I.E.S. MARÍA BELLIDO - BAILÉN

BILINGUAL SECTION – MARÍA ESTHER DE LA ROSA

PROPORTIONALITY AND PERCENTAGES

1. RATIO

A ratio is the quotient between two numbers or two comparable quantities which is

expressed as a fraction.

The terms of a ratio are called the antecedent and the consequent. The antecedent is the

numerator and the consequent is the denominator.

Example

The ratio of girls in a group of students is 5 for every 10 boys: or 1:2). This ratio of girls in a

group expresses how many girls there are in relation to boys, however, it can also be

expressed in terms of the total number of people in a group (1+2=3). For example, 1 of every

3 students is a girl.

It is important to remember that ratios and rational numbers are not the same. If is a

rational number a and b are integers con b≠0, while in the ratio, a and b can be decimal

numbers.

2. PROPORTIONS AND CONSTANT OF PROPORTIONALITY

A proportion is an equality between two ratios.

K IS A CONSTANT OF PROPORTIONALITY

3. PROPERTIES OF PROPORTIONS

1. In a proportion, the product of the means equals the product of the extremes (cross

multiplication).

2. In a proportion of two or more equal ratios, the sum of the antecedents divided by the sum of

the consequents is equal to any one of the ratios.

3. If the means and extremes in a proportion swap positions, the proportion does not vary.

4. CALCULATING THE UNKNOWN TERM OF ONE PROPORTION

To calculate the unknown term of one proportion we use the first property of the proportion:

Example 1

Example 2

Example 3

Example 4

5. DIRECT PROPORTIONS

Two variables are directly proportional when one variable is multiplied or divided one

by any number, the other variable is also multiplied or divided by that same number.

y = kx. k is the constant of proportionality.

Example 1: the weight of a product and its price are directly proportional: If 1 pound of tomatoes cost

$2, 2 pounds will cost $4 and ½ pound cost $1.

Example 2: A car travels 240 miles at a constant speed in 3 hours. How many miles will be travelled

in 2 hours?

In this example, the distance and time are directly proportional, because when the distance increases,

the total time increases and when the distance decreases, the total time decreases.

Follow these steps to complete a direct proportionality word problem:

Write down the ratio using one type of term (number of miles).

Write down the ratio with the second type of term (number of hours).

Cross-multiply, divide and solve

Example 3 : A grandfather shares $450 between his three grandchildren who are 8, 12 and 16 years of

age. If he distributes the money in proportion to their ages, how much will each one receive?

In this example, the age and the money are directly proportional because the older the children are,

the more money they will receive and the younger the children are, the less money they will receive.

Let x, y, z represent the amounts that each will receive.

For the properties of proportions

Each grandchild will receive:

6. INVERSE PROPORTIONS

Two quantities are inversely proportional when one is multiplied or divided by any

number, the other is divided or multiplied by the same number.

Example 1: Speed and time are inversely proportional because as the speed increases, the

time it takes to reach the destination decreases.

Example 2: 3 workers build a wall in 12 hours. How long would it have taken for 6 equally

productive workers?

In this example, the number of workers and the time are inversely proportional, because

when the quantity of people decreases, the total time increases and when the quantity of

people increases, the total time decreases.

Follow these steps to complete an inverse proportionality word problem:

Write down the ratio using one type of term (number of workers).

Write down the ratio with the second type of term (number of hours).

Invert one of the ratios (flip it upside down).

Cross-multiply, divide and solve (the same method used for direct proportions).

Example 3: It takes 14 hours for a faucet with a flow of 18 liters per minute to fill a reservoir with

water. How long will it take if its flow is reduced to 7 liters per minute?

In this example, the flow and time are inversely proportional, because when the flow decreases,

the total time increases and when the flow increases, the total time decreases.

7. PERCENTAGE

A percentage is a ratio where the consequent is 100.

It is denoted by the % symbol and 25% is read as 25 percent which is equal to 25/100 (a

fraction) or 0.25 (a decimal number).

PERCENTAGE INCREASE : A motor bike which originally cost $5,000 now costs $250 more.

What is the percentage increase?

5% increase in price

PERCENTAGE DECREASE: When purchasing a vehicle priced at $8,800, the sales clerk

awards a discount of 7.5% to the customer. How much will the final cost of the vehicle be?

$8,800 − $660 = $8,140

8. SIMPLE INTEREST

The interest is the amount of money that is paid in addition to the amount borrowed. The

total interest is directly proportional to the amount borrowed and the duration of the loan.

Amount borrowed = PRINCIPAL = P Loan duration = Time = t

Percentage charged = Rate = r Additional amount = Interest = I

If time is expressed in years:

If time is expressed in months:

If time is expressed in days:

Example 1: Calculate the amount of simple interest that is paid over a period of five years on a

principal of $30,000 at a simple interest rate 6%.

Example 2: Calculate the total amount paid in six months on a principal of $10,000 at a simple

interest rate of 3.5%.

Example 3: How long will it take a principal of $25,000 at a simple interest rate of 5% to become

$30,000?