive terms is the same, this number is represented by d (the difference)

Examples:

a) 12 , 15 , 18 , 21 , 24 , … d = +3

b) 100, 95, 90, 85, 80 , … d = -5

The difference can be obtained by subtracting two consecutive terms.

General term of an arithmetic sequence

The Nth term or general term of an arithmetic sequence with a first term (

and a difference of “d” is obtained as follow:

Let’s find the general term of an arithmetic sequence:

Example: 8, 18, 28, 38, 48, …

The first term in this sequence is 8 and the difference, d, between consecutive

terms is 10.

Exercise 8:

Write the first five terms in each of the following sequences, whose first

terms and differences are provided. Determine the Nth terms of these

sequences:

a)

b)

c)

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Exercise 9:

Find the Nth term of each of these arithmetic sequences. First calculate the

difference:

a) 54, 65, 76, 87, 98,….

b) 114, 91, 68, 45, 22, ….

c) 18.2, 20, 21.8, 23.6, 25.4,…..

The following can also be used if we don´t know the first term:

Exercise 10:

Find the 86th term of the sequence −11, − 7, − 3, 1, ...

Exercise 11:

Find the difference in an arithmetic sequence where a5= −5 and a19= 65.

Exercise 12:

Find the 60th term of an arithmetic sequence where the fifteenth term is 21

and the difference is 1/3.

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Exercise 13:

Find the number of terms (n) in an arithmetic sequence where the first term

is 7, the last term is 112, and the difference is 3.

Exercise 14:

Find the value of the 30th term of an arithmetic sequence if a7= 24 and

a13=72.

Exercise 15:

Jo has £20 in her piggy bank. In each case, find a rule for the amount of

money she will have in the piggy bank after n weeks if she saves:

a) £3 a week b) £5 a week c) £10 a week

Exercise 16:

Caroline has won a prize of 1000 tins of dog food. In each case, find a

formula for the number of tins she will have left after n weeks if her dog

eats:

a) 5 tins a week b) 7 tins a week c) 14 tins a week

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Exercise 17:

An author has signed a contract to write a book of 400 pages.

In each case, find a formula for the number of pages left to write after n

days if the author writes:

a) 17 pages a day b) 20 pages a day c) 25 pages a day

Exercise 18:

The first row in a theatre is 4.5 m from the stage, and the eighth row is

9.75 m.

a) What is the distance between two rows?

b) How far is the 17th row from the stage?

Exercise 19:

Calculate the difference and the Nth term in an arithmetic sequence

containing the terms a1=3 and a4=33

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Sum of the n first terms of an arithmetic sequence

Let’s start solving this problem:

How do you add the numbers from 1 to 5000 without actually doing it or using a

calculator?

If you wrote out all the numbers from 1 to 5000 and then wrote them backwards

underneath, you would have twice as many numbers as you needed, but the problem

is easier, here is why:

1 2 3 4 ……………………… 4998 4999 5000

5000 4999 4998 4997 ……………………… 3 2 1

5001 5001 5001 5001 ……………………… 5001 5001 5001

Notice that if we add the two lists, we get a list that is the same number, 5001,

repeating. In fact, since each of the lists is 5000 numbers long, we have, in the

sums, a list of 5000 numbers that are each 5001.

You have to admit that adding 5000 5001’s is a lot quicker than the other way

since 5000 × 5001 = 25,005,000.

But wait, you say, that’s too much. We were only supposed to add one list and we

added two. Okay, then the answer must be half as much:

1+2+3 +4 +……. +5000 =

=12,502,500

In general, if you have an arithmetic sequence of n numbers and you know the first

and

last one, you can find the sum, Sn , by:

Example: Find the sum of the first 30 terms of the sequence 4, 7, 10, 13, 16,

Step 1. Identify a1 =4 and n = 30.

Step 2. Calculate a30 using the general term formula:

Step 3. Substitute:

Exercise 20:

The first term of an arithmetic sequence is 16, and the tenth term is 43.

Calculate a20 and S20.

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Exercise 21:

Calculate the sum of the first ten terms of an arithmetic sequence where

a10= 58 and d = 6.

Exercise 22:

Calculate the number of terms (n) in an arithmetic sequence knowing that the

first term is 7,an =53 and Sn=300.

Exercise 23:

Find the sum of the first ten terms in an arithmetic sequence with a first

term of 20 and a difference of 12.

Exercise 24:

The medicine dose is 100 mg the first day and 5 mg off every following day.

The treatment lasts 12 days. How many milligrams must the sick person have

during the whole treatment?

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Exercise 25:

A ball rolling on an inclined plane covers 1 m in the 1st second, 4 m in the

2nd, 7 m in the 3rd, and so on. How many metres does it cover in 20

seconds?

3. GEOMETIC SEQUENCES

Sequences of numbers that follow a pattern of multiplying a fixed number from

one term to the next are called geometric sequences. This fixed number is called

ratio. The following sequences are geometric sequences:

Sequence A: 1, 2, 4, 8, 16, … ratio = 2

Sequence B: 0.01, 0.06, 0.36, 2.16, 12.96, … ratio = 6

Sequence C: 16, -8, 4, -2, … ratio = -1/2

The common ratio can be found dividing each term by the previous term.

General term of a geometric sequence

Let’s find the general term or Nth term of a geometric sequence:

Example: 5, 10, 20, 40, 80, …

The first term in this sequence is 5 and the ratio, r, between consecutive terms is

r = 2

a1 = 5

a2 = 5·2 = 10

a3 = 5·2·2 = 5·22 = 20

a4 = 5·2·2·2 = 5·23 = 40……………..an = 5 · 2n-1

The general term of a geometric sequence can be calculated from the first term

and the ratio using this expression:

an = a1 · rn-1

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The following expression can also be used if we don’t know the first term:

an = ap · rn-p

Example: Find the general term of the following geometric sequences: a) 2, 6, 18, 54, … a1=2; r = 3↔an = 2·3n-1

b) 200, 100, 50, 25, …a1 = 200; r = ½; ↔an = 200 ·(1/2)n-1

Exercise 26:

Find out which of the following sequences are geometric. Find the ratio and

the nth term for the geometric ones.

a) 2, 4, 8, 16, 32, … b)

c)

d)

e) f) 1, -10, 100, -1000, 10000,….

Exercise 27:

Find the first four terms of the following geometric sequences:

a) b) c)

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Exercise 28:

Find the ratio of a geometric sequence where a15 = 192 and a9 = 2187.

Exercise 29:

Find a7 in a geometric sequence where a4 = 125 and r =

Exercise 30:

In a geometric sequence we know that a11 = 256 and a6

. Find the value

of a16 and the general term.

Sum of the n first terms of a geometric sequence

The sum of the n first terms of a geometric sequence can be expressed by a

formula. Let’s find it:

The sum of the n first terms of a geometric sequence is:

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Exercise 31:

Find the sum of the first 7 terms of a geometric sequence where the second

term is 300 and the ratio is 1/2.

Exercise 32:

The first term of a geometric sequence is 6, the last term is 1458, and the

sum is 2184. Find the ratio and the number of terms.

Sum of the terms of a geometric sequence where -1<r<1

The sum is called the “infinite sum”

Exercise 33:

The first term of a geometric sequence is 8 and the ratio is r= 0.75.

Find the infinite sum.

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Exercise 34:

The fourth term of a geometric sequence is 10 and the sixth one is 0.4. Find

the ratio, the first term, the sum of the first 8 terms and the infinite sum.

Exercise 35:

Maria has been studying the growth of a tree that was 40 m tall when she

began her study. The first year it grew 20 cm, the second year it grew 23

cm and the third year it grew 26 cm. Its growth continued each year

following the same pattern.

a) How much did the tree grow in the ninth year?

b) How tall was the tree 9 years after the study began?

Exercise 36:

Ana started training on 1st January. On that day she ran for 20 minutes, and

every day after that she increased her training time 5 minutes.

a) How long did she run for on 15th January?

b) How long did she train for in total between the day she began training and

15th January?

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Exercise 37:

Helen tells two of her friends a secret. The next day, each of her friends

tells the secret to two other friends. On the next day, their friends each

tell the secret to two other friends.

a) How many people are told the secret on the tenth day?

b) Calculate the number of people who were told the secret either before or

on the tenth day.

Product of the n first terms of a geometric sequence

The product of the n first terms of a geometric sequence can be expressed by a

formula.

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UNITs 8 and 9

Plane shapes and solids

1. THE PYTHAGOREAN THEOREM

In a right-angled triangle one of the angles of the triangle measures 90 degrees.

The side opposite the right angle is called the hypotenuse. The two sides that

form the right angle are called the legs or catheti. (cathetus in singular)

In a right triangle the sum of the squares of the lengths of the legs equals the

square of the length of the hypotenuse.

This is known as the Pythagorean Theorem.

For the right triangle in the figure, the lengths of the catheti are “a” and “b”, and

the hypotenuse has length “c”.

Using the PythagoreanTheorem, we can calculate the length of one side of a right-

angled triangle if we know the length of the other two sides.

Exercise 1:

Find the value of the third side of the following right triangles ABC in which

A = 90º.Round the calculations to the nearest hundredth:

a) b = 7 cm, a = 9 cm. b) b = 3 cm , c = 4 cm

c) a = 12.5 cm, c = 8.6 cm d) a = 0.34 cm, c = 0. 27 cm

123

Exercise 2:

One of the sides of a rectangle measures 4 cm and the diagonal 6 cm.

Calculate the other side and the perimeter.

Exercise 3:

In a rectangle the length of its sides are 8.3 cm and 5.4 cm.

Calculate the length of its diagonal.

Exercise 4:

Construct a rhombus with diagonals 4 cm and 7 cm. Calculate the perimeter.

Exercise 5:

Construct a square with a diagonal of 7.5 cm. Calculate the perimeter.

Exercise 6:

Find all the missing sides and angles in the polygons below:

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Exercise 7:

Calculate the apothem of the following pentagon:

125

Exercise 8:

Find the length of the side of a square if its diagonal is 42 cm.

Exercise 9:

Find the value of the diagonal of a rectangle if their sides are 45 cm and 28

cm.

Exercise 10:

Find the length of the apothem of a regular hexagon whose sides measure 10

cm.

Exercise 11:

How long are the sides of an equilateral triangle with a height of 24 cm?

Exercise 12:

Calculate the length of the sides of a regular hexagon with an apothem of 40

cm.

126

Exercise 13:

In a right-angled trapezium, the bases measure 10 cm and 31 cm and the

height is 20 cm, calculate the length of the unknown side.

2. AREAS AND PERIMETER OF PLANE

SHAPES NAME PLANE SHAPE AREA PERIMETER

SQUARE

RECTANGLE

PARALLELOGRAM

TRIANGLE

RHOMBUS

TRAPEZIUM OR

TRAPEZOID

REGULAR

POLYGON

(n sides)

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CIRCLE

CIRCULAR

SECTOR

ANNULUS

Exercise 14:

Calculate the area and the perimeter of a rhombus, whose diagonals are of 6

cm and 8 cm.

Exercise 15:

Calculate the area of these shapes:

128

Exercise 16:

Calculate the area of these shapes

129

Exercise 17:

Find the area of a regular hexagon whose sides measure 20 cm.

130

Exercise 18:

Calculate the area of a rhombus whose side is 17 cm and one of its diagonal

is 16 cm

Exercise 19:

Calculate the area of the shadowed shapes:

131

Exercise 20:

Calculate the area of an isosceles triangle whose equal sides measure 45 cm

and the other side measures 28 cm.

Exercise 21:

Find the area of an isosceles trapezium whose bases are 4 cm and 10 cm and

the other side is 5 cm.

132

3. POLYHEDRA

Polyhedra (in singular polyhedron) are geometric solids whose faces are formed by

polygons

Components:

- Faces are the polygons that bound the polyhedron

- Edges are the lines where two faces join.

- Vertices are the points where three or more

edges meet

- Diagonal is a segment that joins two non-

consecutive vertices

- Dihedron angle is the angle between two faces

Euler’s formula

In a single polyhedron (one without holes), the number of faces (f) plus the

number of vertices (v) is equal to the number of edges (e), plus two.

f + v = e + 2

Regular polyhedra

The regular polyhedra have all their faces formed by identical regular polygons.

They are:

Tetrahedron: Four equilateral triangles

Cube: Six squares

Octahedron: Eight equilateral triangles

Dodecahedron: twelve regular pentagons

Icosahedron: Twenty equilateral triangles

Regular polyhedra:

Tetrahedron Cube Octahedron Dodecahedron Icosahedron

(hexahedron)

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Exercise 22:

Check the Euler’s formula in the regular polyhedra:

NAME OF THE

SOLIDS

Number of faces Number of

vertices

Number of edges

TETRAHEDRON

CUBE

OCTAHEDRON

DODECAHEDRON

ICOSAHEDRON

4. CUBOIDS

A cuboid is a geometric objet with faces that are rectangles (in some cases

squares). The special case in which all the faces are squares is the cube.

They have 6 faces, 12 edges and 8 vertices.

If we name the edges of the cuboid a, b and c, its area is:

A 2ab 2ac 2bc 2ab ac bcAnd the volume is V a b c

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If we name the edges of the cube a, its area is:

A 6a2And the volume is V a3

Diagonal of a cuboid In this cuboid the triangle with sides

a, b and e is a right triangle, so using

the Pythgorean Theorem we can

say , but the triangle with

sides c, e and d is a right triangle as

well in which d is the hypotenuse

and , that is:

And finally

Exercise 23:

Find the area, the volume and the diagonal of each solid shown below:

135

Exercise 24:

Find the area, the volume…..and the missing length of these cuboids:

Exercise 25:

Find the area and the volume of each solid shown below:

136

5. PRISMS

A prism is a polyhedron with two equal and parallel faces that are polygons (bases)

and the other faces are parallelograms.

The distance between the two bases is the height of the prism.

Prisms can be right prisms when the parallelograms faces are perpendicular to the

bases, otherwise they are oblique prisms.

Depending on the polygons of the bases they can be:

Triangular prisms, square prisms, pentagonal prisms,…

This is an hexagonal prism:

The area of a prism is found by adding the areas of its faces, if we call l the side

of the base, a the apothem and h the height of the prism, the total area is

Since the area of each base, which is a regular polygon of n sides, is

and we have n rectangles each one of width l and height h

The volume is the area of base multiplied by the height

137

When the prism is oblique the formula is the same but the height is not the length

of the edge.

Exercise 26:

Find the area and the volume of each prism shown below:

138

6. PYRAMIDS

In a pyramid one of its faces is a polygon called the base and the other faces are

triangles that join at a point that is the apex.

The height of a pyramid (h) is the distance from the base to the apex.

Like the prisms, pyramids can also be right or oblique and depending on the

polygons of the base they can be: triangular, square, pentagonal, hexagonal,etc

This is a square pyramid

The area of a pyramid whose base is a regular polygon of n sides of length b and

apothem g, and calling a to the height of the triangular faces (slant height) is:

The base is a regular polygon with area:

and there are n triangular faces with a total area of

Then the area of the pyramid is

If it is a square based pyramid like the one in the picture, the area is

The volume of a pyramid is

If it is a square based pyramid like the one in the picture, the volume is

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7. TRUNK OF A PYRAMID

A trunk of a pyramid is the part of a pyramid which is

between two parallel planes. The faces of the solid

obtained by cutting it are called bases of the trunk,

and the distance between the two cutting planes is

the height of the trunk. The side faces are

trapeziums.

The area of a trunk of a pyramid with bases which are regular polygons of n sides,

is

being l and l’ the sides of the base polygons and a the

height of the trapeziums.

In the square based pyramid on the right the area of a trapezium is

Exercise 27:

Find the area and the volume of these pyramids:

140

Exercise 28:

Find the area and the volume of the trunk of pyramid shown below:

141

8. SOLIDS OF REVOLUTION

A solid of revolution is obtained by rotating a plane shape around an axis.

They are the cylinder (a rectangle is rotated), the cone (a triangle is rotated) and

the sphere (a semicircle is rotated)

9. CYLINDERS

A cylinder is a curvilinear geometric solid formed by a curved surface with all the

points at a fixed distance from a straight line that is the axis of the cylinder and

by two circles perpendicular to the axis that are the bases.

The curved surface unrolled is a rectangle that measures h (the height of the

cylinder) by 2π· r (the length of the circumference); the radius of the cylinder is

the radius of any of the two bases.

The area of a cylinder is and the volume is

If the cylinder is an oblique cylinder, the formula for the volume is the same, but

the perpendicular height is not equal to the height of the curved surface.

Exercise 29:

Find the area and the volume of these cylinders

a) Radius = 15 cm and height = 12 cm

b) Diameter = 60 cm and height = 25 cm

142

10. CONES

A cone is a solid bounded by a curved surface that has a common point (vertex)

with a line that is the axis of the cone and a circle perpendicular to the axis that

is called the base of the cone.

Vertex or apex is the top of the cone (V)

Generatix of the cone is the straight line that joins the vertex with the circle of

the base.

The area of the curved surface of the cone is the area of a sector and it is given

by the formula AL = π·r·g and the area of the base is so the total area of

the cone is

The volume of the cone is

11. TRUNK OF A CONE

A trunk of a cone is the part of the cone which is between two

parallel planes.

The faces of the solid obtained by cutting it are called trunk

bases, and the distance between the two cutting planes is the

height of the trunk.

The area of any trunk of a cone is the area of the cone minus

the area of the cone that has been removed

143

As can be seen in a vertical cut of the cone on the right the

triangles VBC and VAD are similar and some of the measures

can be calculated from the others.

Exercise 30:

Find the area and the volume of these solids:

a) A cone whose height measures 13.4 cm and the radius measures 6.3 cm

b) A cone whose generatix is 7.8 cm and the radius is 3.4 cm

Exercise 31:

The height of the cone of the picture is 38 cm and the radio r is 15 cm. It

has been cut by a plane at 12 cm from the vertex. Calculate:

a) The area of the trunk of the cone

b) The volume of the trunk of the cone

12. SPHERE In a sphere all points are at the same distance r from the

centre of the sphere C.

The distance from the centre to the surface of the

sphere is called the radius of the sphere r

The area of a sphere of radio r is A = and the volume

is

144

Exercise 32:

Find the area and the volume of the a sphere whose radius is 5 cm

Exercise 33:

Find the area and the volume of the semisphere (hemisphere):

Exercise 34:

Find the area and the volume of an hexagonal prism where the base edge is 5

cm, the apothem 4 cm and the perpendicular height 13 cm

Exercise 35:

Calculate the volume and the area of a squared-based right pyramid. The

edge of the base is 13.2 cm and the perpendicular height is 17 cm long.

Exercise 36:

The volume of this sphere is 32 cm3. Find its radius and its area.

145

Exercise 37:

The volume of a cone is 785.4 cm3 and the perpendicular height is 10 cm.

Find the area.

Exercise 38:

A tomato tin is cylindrical. Each tin has a capacity of 1 litre and a base

radius of 5 cm. Find the height and the area of the tin.

Exercise 39:

A solid consists of a cylinder with a diameter of

15 cm and a height of 7 cm glued together to an

hemisphere, on one side and to a cone, with a

height of 10 cm, on the other. Find the total

volume and area of the solid

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UNIT 11

FUNCTIONS

1. FUNCTIONS

A function is a relation between two variables called x and y in which:

• x is the independent variable and is represented on the horizontal axis (x axis)

• y is the dependent variable and is represented in the vertical axis (y axis)

• Every x -value is associated to one and only one y-value.

Functions can be represented using grids and points. This is important when the

function behaviour needs to be visualized. But be careful, because there are

graphs which are not functions.

For example:

In this case, for example the x-value

-1 has three different

correspondences, so, it is not a

function.

The set of all the values which the variable x

can take is the domain of the function, and the

set of all values that the variable y takes when

x takes values in the domain is called the range.

Exercise 1:

This graph shows the temperature of the

water pouring on a shower head during a

person´s shower.

a) Which variables are associated?

b) What scale is used for each variable?

c) How long does the shower last?

d) What temperature does the water come

147

out at when the shower is turned on?

e) What water temperature does the person like?

f) Someone turns on the water in the kitchen and the temperature drops.

When does this happen?

Exercise 2:

We run water from a tap to fill a water tank. This

graph shows the height the water reaches in the

tank while the tap is running, until the tank is full.

a) What variables are associated?

b) What scale is used for each variable?

c) How long does the water run?

d) What is the height of the tank?

e) How high was the water in the tank before the tap was turned on?

Exercise 3:

Find the domains and ranges of the following functions: a) b) -

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Exercise 4:

This graph shows the distance covered by the

participants on a cycling trip as time progresses:

a) How many kilometers did they cover? How long

did the trip last?

b) What scales are used on the axis?

c) During the trip they make a stop. Say when this

was and how long it lasted.

d) When did they cycle faster, before or after the stop?

Exercise 5:

A balloon is released and rises. This graph shows how the balloon´s height

varies at times passes, until it bursts.

a) How high is it when it bursts? How long

after we release it does burst?

b) Which variable is independent and which

is dependent? What scale is used for each

variable?

c) Say how high the balloon is flying after

2 min and after 7 min.

Exercise 6:

Find the domain of the following functions:

a) b) c)

d) y=3x+2 e)

f)

149

2. INCREASING AND DECREASING

FUNCTIONS

The graph of a function has to be studied from the left to the right, that is to

say, how the y-coordinate varies when the x-coordinate increases. If the graph of

a function is going up from left to right, then it is an increasing function,( when

the independent variable increases, the dependent variable also increases).

If the graph is going down from left to right, then it is a decreasing function.

(when the independent variable increases, the dependent variable decreases)

Exercise 7:

The graph shows the trend of a boy´s weight between the ages of 2 and 20 years. a) Complete the following table:

Age (years) 2 8 14 20

Weight (kg)

b) Explain whether the function is increasing or

decreasing.

c) Specify in which of these time periods his weight

increased the most: between 2 and 8 years of age,

between 8 and 14 years or between 14 of 20 years.

Exercise 8:

Mark the increasing and decreasing intervals on each graph:

a) b) c)

150

3. MAXIMUM AND MINIMUM

A function’s maximum is the point where its coordinate, y, is higher than the

ordinate of other nearby points.

A function’s minimum is the point where its coordinate, y, is lower than the

ordinate of other nearby points.

Exercise 9:

The graph shows a patient’s temperature over four days. The temperature

was taken every six hours.

a) What was the maximum temperature and

when did it occur?

b) What was the minimum temperature and

when does it occur?

c) Specify the intervals in which the

function is increasing and the intervals in

which is decreasing.

Exercise 10:

The graph studies the number of cars parked in

a public car park located in the centre of a

city.

a) Over what period was the study carried out?

b) When did the number of cars parked reach a

minimum? How many cars were there at that

time?

151

c) When did the number of cars parked reach a maximum? How many cars

were there at that time?

Exercise 11:

Mark the increasing and decreasing intervals on the graph. Mark the points

that represent a maximum or minimum.

4. PERIODICITY

Periodic functions are those whose behavior is repeated each time the

independent variable covers a certain interval. The length of this interval is called

a period.

Exercise 12:

The graph shows the distance that separates John from his daughter, who is

spinning round a carousel.

a) How long does it take her to complete

a full spin? How many spins are shown?

b) What is the maximum distance

between John and his daughter? When

do these maxima occur?

c) If the carousel continues to spin, what will the distance be in 90 seconds?

d) Explain why the function is periodic and say what its period is.

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Exercise 13:

A tank of water used for watering

plants is filled in 4 minutes, then 2

minutes later it is emptied in 10

minutes, after which it is filled and

emptied again. The graph describes

this process.

a) What is the tank’s capacity?

b) When is it full?

c) Continue the function from minute 32 to minute 48.

d) Specify the function’s period.

Exercise 14:

The graph describes the distance that separates a comet from the Sun over

the course of time.

a) What is the period?

b) Complete the graph to show the distance covered as the comet completes 3

periods.

5. CONTINUOUS FUNCTIONS

A continuous function is a function whose graph can be drawn without lifting the

chalk from the blackboard, (or the pen from the notebook). Otherwise, the

function is discontinuous. This is only an intuitive definition.

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Exercise 15:

A car park charges the following rates:

-The first two hours are free.

-each further hour or fraction of an hour: €1.5

a) How much does it cost to park for 3 hours? And for 3 and a quarter

hours?

b) Plot the function time ↔ cost

c) Explain why the graph makes a jump at x = 2, x=3, …

d) Is the function continuous or discontinuous?

Exercise 16:

Charles takes one hour to walk from his house to Maria’s house, which is 5 km

away. He stays there for two and a half hours and goes home by bike. His

trip home takes 15 minutes.

a) Plot the function time ↔ distance home for Charles’ trip.

b) Calculate the speed of his trip home in km/h.

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Exercise 17:

A tank of water is emptied at a constant rate of 50 litres per minute. The

tank contains 4 000 l.

a) Complete the table that shows the amount of water in the tank as minutes

pass.

Time (min) 0 5 10 20 30 50

Volume (l) 4 000 3 750

b) Plot the functions time ↔ amount of water. To do this use the scale: X-

axis: 2 cells = 5 minutes; Y-axis: 2 cells = 1 000 l

Exercise 18:

We unplug an iron whose temperature is 100ºC and observe that its

temperature decreases to 50ºC in the first two minutes and then continues to

fall more slowly until reaching room temperature, 18ºc, in 15 minutes.

a) Plot the function time ↔ temperature

b) Can you observe any trend in this function?

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6. POINTS WHERE THE FUNCTION CUTS

WITH THE AXIS:

The y-intercept is the point where the function cuts the y-axis, its coordinates

y=are (o, b), the value “b” is calculated by the equation f(0) = b.

The x-intercept is the point where the function cuts the x-axis, its coordinates

are (a, 0), the value “a” is calculated solving the equation f(x) = 0.

Example: Find the intercept points of the function with the axis

y- intercept↔ ↔ the point is (0 , 6)

x- intercept ↔

Exercise 19:

Find the intersect points with the axis of the following functions:

a) b) c)

d) e) f)

g) h) i)

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Exercise 20:

This is the price list of a car park:

Open from 9 h. to 22 h.

First and second hours.................................Free

Third hour and consecutive or fraction.........1 € each.

Daily maximum.............................................12 €

Draw the graph of the function which relates park timing to its price.

Is it a continuous function?

7. WAYS OF EXPRESSING A FUNCTION

1.- By a sentence.

2.- By a table of values.

3.- By an analytical expression.

4.- By a graph.

A function can be studied by a graph which allows us to know some characteristics

of the function. But there is a great amount of functions given by an analytical

expression which connects x and y variables algebraically.

We generally use the letter f to represent a function. If a function tell us to

triple a number and add 1, i.e. 3x + 1, the function may be written in any of these

ways:

i) f (x) = 3x + 1

ii) y = 3x + 1

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To draw the graph of a function:

• Write a table of values (Calculate the value of y for each value of x)

x -1 0 1 2 3

y -2 1 4 7 10

• Draw a suitable grid. Plot the pairs (x,y) and join them with a line, in this case a

straight line.

Example

Make a table of values of the function y = x2. Then draw the graphs in a suitable

grid.

x -2 -1 0 1 2 3

y 4 1 0 1 4 9

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Example Complete this table relating the base and height of rectangles whose area is

12m²: Represent the function in the form of a graph. Find which expression

corresponds to this function

Base x (m) 1 2 3 4 6 12 x

Height y (m)

Exercise 21:

Copy and complete the table below for the functions:

a) y = x + 2

b) y = x2

x -2 -1 0 1 2 3

Y=x+2

Y=x2-4

Then draw both graphs on a suitable grid

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UNIT 12

LINEAR FUNCTIONS

1. STRAIGHT LINE GRAPHS

Functions with the expression y = mx

Draw the graph of y = 2x and on the same grid the graph of y = 3x and compare

them.

First we make a table with some points of y = 2x

x -2 -1 0 1 2

y -4 -2 0 2 4

If we make a table with some points of y = 3x

x -2 -1 0 1 2

Y -6 -3 0 3 6

And then we represent all the points

As it can be seen, the line y = 3x is steeper than y = 2x , the x-coefficient is the

gradient and indicates how steep the line is.

All the functions with an expression in the form y = mx go through the origin (the

point O =(0,0)).

If we represent the lines y = −3x and y = −2x , we get: We get decreasing

functions and they go through the origin too.

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Functions with the expression y = mx+n

If we represent the functions y = 2x + 1 and y = 2x − 2 we get:

Comparing their graphs we can see that they are straight lines, their gradient is

2, so they are parallel lines, and they cut the y-axis in the points (0,1) and (0,-2)

1 and -2 are called y-intercept.

Summarising all this we make the following definitions:

All straight lines have a similar expression:

y = mx + n where:

• m is the gradient or slope, and indicates how steep it is.

• n is the y-intercept, the point where the line cuts the y-axis. (The line goes

through the point (0, n)

Parallel lines have the same gradient or slope.

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If the equation of a straight line has no y-intercept, the line goes through the

origin and can be related to ratios.

Any function of the form y =mx + n is called a linear function. Its graph is a

straight line.

Examples: y = 2x + 3, y = 2x , y = 1

Linear functions in which n = 0, that is,

y =mx , are called proportionality

functions. The variable “y” is directly

proportional to “x”. The constant ratio,

, is called proportionality

constant (or constant of

proportionality). Their graphs pass

through the point (0, 0) . Examples: y =

1.60x , y = 5x

• Linear functions in which m = 0, that

is, y = n , are called constant functions. Their graphs are horizontal lines.

Example: y = 1

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Up to now, you have seen diagonal and horizontal lines.

• Diagonal lines have equations of the form y =mx + n, where m ≠ 0.

• Horizontal lines have equations of the form y = n.

But there is another kind of straight lines: vertical lines. They are not functions because there are infinite values of “y” corresponding the same value of “x”.

Vertical lines have equations of the form x = k .

Exercise 1:

Which of these equations have straight line graphs? Indicate if each equation

is that of a horizontal, vertical or diagonal line, or none of these.

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Exercise 2:

Match each line with its equation.

2. GRADIENTS AND INTERCEPTS

The gradient (or slope) of a straight line tells you how steep it is.

To work out the gradient find how many units the line rises for each unit it runs

across the page.

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For the line y = 2x - 2 , Gradient= m =

For the line y = -3x + 4 , Gradient = m =

The gradient is the coefficient of x in the equation of the line. (m)

- Straight lines with positive gradient are increasing functions.

- Straight lines with negative gradient are decreasing functions.

- The gradient of constant functions is 0.

The intercept is the distance from the origin to where the line cuts the

y -axis.

The line y = 2x − 2 cuts the y-axis at (0, − 2) . The y-intercept is −2 .

The line y = −3x + 4 cuts the y-axis at (0, 4) . The y-intercept is 4.

The intercept is the constant term in the equation of the line. (n)

Exercise 3:

Find the gradient and intercept of the lines

a) y = −5x + 2 b) y = −2 c) y = 7x d) 3x + 2y = 12

How to calculate the gradient of a line if you know two

points of it?

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Example: The gradient of the line joining P (−3, 2) and Q(5, 1) is:

Since the gradient of a straight line tells you how steep it is, parallel lines will

have the same gradient.

3. THE EQUATION y =mx + n

You can write the equation of any straight line in the form y =mx + n

For example: x + y = 5 ↔ y = −x + 5

4x − y = 6 ↔ y = 4x − 6

The gradient of the line y = mx + n is m.

The y-axis intercept of the line y = mx + n is n.

Example: What are the gradient and intercept of each of these lines?

a) y = 2x + 5 b) y = 1 − 3x

a) y = 2x + 5 ↔ m = 2, c = 5 b) y = 1 − 3x ↔ m = −3, c = 1

Exercise 4:

Match each line with its equation:

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Exercise 5:

Find the gradient and the intercept of these graphs. Write the equation of

each line.

Exercise 6:

Write the equation of a straight line that is parallel to y = 7 − 2x and cuts

the y-axis at (0, 3).

4. FINDING THE EQUATION OF A

STRAIGHT LINE GRAPH

1) If you know the gradient of a line and the y-axis intercept you can write the

equation of the line.

Example: What is the equation of a line with gradient 9 passing through

(0,5)?

Gradient = 9 and y-intercept = 5 , so the equation of the line is y = 9x + 5 .

2) If you know the gradient and a point on the line you can find the equation of

the line.

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Example: What is the equation of a line with gradient 8 that passes through

the point (2, 7)?

Gradient = 8 , so the equation of the line is y = 8x + n .

The line goes through (2, 7) so,

7 = 8 ×2 + n (Put x = 2 and y = 7 in the equation y = 8x + n )

7 = 16 + n ↔ n = −9

The equation of the line is y = 8x – 9

3) If you know two points on a line you can find the equation of the line.

Example: Find the equation of the line joining (1, 2) and (4, 3).

Gradient= m =

The line goes through (1, 2) so substitute 1 for x and 2 for y in the equation

Then the equation of the line is:

Exercise 7:

Find the equations of the ten lines described and represent:

a) Gradient of 7 and intercepts y-axis at (0,5)

b) Gradient of 0.5 and passes through (0, 3)

c) Parallel to a line with gradient 4 and passing through (3, 8)

d) Gradient of 3 and passing through (4, 7)

e) Gradient of -2 and cutting through (4, -3)

f) Parallel to

and passing through (0, -2)

g) Passing through (0, 1) and (1, 5)

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h) Passing through (0, 2) and (5, 7)

i) Passing through the midpoint of (1, 7) and (3, 13) with a gradient of 8

j) Passing through the origin and (4, -3)

Exercise 8:

Where does the line 2y = 9x − 5 cross

a) the y-axis

b) the x-axis

c) the line 4y = x + 24

You can use a graph to represent a real-life situation.

Exercise 9:

The price of 1 kg of rice is €1.50.

a) What is the equation of the function “amount bought ↔ cost”? Draw a

graph of it.

b) Give the slope. What does it mean?

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Exercise 10:

A telephone company charges 30 cents to connect a call and 15 cents per

minute of call time.

a) Write the function that associates call timed with call cost. Draw a graph

of it.

b) What is its slope? What does it mean?

c) How much would we pay for a call that last 8 minutes?

d) How long did a call last if we had to pay €3 for it?

Exercise 11:

A bar uses 2 kg of flour every day to make pasties. It currently has a 30 kg

bag of flour.

a) How much flour will it have after x days have passed?

b) Plot the function “time ↔ amount of flour”.

c) How many days does a bag of flour last? What is the function`s domain of

definition?

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Exercise 12:

For a ticket to travel 10 km a local train company charges €2. To travel 15

km we pay €2.50.

a) What is the function that tells us how much we have to pay depending on

how many kilometers we travel?

b) How much does it cost to travel 30 km?

Exercise 13:

Yesterday to connect to the internet for 3 hours I paid €3, and today in the

same Internet café I paid €2.50 for 2 hours of internet use. Find the

equation of the function “connection time ↔ cost”.

Exercise 14:

A tank containing 100 liters of water is emptied at a constant rate of

10l/min.

a) Complete the table of values for the function “time ↔ amount of water in

tank”.

Time (min) 1 2 3 4 5 6 7 8 9 10

Amount of water (l)

b) Plot the function.

c) What is the function`s equation?

d) Write the domain of definition.

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