FractionsObjectives pursued consists of having participants to:

Section 1

• Distinguish between various types of fractions

• Converting improper fractions to whole or mixed

numbers

• Converting mixed numbers to improper fractions

• Reducing fractions to lowest terms

• Raising fractions to higher terms

1

Fractions(contd)

Section 2

• Adding fractions and mixed numbers

• Subtracting fractions and mixed numbers

• Multiplying fractions and mixed numbers

• Dividing fractions and mixed numbers

2

Fraction (contd)

A mathematical way of expressing a part of a whole thing. ¼ is a

fraction expressing one part out of a total of four parts.

numerator

The number on top of the division line of a fraction. It

represents the dividend in the division. In the fraction ¼,

1 is the numerator.

denominator

• The number on the bottom of the division line of a

fraction. It represents the divisor in the division. In the

fraction ¼, 4 is the denominator

3

Fraction (contd)

Various types of fractions

common or proper fraction

• A fraction in which the numerator is less than the

denominator. It represents less than a whole unit. The fraction

¼ is a common or proper fraction.

improper fraction

• A fraction in which the denominator is equal to or less than the

numerator. It represents one whole unit or more. The fraction 4/1

is an improper fraction.

mixed number

• A number that combines a whole number with a proper

fraction. The fraction 10¼ is a mixed number.

4

Fraction (contd)

• To convert improper fractions to whole or mixed number

• STEP 1 Divide the numerator of the improper fraction by

the denominator.

• STEP 2a If there is no remainder, the improper fraction

becomes a whole number.

• STEP 2b If there is a remainder, write the whole number

and then write the fraction as

𝑤ℎ𝑜𝑒 𝑛𝑢𝑚𝑏𝑒𝑟𝑟𝑒𝑚𝑖𝑛𝑎𝑑𝑛𝑒𝑟

𝑑𝑖𝑣𝑖𝑠𝑜𝑟

5

Fraction (contd)

Converting improper fraction to whole and mixed numbers examples25

5=5

4

3=1

1

3

66

10=6

6

10

6

Fraction (contd)

• Converting mixed numbers to improper fractions

STEP 1 Multiply the denominator by the whole number.

STEP 2 Add the numerator to the product from Step 1.

STEP 3 Place the total from Step 2 as the “new” numerator.

STEP 4 Place the original denominator as the “new”

Denominator

Examples:

3 2

5= 5∗3+2

5=

17

5

11

2=2∗1+1

2=

3

2

226

7=22∗7+6

7=

160

7

7

Fraction (contd)

Reducing Fractions to Lowest Terms

reduce to lowest terms

The process of dividing whole numbers, known as common

divisors or common factors, into both the numerator and

the denominator of a fraction.

• Raising Fractions to Higher Terms

raise to higher terms

The process of multiplying the numerator and denominator

of a fraction by a common multiple.

Needed to have fractions with differing denominators to

have the same denominator

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Fraction (contd)

Examples

2

5=

?

10one does as follows

10÷5 =2, then 2*2 =4 so ?=4

The answer becomes:

2

5=4

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Adding up fractions:

Like fractions: these are fractions with the same

denominator but different numerators

Example: 2

7+5

7+

3

7+

1

7=11

7= 1

4

7

. 9

Fraction (contd)

When adding like fractions you add up the numerators and

the denominator remain the same as for individual fractions

The same rule applies when subtracting fractions

unlike fractions: these are fractions with the different

denominators

Example

2

5+3

6+

1

4=

48+60+30

120=138

120=1

18

120=1

9

60

Adding unlike fractions requires fining a common

denominator. The easy way to get it is multiply all the

denominators. The numerator is found by dividing the

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Fraction (contd)

common denominator by each individual denominators and

multiplying the result with the numerator of individual

fractions. Adding mixed numbers

STEP 1 Add the fractional parts. If the sum is an improper

fraction, convert it to a mixed number.

STEP 2 Add the whole numbers.

STEP 3 Add the fraction from Step 1 to the whole number

from Step 2.

STEP 4 Reduce the answer to lowest terms if necessary.

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Fraction (contd)

1

2*3

4=3

8

To multiply mixed number, convert each mixed number into

an improper fraction before multiplying

Example:

21

2*3

3

4=

5

2*15

4

Dividing fractions: multiply the first fraction with the inverse

of the second fractions

Example: 3

7÷*

4

9=

3

7*9

4

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Fraction (contd)

Fractions: applications

A certain ward received money from the government to be

spent on activities aimed at uplifting the community . The

municipality decided to spend 1/3 of the money on

assisting the disabled. It then spent 1/9 of the money on

the youth empowerment by informing them on how to get

jobs

a) What fraction of the money received did the municipality

use

b) If the municipality received R360,000 what amount was

spent on community uplifiment

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Fraction (contd)

c) How much was spend on the disabled

d) How much was spend on the youth empowerment

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Fraction (contd)

solutions

a)1

3+1

9=3

9+1

9=

4

9the fraction of the money is 4/9

b) The money spent on community upliftment was

R360,000* 4/9=160,000

c) The money spent on the disabled is

1/3*R360000=120,000

d) Money spend on the youth empowerment was R

1/9*360000= R40,000

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• EquationsThe objectives of this topic is to have participants:

• Be aware of the concept, terminology, and rules of

equations

• Solving equations for the unknown and providing the

solution

• Writing mathematical expressions and equations from

written statements

• Setting up and solving business-related word problems

by using equations

• Understanding and solving ratio and proportion problems

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Basic equation concepts

Formula: A mathematical representation of a fact, rule, principle, or other logical relation in which letters represent number quantities.

Equation: A mathematical statement expressing a relationship of equality; usually written as a series of symbols that are separated into left and right sides and joined by an equal sign. X+ 7 = 10 is an equation.

Expression: A mathematical operation or a quantity stated in symbolic form, not containing an equal sign. X + 7 is an expression.

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Variables(Unknowns):The part of an equation that is not given. In equations, the unknowns are variables (letters of the alphabet), which are quantities having no fixed value. In the equation X+ 7 = 10, X is the unknown or variable.

Constants (Knowns):The parts of an equation that are given. In equations, the knowns are constants (numbers), which are quantities having a fixed value. In the equation X + 7 = 10, 7 and 10 are the knowns or constants.

Terms: The knowns (constants) and unknowns (variables) of an equation. In the equation X + 7 = 10, the terms are X, 7, and 1

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Solve an Equation: The process of finding the numerical

value of the unknown in an equation.

Solution(or Root):The numerical value of the unknown that

makes the equation true. Example: X + 7 = 10, 3 is the

solution, because 3 + 7 = 10.

Coefficient: A number or quantity placed before another

quantity, indicating multiplication. For example, 4 is the

coefficient in the expression 4C. This indicates 4 multiplied

by C.

Transpose:To bring a term from one side of an equation to

the other, with a corresponding change of sign.

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20

steps for solving equations and providing the solution

step 1

Transpose all the unknowns to the left side of the equation and all the knowns to the right side of the equation by using the following “order of operations” for solving equations.

• Parentheses, if any, must be cleared before any other operations are performed. To clear parentheses, multiply the coefficient by each term inside the parentheses. 3(5C+ 4) = 2 3(5C) + 3(4) = 2 15C+ 12 = 2

• To solve equations with more than one operation:

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■First, perform the additions and subtractions.

■Then perform the multiplications and divisions.

step 2

Prove the solution by substituting your answer for the letter or letters in the original equation. If the left and right sides are equal, the equation is true and your answer is correct

Tips for solving equations

•Whatever you do to one side of the equation, do to the other.

. Use the opposite operation to get rid of (i.e., move) a term from one side of the equation to the other.

•

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• Isolate the unknown (variable).

•Combine like terms

23

24

25

26

• STEP 1

To combine unknowns, they must be on the same side of

the equation.

If they are not, move them to the same side.5X= 12 + 2X

5X–2X= 12

STEP 2

Once the unknowns are on the same side of the equation,

add or subtract their coefficients as indicated.

5X–2X=12; 3X= 12

27

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Transforming a word problem into an equation

STEP 1 Read the written statement carefully.

STEP 2 Using the list on the following slide, identify and

underline the key words and phrases.

STEP 3 Convert the words to numbers and mathematical

symbols.

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30

Writing Expressions Example

A number increased by 18

Y + 18

13 less than P

P – 13

The difference of R and 25

R - 25

6 more than 4 times T

4T + 6

31

Writing Equations

A number increased by 33 is 67

• X + 33 = 67

A number totals 5times B and C

• X = 5B + C

12 less than 4G leaves 36

• 4G - 12 = 36

The cost of R at R5.75 each is R28.75

• 5.75R = 28.75

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Steps for solving word problem using equations

STEP 1 Understand the situation. If the problem is written, read it

carefully, perhaps a few times. If the problem is verbal, write down

the facts of the situation.

STEP 2 Take inventory. Identify all the parts of the situation. These

parts can be any variables, such as dollars, people, boxes, tons,

trucks, anything! Separate them into knowns and unknowns.

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STEP 3 Make a plan—create an equation. The object is to

solve for the unknown. Ask yourself what math relationship

exists between the knowns and the unknowns. Use the

chart of key words and phrases on page 132 to help you

write the

equation.

STEP 4 Work out the plan—solve the equation. To solve an

equation, you must movethe unknowns to one side of the

equal sign and the knowns to the other.

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STEP 5 Check your solution. Does your answer make

sense? Is it exactly correct? It is a good idea to estimate an

approximate answer by using rounded numbers.

This will let you know if your answer is in the correct range.

If it is not, either the equation is set up incorrectly or the

solution is wrong. If this occurs, you must go back and start

again.

35

Gene and Fatima work in an electronics store. During a

sale, Gene sold 8 less items than Fatima. If they sold a

total of 86 items, how many did each sell?

Fatima = X

Gene = X – 8

X + X – 8 = 86

2X = 86+8

2x= 94

X=47

X -8= 47 – 8 = 39

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One-third of the products made by Vega Inc. are produced

by the soap division. If the soap division makes 23

products, how many total products are made by Vega Inc.?

Total employs =x1

3X = 23

X=23*3=69

37

Solving ratios and proportions problems

ratio

• A fraction that describes a comparison of two

numbers or quantities. For example, five cats for every

three dogs would be a ratio of 5 to 3, written as 5:3.

proportion

• A mathematical statement showing that two ratios

are equal.

• For example, 9 is to 3 as 3 is to 1, written 9:3 =

3:1.

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steps for solving proportion problems

using cross-multiplicationSTEP 1 Assign a letter to represent the unknown quantity.

STEP 2 Set up the proportion with one ratio (expressed as

fraction) on each side of the equal sign.

STEP 3 Multiply the numerator of the first ratio by the

denominator of the second and place the product to

the left of the equal sign.

.

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STEP 4 Multiply the denominator of the first ratio by the

numerator of the second and place the product to the right

of the equal sign.

STEP 5 Solve for the unknown

If the interest on a R4,600 loan is R370, what would the

interest on a loan of R9,660 be?

4600

370=9660

𝑥

4,600X = 370(9,660)

4,600X = 3,574,000

X = 777

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• Applications to municipalities Application 3

A municipality is planning to purchase 2 tables and 3

chairs. The procurement officer at the municipality call the

supplier to enquire the price of each item. The supplier is

reluctant to disclose the price of each but states that the

total costs would be R7050. When the procurement officer

insists on the price, the supplier says that the price of a

table is only R40 more than the cost of the chair. How

could the procurement officer know the cost of one chair

and one table (see solutions at the end of the notes after

trying it )

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