PCSR REVIEW SERIES WEEK 7: Conversion of Decimals, Percent, and Fractions Operations

After learning about solving quations, let’s learn about operations on decimals. Let’s also learn the conversion among decimals, fractions, and percent. Below are the articles and videos about these topics. Exercises and problems will be posted later.

ARTICLES

Operations on Decimals

Conversion

YOUTUBE VIDEOS

Conversion

Enjoy!

Week 6 Review: Answers and Solutions

PCSR WEEK 6 Review: Solving Equations
Practice Exercise: Find the value of x.
1.) x + 5 = 8 => x = 8 – 5 => x = 3
2.) x – 3 = 6 => x = 6 + 3 => x = 9
3.) x + 8 = 0 => x = 0 – 8 => x = -8
4.) 4x = 12 => x = 12/4 => x = 3
5.) x/2 = -6 => x = -6(2) => x = -12

PCSR WEEK 6 Review: Solving Equations. In each equation, find the value of x.
1.) 2x – 1 = 5

2x = 5 + 1
2x = 6
x = 6/2
x = 3

2.) x – 12 = – 2x

x + 2x = 12
3x = 12
x = 12/3
x = 4

3.) x + 6 = 3x – 5

x – 3x = -5 – 6
-2x = -11
x = -11/-2
x = 11/2 or 5 1/2

4.) 5x + 12 = 3x – 6

5x – 3x = -6 – 12
2x = -18
x = -18/2
x = -9

5.) 2(5 – x) = 13

By distributive property, (2)(5) – (2)(x) = 13
10 – 2x = 13
-2x = 13 – 10
-2x = 3
x = 3/(-2)
x = -1 1/2

6.) 3(x + 8) = 15 + 6x

(3)(x) + (3)(8) = 15 + 6x
3x + 24 = 15 + 6x
3x – 6x = 15 – 24
-3x = -9
x = -9/-3
x = 9/3 or 3

7.) -2(3x – 4) = 2(1 – x)

(-2)(3x) – (-2)(4) =(2)(1) -(2)(x)
-6x – (-8) = 2 – 2x
-6x + 8 = 2 – 2x
-6x + 2x = 2 – 8
-4x = – 6
x = -6/(-4)
x = 6/4 or 3/2 or 1 1/2

8.) 4(x + 2) – 5 = x + 6

4(x) + 4(2) – 5 = x + 6
4x + 8 – 5 = x + 6
4x + 3 = x + 6
4x – x = 6 – 3
3x = 3
x = 3/3 or 1

9.) 3x/4 = 18

3x = 18(4)
3x = 72
x = 72/3
x = 24

10.) x/4 + 6 = 16

x/4 = 16 – 6
x/4 = 10
x = 10(4)
x = 40

11.) x/2 – 7 = 5 – 2x

To eliminate the fraction, we multiply both sides of the equation by 2.
2(x/2 – 7) = 2(5 – 2x)
2(x/2) – 2(7) = 2(5) – 2(2x)
x – 14 = 10 – 4x
x + 4x = 10 + 14
5x = 24
x = 24/5 or 4 4/5

12.) (x + 5)/2 = x – 3

To eliminate the fraction, we multiply both sides of the equation by 2.

2[(x + 5)/2] = 2(x – 3)
x + 5 = 2(x) – 2(3)
x + 5 = 2x – 6
x – 2x = -6 – 5
-x = -11
x = -11/-1
x = 11

13.) (2x – 3)/2 = (x + 2)/3

To eliminate the fraction, we multiply both sides of the equation by the LCM of 2 and 3 which is 6.

6[(2x – 3)/2] = 6[(x + 2)/3]
(6/2) (2x – 3) = (6/3) (x + 2)
(3)(2x – 3) = (2)(x + 2)
(3)(2x) – (3)(3) = (2)(x) + (2)(2)
6x – 9 = 2x + 4
6x – 2x = 4 + 9
4x = 13
x = 13/4
x = 3 1/4
14.) 8 – (x + 3)/4 = (x + 8)

4(8) – 4[(x + 3)/4] = 4(x + 8)
(32) – (x + 3) = (4)(x) + (4)(8)
32 – x – 3 = 4x + 32
29 – x = 4x + 32
-x – 4x = 32 – 29
-5x = 3
x = 3/(-5)
x = – 3/5

15.) 3(x -9)/4 = 2(x + 6)/5

[(3)(x) – (3)(9)]/4 = [(2)(x) + (2)(6)]/5
(3x – 27)/4 = (2x + 12)/5

To eliminate the fraction, we multiply both sides of the equation by the LCM of 2 and 3 which is 6.

20 [(3x – 27)/4 = (2x + 12)/5]

(20/4)(3x – 27) = (20/5)(2x + 12)
5(3x – 27) = 4(2x + 12)
(5)(3x) – (5)(27) = (4)(2x) + (4)(12)
15x – 135 = 8x + 48
15x – 8x = 48 + 135
7x = 183
x = 183/7 or 26 1/7

Week 6 Review: Practice Exercises and Problems

PCSR WEEK 6 Review: Solving Equations
Practice Exercise: Find the value of x.
1.) x + 5 = 8
2.) x – 3 = 6
3.) x + 8 = 0
4.) 4x = 12
5.) x/2 = -6

Practice Problems
Find the value of x.

1.) 2x – 1 = 5

2.) x – 12 = – 2x

3.) x + 6 = 3x – 5

4.) 5x + 12 = 3x – 6

5.) 2(5 – x) = 13

6.) 3(x + 8) = 15 + 6x

7.) -2(3x – 4) = 2(1 – x)

8.) 4(x + 2) – 5 = x + 6

9.) 3x/4 = 18

10.) x/4 + 6 = 16

11.) x/2 – 7 = 5 – 2x

12.) (x + 5)/2 = x – 3

13.) (2x – 3)/2 = (x + 2)/3

14.) 8 – (x + 3)/4 = (x + 8)

15.) 3(x -9)/4 = 2(x + 6)/5

PCSR REVIEW SERIES WEEK 6: Solving Equations

Solving equations is the most important part of problem solving and algebra in general. In solving word problems, you will have to set up equations and solve for unknowns. Be sure to master this concept.

ARTICLES

VIDEOS (Taglish)

More videos

In the next post, we are going to answer some problems and exercises.

Week 5 Review Answers and Solutions

This is the full solutions for the problems and exercises about operations on integers, order of operations, and PEMDAS rules.

Practice Exercises 1

a.) 12 + (-4) = 8
b.) (-9) + 3 = – 6
c.) (-7) + (- 5) = -12
d.) 8 + 3 + (-11) = (8+3) + (-11) = (11) + (-11) = 0
e.) 6 + (-10) + (-2) = 6 + (-10 + -2) = 6 + (-12) = -6

Practice Exercises 2

a.) 3 – 5 = (3) + (-5) = -2
b.) -9 – 4 = (-9) + (-4) = -13
c.) (-7) – (- 8) = (-7) + (8) = 1
d.) – 2 – 6 = (-2) + (-6) = -8
e.) 1 – (-10) = (1) + (10) = 11

Practice Exercises 3

a.) 4 × (- 5) = -20
b.) (-2) × (- 4) = 8
c.) 6 × (- 3) = -18
d.) 8 × 2 × (-1) = -16
e.) (-3) × (2) × (-7) = 42

Practice Exercises 4
a.)-20 ÷ 4 = -5
b.) 18 ÷ (- 6) = -3
c.) (-16) ÷ (- 2) = 8
d.) 0 ÷ 8 = 0
e.) 9 ÷ 3 = 3

1.) 2 + 3 × 5

2 + 3 × 5
= 2 + 15
= 17

2.) (2 + 3) × 5

(2 + 3) × 5 = (5) × 5
= 25

3.) 3 × (-3) + 4 × (-2)

= 3× (-3) + 4× (-2)
= (-9) + (-8)
= -17

4.) 3(5^2 – 8)

3(5 × 5 – 8)
= 3(25 – 8)
= 3(17)
= 51

5.)  2(5 – 8)^2
= 2(-3)^2
= 2(-3 × -3)
= 2(9)
= 18

6.) 16 + (-4) + 12 + (-8 x 3)

= 16 + (-4) + 12 + (-24)
= (16 + 12) + (-4 + -24)
= (28) + (-28)
= 0

7.) (3^2 + 2^2)^2 = ?

= (3 × 3 +2× 2)^2
= (9 + 4)^2
= (13)^2
= 13 × 13
= 169

8.) 6 + 3 × 2 – 5 = ?

= 6 + (3 × 2) – 5
= 6 + 6 – 5
= 12 – 5
= 7

9.) 8 – 12(3 – 4) + (-5 × 2)

= 8 – 12(3 – 4) + (-5 × 2)
= 8 – 12(-1) + (-10)
= 8 – (-12) + (-10)
= 20 + (-10)
= 10

10.) 7 + 3 × (-5) – 9 / 3 = ?

= 7 + 3 × (-5) – 9/3
= 7 + (-15) – 3
= 7 + (-18)
= -11

Week 5 Review: Practice Exercises and Problems

In the previous post, we learned about operations on integers, order of operations, and PEMDAS rules. Below are the exercises and problems about these topics.

PCSR WEEK 5 Review: Operations on Integers

Practice Exercises 1

a.) 12 + (-4)
b.) (-9) + 3
c.) (-7) + (- 5)
d.) 8 + 3 + (-11)
e.) 6 + (-10) + (-2)

Practice Exercises 2

a.) 3 – 5
b.) -9 – 4
c.) (-7) – (- 8)
d.) – 2 – 6
e.) 1 – (-10)

Practice Exercises 3

a.) 4 × (- 5)
b.) (-2) × (- 4)
c.) (6) × (- 3)
d.) 8 × 2 × (-1)
e.) (-3) × (2) × (-7)

Practice Exercises 4
a.)-20 ÷ 4
b.) 18 ÷ (- 6)
c.) (-16) ÷ (- 2)
d.) 0 ÷ 8
e.) 9 ÷ 3

Practice Problems

1.) 2 + 3 × 5
2.) (2 + 3) × 5
3.) 3 x (-3) + 4 ×(-2)
4.) 3(5^2 - 8)
5.) 2(5 - 8)^2
6.) 16 + (-4) + 12 + (-8 × 3)
7.) (3^2 + 2^2)^2
8.) 6 + 3 × 2 – 5
9.) 8 – 12(3 – 4) + (-5 × 2)
10.) 7 + 3 × (-5) – 9 / 3

PCSR REVIEW SERIES WEEK 5: Operations on Integers and PEMDAS

Below are the articles and videos that you should read and watch about operations on integers, order of operations, and PEMDAS. Exercises and problems will be posted soon.

PART 1: OPERATIONS ON INTEGERS

Articles

  1. How to Add Positive and Negative Integers
  2. How to Subtract Positive and Negative Integers
  3. How to Multiply Positive and Negative Integers
  4. How to Divide Positive and Negative Integers

Videos

  1. How to Add Integers Using the Number Line Part 1
  2. How to Add Integers Using the Number Line Part 2
  3. Addition of Integers Using Chips
  4. How to Subtract Integers Using the Number Line (Part 1)
  5. How to Subtract Integers Using the Number Line (Part 2)
  6. Subtraction of Integers Worked Examples Part 1
  7. Subtraction of Integers Worked Examples Part 2
  8. Subtraction of Integers Using Chips
  9. Addition and Subtraction of Integers Worked Examples
  10. Introduction to Multiplication of Integers
  11. Multiplication of Integers Worked Examples
  12. Division of Integers
  13. Division of Integers Worked Examples

PART 2: PEMDAS

Articles

Videos

  1. PEMDAS Order of Operations Part 1
  2. PEMDAS Order of Operations Part 2
  3. PEMDAS Order of Operations Part 3

More video

Introduction to PEMDAS

Week 4 Review Answers and Solutions

Below are the solutions and answers to the Week 4 Practice Problems and Solutions.

Practice Exercises 1

Note: In multiplying fractions, we multiply the numerator by the numerator of the other fraction, and then multiply the denominator by the denominator of the other fraction. For whole numbers, we can put 1 as the denominator. All fractions must be in lowest terms.

A. 1/2 × 1/3 = 1/6
B. 2/3 × 4/5 = 8/15
C. 8/1 × 5/6 = 40/6 = 6 4/6 or 6 2/3
D. 2 5/8 × 3 = 21/8 × 3/1 = 63/8 = 7 7/8
E. 3 1/8 × 4/5 = 25/8 × 4/5 = 100/40 = 2 20/40 or 2 1/2
F. 1 2/3 × 2 3/4 = 5/3 × 11/4 = 55/12 = 4 7/12

Practice Exercises 2

A. When dividing fractions you get the reciprocal of the divisor, and then multiply. In 1/5÷ 3/10, the divisor 3/10 and the reciprocal of 3/10 is 10/3. So, 1/5 × 10/3 = 10/15 or 2/3

B. 1/2 ÷ 3/8 = 1/2 × 8/3 = 8/6 = 1 2/6 or 1 1/3

C. 9 ÷ 3/7 = 9 × 7/3 = 63/3 or 21

D. 2 5/8 ÷ 2

First, we convert 2 5/8 to improper fraction as follows. That is 2\frac{5}{8} = \frac{8 \times 2 + 5}{8} = \frac{21}{8}. Don’t forget that the denominator of the mixed fraction is the same as the denominator of the improper fractions.

Here, the reciprocal of 2 is 1/2. So, 21/8 × 1/2 = 21/16 = 1 5/16

E. 3 1/8 ÷ 3/5 = 25/8 × 5/3 = 125/24 = 5 5/24

F. 2 3/4 ÷ 1 1/8 = 11/4 ÷ 9/8 = 11/4 × 8/9 = 88/36 = 2 16/36 or 2 4/9

Practice Problems

Note: In multiplication and division of fractions, all mixed fractions must be converted to improper fractions (see Practice Problem 2D above).

1.) 2/3 × 1/4 = 2/12 or 1/6

2.) 3/5 × 35/1=105/5 = 21 (women)
2/5 × 35/1 = 70/5 = 14 (men)

3.) 2 3/4 × 7 = (11/4) ×  7 = 77/4 = 19 1/4

4.) A = L × W
A = (35 1/4) × (20 1/2) = (141/4) × (41/2)
= 5781/8= 722 5/8

5.) 2 4/5 × 5/1= 14/5 × 5/1 = 14

6.) 1 1/2 L juice is to be shared equally by 6 friends, so 1 1/2 ÷ 6.

The mixed fraction  1 1/2 is 3/2 in improper form.

Dividing by 6 is the same as multiplying by 1/6, so 3/2 × 1/6 = 3/12

3/12 is not yet in its lowest term. So get its lowest term, divide the numerator and denominator by their GCF which is 3.  So, 3/12 will become 1/4.

The final answer is 1/4 L.

7.) 8 ÷ 2/5 = 8 × 5/2 = 8/1 × 5/2 = 40/2 = 20.

Answer: 20 cakes

8.) Bookshelf length divided by book’s thickness = number of books that will fit in the bookshelf

5 1/4 feet ÷ 1 1/2 inches = ?

Notice that the units are in feet and inches. We cannot proceed until the units are the same, so we need  to convert feet into inches. (1 ft = 12 in) So 5 ft × 12 in per ft = 60 in. We still have 1/4 ft, so 1/4 of 12 which is 3 in. All in all, we have 63 in. Now our equation is

63  ÷ 3/2 = 63/1 × 2/3

= 126/3 = 42.

Answer: 42 books

9.) 5 pumpkin pies are to be shared equally among 12 persons, equals 5/12.

10.) We have 8 3/4 hectares ÷ 4 children

35/4 ÷ 4 =35/4 × 1/4 = 35/16

Converting 35/16 to mixed fraction, we have 2 3/16.

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