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## Solving Fractions

In this lesson, we will learn how to solve fractions. Just like any other number, fractions can be added, multiplied, divided and subtracted.

Let us first define what a fraction is. Fraction is a number having a numerator and denominator. Take the example 4/5. The number 4 is the numerator (the number above the bar) and 5 which is the denominator (the number below the bar).

Now that we know what a fraction is, let us learn its different types:

1. ### Proper fraction

This fraction has a smaller numerator number compared to the denominator.

Examples: 1/2 , 2/3 , 4/5 ,3/4 , 9/10

2. ### Improper fraction

This fraction has a bigger denominator than the numerator.

Examples: 5/4 , 7/4 , 34/6, 3/2 , 7/3

3. ### Mixed fraction

This fraction consists of a whole number and a fraction. The fraction can either be proper or improper.

Examples:1  2/3 , 2  4/5 , 1  9/8 , 6  4/3 , 3  9/4

### How to solve fractions:

For proper fractions, just add the numerator and copy the denominator.
If possible, simplify or reduce to its lowest terms.

Example:

1. 2/7 + 1/7 = 3/7

2. 4/5 + 1/5
= 5/5
= 1

3. 2/9 + 8/9
= 10/9
= 1  1/9

4. 5/12 + 5/12
= 10/12
= 5/6

5. 3/8 + 1/8
= 4/8
= 1/2

For fractions with different denominators you have to:

1. Find the LCD (Least Common Denominator).

2. Change the fractions to have the same LCD.

4. Reduce to its lowest term.

Example:

1. 1/6 + 1/12
= 2 + 1
12
= 3/12
= 1/4

2. 1/3 + 1/6
= 2 + 1
6
= 3/6
= 1/2

3. 3/10 + 4/15
= 9 + 8
30
= 17/30

4. 2/9 + 1/18
= 4 + 1
18
= 5/18

5. 4/7 + 3/5
= 20 + 21
35
= 41/35
= 1  6/35

For mixed fractions with the same denominator you have to:

2. If the resulting fraction is improper (the numerator is equal to or bigger than the denominator), convert the fraction into a mixed fraction.

Example:

1. 1  2/5 + 2  3/5
= 3  5/5
= 3 + 1
= 4

2. 2  3/7 + 3  3/7
= 5  6/7

For mixed fractions with different denominators, you have to:

1. Change the mixed fraction into an improper fraction.

2. Find the LCD (Least Common Denominator).

3. Change the fractions to get the same LCD.

5. Reduce to its lowest term.

Example:

1. 2  5/6 + 3  5/18
= 17/6 + 59/18
= 51 + 59
18
= 110/18
= 6  2/18
= 6  1/9

2. 2  8/9 + 1  2/3
= 26/9 + 5/3
= 26 + 15
9
= 41/9
= 4  5/9

### Subtraction

In subtracting fractions, again one must consider the type of fraction you are about to subtract.

For fractions with the same denominator, just subtract the numerator and copy the denominator.

Example:

1. 2/5 – 1/5
= 1/5

2. 3/4 - 1/4
= 2/4
= 1/2

For fractions with different denominators you have to:

1. Find the LCD (Least Common Denominator).

2. Change the fractions to have the same LCD.

3. Subtract the numerators.

4. Reduce to its lowest term.

### Example:

1. 3/4 - 1/8
= 6 – 1
8
= 5/8

For mixed fractions with the same denominator you have to:

1. Look at the fractions that you are about to subtract. If the first numerator is smaller than the second, make the first numerator bigger than the second.

2. Then, find the difference between the numerators and place it over the common denominator.

3. You can now find the difference between the integers.

4. Simplify the resulting fraction by reducing it to its lowest term.

Example:

1. 2  6/7 – 1  1/7

= 1  5/7

### Multiplication

The process of multiplication is just like doing addition over and over again, repeatedly.

Multiplication involves 3 steps:

1. Multiplying the numerator.

2. Multiplying the denominator.

3. Reducing to lowest terms.

### Examples:

1. 1/2 x 2/4 = 1 x 2
2 x 4
= 2/8
=1/4

2. 3/5 x 2/8 = 3 x 2
5 x 8
= 6/40
= 3/20

3. 2/5 x 5/9 = 2 x 5
5 x 9
= 10/45
= 2/9

### Division

The division of fraction requires only 4 basic steps:

1. Look at the fraction you are about to solve

2. Change the second fraction into its reciprocal. The reciprocal of a fraction in the fraction inverted. For example, reciprocal of ½ is 2/1.

3. Multiply the fractions.

4. If possible, simplify the resulting fraction or reduce to its lowest term.

1. 2/3 ÷ 6/7
= 2/3 x 7/6
= 2 x 7
3 x 6
= 14/18
= 7/9

2. 3/8 ÷ 4/5
= 3/8 x 5/4
= 3 x 5
8 x 4
= 15/40
= 3/8

3. 8/9 ÷ 5/6
= 8/9 x 6/5
= 8 x 6
9 x 5
= 48/45
= 16/15
= 1  1/15