Solving Equations Exercises – Set 1A

Below are exercises on solving equations.

Part I

1. x + 2 = 10

2. x - 7 = -14

3. 4x = 20

4. -3x = 18

5. \frac {x}{4} = 20

6. \frac {x}{9} = -10

7. 2x + 1 = 15

8. 4x - 7 = 13

9. -6x + 2 = -14

10. 3x - 1 = 8

Part II

1. 9 - x = 17

2. 4x - 7 = -15

3. 7x + 36 = 4x

4. -2x - 11 = -x

5. 7x = 5x - 6

6. 2x - 5 = x + 4

7. 9x + 6 = 7x - 8

8. 9 - x = 2 + 6x

9. \frac {x}{2} = x + 7

10. -\frac {1}{4}x = 3x - 12

Part III

1. 2(x - 5) = -8

2. 6(2x - 1) = -8 + x

3. 3(4 - 3x) = 3x

4. 5(x- 7) - 2 = x - 1

5. 4(x - 1) = 3(x + 1)

6. 7(4 - 2x) = x - 2

7. 4 - 6(x - 7) = -x - 4

8. -\frac {x}{4} = x + 10

9. 3x + \frac {1}{2} = 12

10. \frac {3x}{5} = 4x - 8

PEMDAS Exercises – Set 1

Below are exercises on division of signed numbers

Part I

1. 5 + 3 - 2

2. 9 - 6 + 4

3. 7 + 4 \times 3

4. 6 \times (-2) + 3

5. 2 \times (-9 + 4)

6. 60 \div (-6 + 2)

7. (-3 - 11) \times (-7)

8. 12 \div 3 \times 5

9. 4 \times (-1 - 6)

10. -5 + (13 - 7) \div 3

Part II

1. 4 \times (-3 - 5)

2. -2 \times (3 + 6)

3. (9 - 13) \div (-1)

4. (4 + 6)^2 - 7

5. (-4)^2 \times (-2)^3

6. -3 - 7 \times 2

7. 3 - (-2) + 8

8. –12 - 8 \div 4

9. 9 - (-4) \times (-4)

10. 10 \div (-2) - (-3 \times 4)

Part III

1. 3 - (-2) \times 5

2. -4 \times 3 + 6 \times 2

3. 16 \div (-2) + 12 \div 4

4. 36 \div (-13 + 4)

5. 18 \div (-3)^2 + (-4)

6. 4 \times (-2) + (-3^2)

7. 3 \times [-4 - (12 - 5)]

8. (-3)^2 + 2^3 \div (-4)

9. 9 - (-4^2) \times (-2)

10. -2 \times (-3 \times 2)^2 - (-2)^2

The solution can be found here.

Division of Integers Exercises – Set 1

Below are exercises on division of signed numbers. Part I and II are exercises for divison of integers, while part III is an extension to fractions and decimals.

Part I

1.  12 \div (-3)

2. -16 \div 4

3.  -14 \div (-2)

4. 15 \div (-3)

5. 0 \div (-5)

6. -18 \div (-1)

7. -25 \div 5

8. 6 \div (-3)

9. -8 \div (-4)

10. 12 \div 2

Part II

1.  -34 \div 17

2.  45 \div -9

3. -56 \div (-8)

4. -63 \div 9

5. 0 \div (-100)

6. -54 \div (-9)

7. 72 \div (-8)

8. -50 \div 10

9. -17 \div (-17)

10. 36 \div (-4)

Part III

1. \frac {1}{2} \div (- \frac {1}{4})

2. -\frac {2}{5} \div(- \frac {3}{4})

3. -\frac {3}{4} \div \frac {3}{8}

4. \frac {4}{5} \div (-\frac {1}{5})

5. -\frac {5}{7} \div \frac {1}{2}

6. 0.4 \div (-0.2)

7. -0.9 \div (-0.3)

8. -4.5 \div 0.9

9. -0.8 \div 0.2

10. 0 \div 0.7

Multiplication of Integers Exercises – Set 1

Below are exercises on multiplication of signed numbers. Part I and II are exercises for multiplication of integers, while part III is an extension to fractions and decimals. I have written the exercises using the \times notation as well as the parenthesis to familiarize you with both notations.

Solutions and answers can be found here.

Part I

1. 2 \times (-3)

2. -4 \times -7

3. -9 \times 3

4. 7 \times 7

5. -11 \times 3

6. (12)(-4)

7. (-6)(-3)

8. (9)(0)

9. (-5)(3)

10. (13)(3)

Part II

1. 2 \times 3 \times (-4)

2. -4 \times 8 \times 1

3. -4 \times (-3) \times 3

4. -6 \times 2 \times (-5)

5. -12 \times 13 \times 0

6. (-6)(-1)(2)

7. (-5)(8)(3)

8. (-7)(-1)(-3)

9. (11)(8)(-1)

10. (-2)(-1)(7)(3)

Part III

1. \frac {1}{2} \times (-\frac {1}{3})

2. -\frac {1}{5} \times (-\frac {5}{8})

3. -\frac {3}{4} \times \frac {1}{6}

4. \frac {2}{7} \times \frac {1}{4}

5. -\frac {6}{11} \times \frac {2}{3}

6. 0.5 \times (-3)

7. -0.2 \times (0.4)

8. -3 \times (-0.1)

9. -0.6 \times (-0.4)

10. -0.7 \times 0

You might also want to practice division of integers.

Subtraction of Integers Exercises – Set 1

Below are exercises on subtraction of signed numbers. Part I and II are exercises for subtraction of integers, while part III is an extension to fractions and decimals. Solutions and answers can be found here.

Part I

1. 8 – (-7)

2. -4 – (-10)

3. -6 – 8

4. 0 – (-5)

5. -17 – (-13)

6. 0 – 18

7. 12 – 19

8. -11 – 18

9. 21 – (-22)

10. -14 -(-14)

Part II

1. 31 – (-14)

2. -17 – (-11)

3. -19 – 12

4. 0 – (-17)

5. -34 – (-21)

6. 0 – 47

7. 36 – 42

8. -25 – 35

9. 28 – (-30)

10. -45 – (-45)

Part III

1. \frac {1}{7} - ( -\frac {3}{7})

2. - \frac {3}{5} - (- \frac {4}{5})

3. -\frac {3}{4} - \frac {1}{4}

4. \frac {7}{11} - \frac {9}{11}

5. - \frac {5}{12} - (- \frac {5}{12})

6. 0.4 – (-0.3)

7. -0.8 – (-0.7)

8. -1.2 – 0.4

9. 0.3 – 0.9

10. -0.6 – (-0.6)

You might also want to practice multiplication of integers.

Addition of Integers Exercises – Set 1

Below are exercises on addition of signed numbers. Part I and II are exercises for addition of integers, while part III is an extension to fractions and decimals. Solutions and answers will be can be found here.

Part I

1. -7 + 13

2. (-8) + (-9)

3. 5 + (-18)

4. 34 + (-38)

5. (-34) + 34

6. 0 + (-25)

7. 5 + (18)

8. -14 + (-12)

9. 13 + (-13)

10. 16 + 18

Part II

1. 13 + 12 + (-15)

2. -18 + (-2) + 14

3. 14 + 8 + (-14)

4. 21 + 7 + (-28)

5. -23 + 0 + 14

6. -19 + 22 + (-6)

7. 32 + (-11) + (-27)

8. 12 + (-11) + 10

9. 11 + (-22) + 21 + (-10)

10. 16 + (-16) + 8 + 3

Part III

1. \frac {1}{2} + (- \frac {3}{2})

2. \frac {1}{4} + (- \frac {2}{4})

3. \frac {3}{7} + (-\frac {3}{7})

4. - \frac {1}{2} + \frac {3}{4}

5. -\frac {4}{11} + (-\frac {3}{11})

6. 0.7 + (-0.3)

7. 4.8 + (-1.2)

8. -3.7 + 2.2

9. 12.6 + (-11.1)

10. 13.75 + (-15.2)

You might also want to practice subtraction of integers.

Week 11 Review: Practice Exercises and Problems

After learning about work problems, let’s solve the following exercises. Solutions and answers will be posted soon.

Week 11 Review: Practice Exercises and Problems

1.) Aria can do a job in 7 days. What part of the job is finished after she worked for 3 days?

2.) Katya can do a job in 5 days. Marie can do the same job in 6 days. If they both worked for 1 day, what part of the job is finished?

3.) Ramon can paint a house in 6 days. Ralph can do the same job in 10 days. If they both worked for 2 days, what part of the job is done?

4.) One hose can fill a pool in 3 hours and a smaller hose can fill the same pool in 4 hours. How long will it take the two hoses to fill the entire pool?

5.) Marco can dig a ditch in 5 hours and he and Jimmy can do it in 2 hours. How long would it take Jimmy to dig the same ditch alone?

6.) Maria can paint a fence in 6 days and Leonora can do the same job in 7 days. They start to paint it together, but after two days, Leonora left, and Maria finishes the job alone. How many days will it take Leonora to finish the job?

7.) An inlet pipe can fill a pool in 4 hours. An outlet pipe can fill the same pool in 6 hours. One day, the pool was empty. The owner opened the inlet pipe but forgot to close the outlet pipe. How long will it take to fill the pool?

Week 9 Review: Answers and Solutions

Below are the solutions to the exercises and problems about age problems.

Exercises

1.) Leah is 3 years older than Lanie. The sum of their ages is 29. What are their ages?

Let x = Lanie’s age
x + 3 = Leah’s age

The sum of their ages is 29.

x + (x + 3) = 29
2x + 3 = 29
2x = 29 – 3
2x = 26
x = 26/2
x = 13 (Lanie’s age)

Leah’s age = x + 3 = 13 + 3 = 16

Answer: Lanie 13, Leah 16

2.) Alfred’s thrice as old as Fely. The difference between their ages is 16. What are their ages?

Let x = Fely’s age
3x = Alfred’s age

The difference between their ages is 16.

3x – x = 16
2x = 16
x = 16/2
x = 8(Fely’s age)

3x = 3(8) = 24 (Alfred’s age)

Answer: Alfred 24, Fely 8

3.) Kaye is 4 years younger than Kenneth. The sum of their ages is 42. What are their ages?

Let x = Kenneth’age
x – 4 = Kenneth’s age

The sum of their ages is 42.

x + (x – 4) = 42
2x – 4 = 42
2x = 42 + 4
2x = 46
x = 46/2
x = 23 (Kenneth’s age)
x – 4 = (23)-4 = 19 (Kaye’s age)

Answer: Kaye 19, Kenneth 23.

Problems

1.) Gina is 5 years older than Liezel. In 5 years, the sum of their ages will be 39. What are their ages?

Present ages
Let x = Liezel’s age
x + 5 = Gina’s age.

In 5 years
(x + 5) = Liezel’s age
(x + 5) + 5 = Gina’s age.

The sum of their ages will be 39.

(x + 5) + (x + 5) + 5 = 39
2x + 15 = 39
2x = 39 – 15
2x = 24
x = 24/2
x = 12 (Liezel’s age)
(x + 5) = 12 + 5 = 17 (Gina’s age)

Answer: Gina 17, Liezel 12.

2.) Alex is 7 years older than Ben. Three years ago, the sum of their ages was 29. What are their ages?

Present ages
Let x = Ben’s age
x + 7 = Alex’s age

3 yrs ago
x – 3 = Ben’s age
x + 7 – 3 = Alex’s age

The sum of their ages was 29.

(x – 3) + [(x + 7) – 3] = 29
x – 3 + x + 4 = 29
2x + 1 = 29
2x = 29 – 1
2x = 28
x = 28/2
x = 14 (Ben’s age)
(x + 7) = 14 + 7 = 21 (Alex’s age)

Answer: Alex 21, Ben 14

3.) Yna is 18 years older than Karl. In 8 years, she will be as twice as old as Karl. What are their ages?

Let x = Karl’s age
x + 18 = Yna’s age

In 8 years…
Karl = x + 8
Yna = (x + 18) + 8

…she (Yna) will be as twice as old as Karl

Yna’s age = 2 times Karl’s age

(x + 18) + 8 = 2(x + 8)
x + 26 = 2x + 16
x – 2x = 16 – 26
-x = -10
x = 10 (Kar’s age)
x + 18 = 10 + 18 = 28 (Yna’s age)

4.) Peter’s age is thrice Amaya’s age. In 5 years, his age will be twice Amaya’s age. How old is Peter?

Let x = Amaya’s age
3x = Peter’s age

In 5 years…
Amaya = x + 5
Peter = 3x + 5

…his age will be twice as Amaya’s age

3x + 5 = 2(x + 5)
3x + 5 = 2x + 10
3x – 2x = 10 – 5
x = 5 (Amaya’s age)
3x = 3(5) = 15 (Peter’s age)

5.) Martin is thrice as old as Kaye. If 7 is subtracted from Martin’s age and 5 is added to Kaye’s age, then the sum of their ages is 34. What are their ages?

Let x = Kaye’s age
3x = Martin’s age

If 7 is subtracted from Martin’s age…
3x – 7

…and 5 is added to kaye’s age…
x + 5

…then the sum of their ages is 34.

(3x – 7) + (x + 5) = 34
4x – 2 = 34
4x = 34 + 2
4x = 36
x = 36/4
x = 9 (Kaye’s age)

3x = 3(9) = 27 (Martin’s age)

Answer: Kaye 9, Martin 27.

6.) James is 9 years older than Kevin. Two years ago, his age was twice that of Kevin’s age. How old is James?

Present ages
Let x = Kevin’s age
x + 9 = James’ age

2 years ago

x – 2 = Kevin’s age
(x + 9) – 2 = x + 7 = James’ age

…his age was twice of Kevin
x + 7 = 2(x – 2)
x + 7 = 2x – 4
x – 2x = -4 – 7
-x = -11
x = 11 (Kevin’s age)
x + 9 = 11 + 9 = 20 (James’ age)

Answer: James is 20 years old.

7.) Mark is twice as old as Lorie. Rey is 6 years younger than Mark. Three years ago, the average of the ages of the three of them is 20. What are their present ages?

Present ages
Let x = Lorie’s age
2x = Mark’s age
2x – 6 = Rey’s age

3 years go
x – 3 = Lorie’s age
2x – 3 = Mark’s age)
(2x – 6) -3 = (2x -9) = Rey’s age

The average of their ages was 20.

(Lorie’s age + Mark’s age + Rey’s age ) / 3 = 20
[(x – 3) + (2x – 3) + (2x – 9)]/3 = 20.

Multiplying both sides by 3,

x – 3 + 2x – 3 + 2x – 9 = 20(3)
5x – 15 = 60
5x = 60 + 15
5x = 75
x = 75/5.

x = 15 (Lorie’s age)
2x = 2(15) = 30 (Mark’s age)
2x – 6 = 2(15) – 6 = 30 – 6 = 24 (Rey’s age)

Answer: Lorie 15, Mark 30, Rey 24.

8.) Sam is thrice as old as Vina. Rio is half as old as Vina. The sum of their ages is 54. What are their ages?

Let x – Vina’s age
3x = Sam’s age
x/2 = Rio’s age

The sum of their ages is 54.
x + 3x + x/2 = 54

Multiply both sides by 2.
2(x + 3x + x/2 = 54)2
2(x) + 2(3x) + 2(x/2) = 2(54)
2x + 6x + x = 108
9x = 108
x = 108/9
x = 12(Vina’s age)

3x = 3(12) = 36 (Sam’s age)
x/2 = 12/2 = 6 (Rio’s age)

Answer: Vina 12, Sam 36, Rio 6.

9.) Four years from now, Tina’s age will be equal to Kris’ present age. Two years from now, Kris will be twice as old as Tina. What are their present ages?

Present Ages

x = Kris’age
x – 4 = Tina’s age

2 years from now

x + 2 = Kris’ age
x – 4 + 2 = x – 2 = Tina’s age

4 years from now
x + 4 = Kris’ age
x = Tina’s age

Two years from now, Kris will be twice as old as Tina.
x + 2 = 2(x – 2)
x + 2 = 2x – 4
x – 2x = -4 – 2
-x = -6
x = 6 (Kris’ present age)
x – 4 = 6 – 2 = 4 (Tina’s age)

Answer: Tina 2, Kris 6.

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