Exercises and Activities

Exercises and Activities

in

Elementary Algebra

Exercise No. 1

Objective: Write an algebraic expression for each English phrase.

Direction: Write an algebraic expression for each of the following English phrase.

1. Seven times x _____________________________________2. The product of 5 and T _____________________________________3. Twice the number x _____________________________________4. The product of A and 10 _____________________________________5. Fifty time a number L _____________________________________6. Thrice the number P _____________________________________7. Multiply S by 4 _____________________________________8. The product of B and W _____________________________________9. 2x divided by y _____________________________________10.The sum of 100 and Y _____________________________________11.The sum of C and T _____________________________________12.Combine L with 12 _____________________________________13.The sum of m and n _____________________________________14.33 increased by a _____________________________________15.12 more than a number _____________________________________16.Twice the sum of x and y _____________________________________17.The product of 2a and 3b _____________________________________18.The sum of 4x and 7y _____________________________________19.Add 4 to 5q _____________________________________20.Seven decreased by N _____________________________________21.N decreased by 7 _____________________________________22.9 less than x _____________________________________23.9 more than x _____________________________________24.32 diminished by x _____________________________________25.5 added to 5f _____________________________________26.X divided by 17 _____________________________________27.The number 5 divided by q _____________________________________28.X less than 9 _____________________________________29.The quotient of 6 and twice x _____________________________________30.The difference between x and y _____________________________________31.Twice the difference of m and n _____________________________________32.The sum of 11 and an unknown numver V ________________________________33.Twice the quotient of x and y _____________________________________34.Six decreased by twice a number represented by 3 _____________________35.The sum of two unknown numbers (Choose your own letters.) ________________

Exercise No. 2

Objective: Translate each phrase into an algebraic expression.

Direction: Translate each of the following into an algebraic expression. Use any letter to represent the unknowns unless otherwise specified.

1. Two more than a certain number

2. Three less than a certain number

3. A certain number increased by 8

4. Seven decreased by a number

5. A number diminished by 4

6. The sum of two different numbers

7. The product of two different

numbers

8. The product of 2 and b diminished

by a

9. 9 less than the product of x and y

10.8 more than the product of 4 and q

11.100 more than 5 times a number

12.The sum of m and 2 multiplied by 5

13.The product of 4 and x divided by

11

14.The product of 7 and a

15.Three times the number x plus y

16.Twice the product of a and b

17.Twice the sum of a and b

18.Twice the sum of a, b and c

19.Three times the sum of x and y

20.48 decreased by the product of x, y

and z

21.45 diminished by the product of x, y

and z

22.The product of four and the sum of

a and b

23.The difference between m and n

divided by 4

24.Four times the number x increased

by 7

25.The sum of two different numbers

divided by 15

26.Ten more than the sum of two

different numbers.

27.The difference between 10 times a

number and y

28.The product of 7 and m increased

by the product of p and q

29.The product of 8 and a number

increased by another number

30.The product of seven and a number

diminished by another number

Exercise No. 3

Objective: Represent the unknowns.

Direction: Represent each of the following algebraically.

1. If Ayie is 24 yars old now, represent her age years ago.

2. If Mark is m years old now, how was he five years ago?

3. Ariel’s age in three years if he is x years old now.

4. Aloha was x years old two years ago. How old will she be three years from now?

5. Aaron was 12 years old four years ago. How old will he be in m years from now?

6. The number of minutes in S seconds.

7. A number added to one-half itself.

8. A number multiplied by Itself

9. Tony has twice as much money as Ben.

10.The length of the rectangle is two more than its width.

11.The perimeter P of a rectangle is twice the sum of the length l and width w.

12.The perimeter P of a triangle is the sum of the length of the sides a, b and c.

13.The area of a circle is the square of the radius (r) times π.

14.The distance (d) traveled is the product of the rate (r) and time (t).

15.The area (A) of a triangle is half the base (b) times the height (h).

16.The average (A) of any two numbers n and m is their sum divided by 2.

17.The diameter (d) of a circle is twice the radius (r).

18.A profit (p) is the difference between the revenue (r) and the cost (c).

19.The circumference (C) of a circle is the product of the diameter (d) and π.

20.A simple interest (I) is the product of the principal (P), the rate (r), and the time (t).

Exercise No. 4

Objective: Translate algebraic expression to English phrase.

Direction: Translate each of the following algebraic expression into an English phrase.

1. x+y

2. 4+m

3. A+4

4. M-N

5. N-M

6.

7.

8.

9.

10.MN

11.2a+3

12.4x-5

13.4-7x

14.

15.

16.

17.

18.x(a-b)

19.y(b-a)

20. (a+b)(b-a)

21.3(a+4) - 32

22. + 4(x+y)

23. (xy)2

Exercise No. 5

Objective: Form an equation and solve each problem.

John wants to surprise his on Valentine’s Day. He wants to give her a flower that can symbolize what he really feels for her. John consulted with her gardener and is his advice, “Hmmmm…the best flower for her is…”.

Directions: Translate and solve each to determine the integers being asked for. From the boxes, get the word that corresponds to your answer. Write the integers and the symbolic meaning of each flower on the column provided.

Equation Integer

1. The sum of four consecutive integers is 50. Find the integers.

2. Find two consecutive integers such that six times the first equals five more than five times the second.

3. Find three consecutive even integers such that the second plus the third is three time the first.

4. Find two consecutive integers such that four times the smaller minus three times the larger equals zero.

5. Find three consecutive odd integers such that two less than the first added to four less than the second is equal to the third integer increased by nine.

Flowers Integers/Symbolic Meaning

1. Cadena de Amor

2. Lilac

3. Gardenia

4. Orchid

5. Chysanthemum

Exercise No. 6

Objective: Write the equation from the word problem.

Directions: Match the equation with the word problem it represents. Write the equation and corresponding letter. Then write the letter of the equation in the circle that has the problem number. Discover the message formed by the letters.

M. = 73 T. x+8 = 21 S. a-12 = 25 D. m-13 = 41

H. c+10 = 35 P. 4t = 360 A. 2s = 18 U. = 55

1. Allan has 8 more marbles than Jon does. If Allan has 21 marbles, how many does Jon has? ____________________

2. Ana is 12 years younger than Ellen. If Ana is 25, how old is Ellen? _______________

3. Owen has scored twice as many goals as Rico has. If Owen’s goal total is 18, how many goals has Rico scored? ____________________

4. Liza delivered 13 fewer magazines this week than last week. If she delivered 41 magazines this week, how many did she deliver last week? ____________________

5. Three friends went out for lunch. They shared the cost of the meal equally. If each person paid P55, what was the total cost of the meal? ____________________

6. Jenny and her friends share a pizza. The cost of pizza is shared equally between them. If each of them pays P73, how much does the pizza cost? __________________

7. Henry sold 10 more cards than Eric did. If Henry sold 35 cards, how many did Eric sell? ____________________

8. Edna paid 4 times as much for a tape as her friend did. If Edna paid P360, how much did her friend pay? ____________________

Exercise No. 7

Objective: To translate verbal phrase to algebraic expressions.

Do you remember me? I pioneered the use of symbols in representing numbers and I am quite well known for that. Answer the matching test to know me.

Directions: Match each verbal phrase in Column A with the algebraic expressions in Column B. Write the letter of your answer on the blank provided before the item number. Read the word formed by the answers to identify the person above.

Answer Column A Column B

1. The square of six minus the number x

2. Two more than six times the number n

3. Three times a number x decreased by five

4. The quotient of sixteen and the number n

5. Five times the product of the number m and three

6. Twenty added to the product of the number a and b

7. Nine increased by the quotient of the number m and n

8. The product of twelve and the number y divided by seven

9. Eight subtracted from the sum of eleven and the number y

10.The sum of product of four and the number m and the quotient of the number n and three

a. ab+20

d. 62-x

h. 5(3m)

i. 6n+2

n. 9+(m/n)

o. 3x-5

p. 16/n

s. 4m+(n/3)

t. 12y/7

u. (y+11) - 8

Exercise No. 8

Objective: To translate verbal phrase to algebraic expressions.

Directions: Match Column A with Column B by performing the indicated statements and operations. Write the letter that corresponds to your answer on the space before each number.

1. Five times the quotient of twelve and a number a. 2(x+20)+4

2. The square of a number added to the cube of b. (8-x)(14-x)

another number

3. The sum of a number and fifteen c. x+(x+20)

4. The sum of eight and a number multiplied by d. 2x+8

the difference of fourteen and a number

5. Four years more than twice the age of Ezra in e. 3(x+5)

twenty years

6. In 5 years, he will be thrice as old as his son f. 5(12/x)

7. The difference of two numbers is 20 g.(8-x)(14+x)

8. The product of a number and 14 h. x-5

9. Twice a number increased by8 i. x2+y3

10.Sara’s age 5 years ago j. x+15

k. 14x

Activity No. 1

Objective: Solve problems accurately.

Directions: Analyze and solve each problem. Answer what is asked in each problem.

The sum of two numbers is 29 and their difference is 5. Find the numbers.

REPRESENT: Let ___ = one of the numbers

RELATE: ___ = the other number

EQUATE: ___ equation (their difference is 5)

SOLVE:

PROVE:

a. Their sum is 29: _____

b. Their difference is 5: _____

The sum of two numbers is 30. The second number is 2 more than thrice the first number. Find the numbers.

REPRESENT: Let ___ = the first number

RELATE: ___ = the other number

EQUATE: ___ equation (their sum is 30)

SOLVE:

PROVE:

a. Their sum is 30: _____

b. The other number is 2 more than thrice the first number: _____

Activity No. 2

Objective: Write the expressions and perform some calculations.

You have heard of number puzzles when someone asks you to choose numbers and perform some calculations, and then the person tells you what the result is, or is able to tell you the number you chose.

Example:

Choose a number _______________________

Add three _______________________

Double the result _______________________

Subtract two _______________________

Divided by two _______________________

Subtract the number you chose _______________________

The result is two _______________________

To see why the result is two, look at the expression for each step.

Choose a number _________ x

Add three _________ x+3

Double the result _________ 2(x+3) = 2x+6

Subtract two _________ 2x+6-2 = 2x+4

Divided by two _________

Subtract the number you chose _________ x+2-2 = x

Direction: Write the expressions that show how the following number puzzles work.

1. Choose a number _______________________

Add the next smaller number _______________________

Add nine _______________________

Divide by two _______________________


The result is 4 _______________________

2. Choose a number _______________________

Double it _______________________

Add six _______________________

Double the result _______________________

Divide by four _______________________


The result is 3 _______________________

Activity No. 3

Objective: Use equations in solving age problems.

Mr. Garcia, a mathematics teacher, was travelling by bus with his family consisting of his mother, wife, son and daughter. The bus conductor, doubting the son’s right age to have a half-fare ticket, asked “How old is your daughter?” Mr. Garcia irritated at having him questioned, confused the conductor with his answer: “My daughter is twice as old as my son. My wife is three times as old as the sum of the age of my son and daughter, and I am as old as my wife and my daughter. My mother, whose age is the total of all our ages, is now 69?”

Can you help the conductor in finding the age of the girl? To help the conductor,

1. Represent the son’s age by x

2. Represent the daughter’s age: (daughter’s age twice as old son’s age) _____

3. Find the sum of the ages of the son and daughter in terms of x _____

4. Represent the age of the wife: (wife’s age is three times as old as the sum of the age of son and daughter) _____

5. Represent the age of Mr. Garcia (Mr. Garcia’s age is as old as his wife and his daughter) _____

6. Represent the total ages of the son, the daughter, the wife and Mr. Garcia _____

7. Translate the following into an equation:

Total age of the son, daughter, the wife = Mother’s age and Mr. Garcia

_______________________________=________________________

8. Solve for x: _____

9. How old is the daughter? _____

10.Does the conductor have the right to doubt Mr. Garcia? _____

Activity No. 4

Rate and Time

Objective: Solve problems involving distance.

Directions: Read and analyze each problem. Do what is being asked.

1. A truck and a car leave the same service at the same time and travel in the same direction. The truck travels at 110 km per hour (kph) and the car 80 kph. They can maintain CB radio contact with a range of 20 kilometers (km). When will they lose contact?

Complete the following table to aid the translation.

distance = rate x time

Distance Speed Time

Truck d 110

Car d t

2. A train leaves a station and travels east at 72 kph. Three hours later, a second train leaves on a parallel track and travels east at 120 kph. When will it overtake the first train?

Distance Speed Time

Slow train d 72

Fast train d T


d = 72( )

d = ( )t

3. Two cars leave town at the same time going in the same direction. One travels at 30 mph and the other travels at 40 mph. in how many hours will they be at 72 miles apart? (Hint, formulate a chart)

4. A private airplane leaves an airport and flies due south at 192 kph. Two hours later, a jet leaves the same airport and flies due south at 960 kph. When will the jet overtake the plane?


Distance Speed Time

Airplane d 192 t+2

Jetplane d 960 t

Exercises and Activities

In

Intermediate Algebra

Exercise No. 1Number Related Problems


A. Represent “twice a given number” if the given is represented by:

1. 8 __________ 6. 3y+2 __________

2. b __________ 7. ½ x __________

3. 3a __________ 8. 3-5x __________

4. a+2 __________ 9. __________

5. 2m+5 __________ 10. __________

B. Represent in terms of x, “4 increased by a given number” if the given number is represented by:

11.2x ______________________________

12.5x ______________________________

13.2x+1 ______________________________

14.5-x ______________________________

15.7x+5 ______________________________

C. Represent in terms of x, “five less than a given number” if the given number is represented by:

16.10 __________ 21. 5-3x __________

17.2x __________ 22. 7x-3 __________

18.3x+1 __________ 23. ½ x __________

19.x-4 __________ 24. ¼ x __________

20.2x+7 __________ 25. -5 __________

D. The sum of two numbers is 60. Represent in terms of x the larger number if the smaller number is represented by:

26. 14 __________ 31. 4x+1 __________

27. x __________ 32. 1-7x __________

28. 3x __________ 33. ____________

29. 2x+1 __________ 34. 2x+4+3x __________

30. 2-3x __________ 35. 5x-4+x __________

E. The sum of two numbers is 120. Represent (in terms of m) the smaller number if the larger number is represented by:

36.80 __________ 41. 4-m __________

37.m __________ 42. 3-4m __________

38.15m __________ 43. __________

39.m+4 __________ 44. __________

40.2m+3 __________ 45. 3(m+1) __________

F. Represent “one-half of a number” if the number is expressed as:

46. 24 __________ 51. 5+k __________

47. t __________ 52. 7k+5 __________

48. 2a+1 __________ 53. m+n __________

49. 7b __________ 54. 3a-2b __________

50. 2-3a __________ 55. ____________

G. Represent “the sum of thrice the first number and the second” if the first and the second numbers are represented by:

56. 4 and 7 ______________________________

57. p and 4 ______________________________

58. p+3 and 3p ______________________________

59. sp and 4p ______________________________

60. 3-p and p ______________________________

Exercise No. 2Number Problems

Objective: Translate mathematical sentences into equations and solve.

1. The sum of five and a number is negative three.

The sum of five and a number _____________________________________

Equation: _____________________________________________________

Answer: _____________________________________________________

2. Twice the number diminished by five is thirteen.

Twice the number ________________________________________________

Twice the number diminished by thirteen __________________________

Equation: _____________________________________________________

Answer: _____________________________________________________

3. The difference of three times a number and seven is negative forty-three.

Three time a number ___________________________________________

The difference of three times a number and seven _____________________

Equation: _____________________________________________________

Answer: _____________________________________________________

4. Fifteen times a number is equal to 45 diminished by 10.

Fifteen times a number ___________________________________________

Fifty five diminished by ten ___________________________________________

Equation: _____________________________________________________

Answer: _____________________________________________________

5. Seventeen more than five times a number is twenty two.

Five times a number ___________________________________________

Seventeen more than five times a number __________________________

Equation: _____________________________________________________

Answer: _____________________________________________________

6. One less than thrice the number is fourteen.

Thrice the number ________________________________________________

One less than thrice the number _____________________________________

Equation: _____________________________________________________

Answer: _____________________________________________________

7. The quotient of a number and four is six.

The quotient of a number and four ________________________________

Equation: _____________________________________________________

Answer: _____________________________________________________

8. Forty five is equal to twice a number decreased by seven.

Twice a number decreased by seven ________________________________

Equation: _____________________________________________________

Answer: _____________________________________________________

9. The difference between eight times a number and nine is negative twenty five.

Eight times a number ___________________________________________

The difference between eight times a number and nine ________________

Equation: _____________________________________________________

Answer: _____________________________________________________

10.Five times a number equals sixty five.

Five times a number ___________________________________________

Equation: _____________________________________________________

Answer: _____________________________________________________

Exercise No. 3Age Problems


A. Mercy is 17 years old now. Represent her age: __________

1. 2 years from now __________

2. 3 years ago __________

3. x years from now __________

4. 2x years ago __________

5. x+7 years hence __________

B. Melchor is m years old now. Represent his age: __________

1. 15 years from now __________

2. 11 years ago __________

3. a years ago __________

4. n+5 years ago __________

5. 5x years from now __________

C. Mr. Matthew Matics is y+5 years old now. Represent his age: __________

1. 3 years ago __________


3. 2y+3 years ago __________

4. 5y years from now __________

5. 3y-5 years from now __________

D. In 4 years, Mr. Parabola will be 17 years. Represent his age: __________

1. Now __________

2. 8 years ago __________


4. x years ago __________

5. a-3 from now __________

E. Arielle will be 57 years old p years from now. Represent her age:__________

1. Now __________

2. 3 years ago __________


4. m years from now __________

5. m years ago __________

F. Five years ago, Edwin was 27 years old. Represent his age: __________

1. At present __________

2. 3 years ago __________


4. q years ago __________

5. q years from now __________

G. P years ago, Mr. Reyes was Q years old. Represent his age: __________

1. At present __________


3. 5 years ago __________

4. R years from now __________

5. P+4 years ago __________

Exercise No. 4Work Problems


A. What part of the job can Aris do per hour if the whole job takes him:

1. 11 hours __________

2. m hours __________

3. 3 days __________

4. (x+1) hours __________

5. (x+3) hours __________

6. 45 minutes __________

7. ½ hour __________

8. 5 days __________

9. 2½ hours __________

10. hour __________

B. If Owen can paint a wall in 5 hours, how much of the wall can he paint in:

1. 5 hours __________

2. 2½ hour __________

3. 3x hours __________

4. (x+7) hours __________

5. hours __________

C. What part of the sink can a faucet fill per minute if the whole sink can be filled in:

1. 45 minutes __________ 4. 7 minutes __________

2. 4 hours __________ 5. 2/3 minute __________

3. 3 minutes __________

Exercise No. 5Motion Problems


A. A bus travelled a distance of 180 kilometers. How long did it take to make this trip if the bus rate was:

1. 60 kph __________ 5. 2x kph __________

2. 30 kph __________ 6. (x+1) kph __________

3. 45 kph __________ 7. 5x kph __________

4. x kph __________

B. Mr. Ramos can travel at a rate of 6 kilometers per hour. Find the distance he can cover in:

1. 2 hours __________ 5. 3 hrs 45 minutes __________

2. 4 hours __________ 6. x hours __________

3. 3½ hours __________ 7. (x-3) hours __________

4. 30 minutes __________ 8. 30x minutes __________

C. An airplane flew 4000 kilometers. Represent the rate of the plane if the time needed to fly is:

1. 4 hours __________ 4. 2y hours __________

2. 6 hours __________ 5. 3(x+1) hours __________

3. x hours _________

D. Give the average rate of the car if it travelled:

1. 3 hours at a rate of 69 kph

2. 1 hour at a rate of 45 kph and 1 hr at a rate of 50 kph

3. 1 hour at a rate of 50 kph and 2 hrs at a rate of 45 kph

4. 3 hours at a rate of 35 kph and 2 hrs at 40 kph

5. 1 hour at m kph and 2 hours at 3m kph

Activity No. 1

Objective: Solve Number Problems

Directions: Analyze and Solve. Answer what is asked in each problem.

The smaller of two numbers is thrice the larger. The larger number is eight more than the smaller one. Find the numbers.

REPRESENT: Let _____ = the larger number

RELATE: ________ = the smaller number

EQUATE:

SOLVE:

Answers:

PROVE: a. The smaller of two numbers is thrice the larger _____

The sum of two even numbers is 30. The larger number is twelve more than one half the smaller number. Find the numbers.

REPRESENT: Let _____ = the smaller number

RELATE: ________ = the larger number

EQUATE:

SOLVE:

Answers:

PROVE: a. The two numbers are even

b. Their sum is 30

c. The larger number is 12 more than one half the smaller number

Activity No. 2

Objective: Solve age problems.

Directions: Complete the table by representing the unknowns. Solve accurately.

1. The sum of Patrick’s age and Wayne’s age is 58. Eight years ago, Patrick was twice as old as Wayne then. How old is Wayne.

REPRESENT: Let _____ = Wayne’s present age

RELATE: _________ = Patrick’s present age

Now Past

Wayne

Patrick

EQUATE:

SOLVE:

ANSWER:

PROVE: The sum of their ages is 58.

2. Alvin is now 21 years older than his son. In 8 years, he will be twice as old as his son’s age. What are their present ages?

REPRESENT: Let _____ = the son’s present age

RELATE: _________ = Alvin’s present age

Now Future

Son

Alvin

EQUATE:

SOLVE:

ANSWER:

PROVE: Alvin’s age is twice his son’s age.

3. Aris is twice as old as Rico while Jay is 24 years younger than Aris. If half of Aris’ age six years ago was three less than one half the sum of Rico’s age in four years and Jay’s present age, find the ages of each.

REPRESENT: Let _____ = Rico’s present age

RELATE:

Now Past (6 years ago) Future (4 years from now)

Rico

Aris

Jay

EQUATE:

SOLVE:

ANSWER:

PROVE: a. Aris is twice as old as Rico: _____

b.Jay is 24 years younger than Aris: _____

Activity No. 3

Objective: Solve motion problems.

Directions: Complete the table by representing the unknowns. Solve accurately.

Charts are very useful in solving problems involving two moving objects whose speed does not change.

1. Pangasinan and Manila are about 240 kilometers apart. A car leaves Manila traveling towards Pangasinan at 65 kph. At the same time, a bus leaves Pangasinan bound for Manila at 55 kph. How long will it take before they meet?

REPRESENT: Let _____ = the number of hours it takes for the vehicles to meet

RELATE:

Rate x Time = Distance

Car

Bus

EQUATE:SOLVE:ANSWER:

2. There are two buses. The rabbit heads north in the expressway at 45 kph. Exactly 12 minutes after, the Express follows at a steady speed of 54 kph. How long does it take the Express to overtake the rabbit?

REPRESENT: Let _____ = the Express’ time

RELATE: _________ = the Rabbit’s time


Rabbit

Express


3. While waiting for a train, Mel takes a bus ride at 60 kph to a certain point and then walks back leisurely at 10kph. The bus ride takes 15 minutes less than his walking time. How far does he walk and for how long?

REPRESENT: Let _____ = the walking time

RELATE: _________ = the walking distance


Walk

Ride


Activity No. 4

Objective: Solve work problems.

Directions: Represent and solve the given problems.

1. Richard can build a doghouse by himself in 3 days. Alvin can build the same doghouse in 6 days. How long will it take them if they work together?

REPRESENT:RELATE:

Rate x Time = Work

Richard x

Alvin x

EQUATE:SOLVE:

ANSWER:PROVE:

2. Marivic can type a committee report in 5 hours. Maricar, who is a slower typist, helped Marivic and together they finished the report in 3 ours. How long would it take Maricar if she worked alone?

REPRESENT: Let t = time it would take Maricar to do the job herself

Rate x Time = Work

Marivic 1/5

Maricar 1/t

EQUATE:SOLVE:

ANSWER:PROVE:

3. A swimming pool has two inlet pipes. One pipe can fill the pool in 6 hours; the other can fill it in 3 hours. The pool has one outlet pipe that can empty the pool in 4 hours. One day, when filling up the pool after it was cleaned, the outlet pipe was left open by mistake. How long did it take to fill the pool?

REPRESENT:RELATE:

Rate x Time = Work

Input Pipe 1 x

Input Pipe 2 x

Outlet Pipe x

EQUATE:SOLVE:

ANSWER:PROVE:

Exercises and Activities

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