# Fractions

## Audio-visual

# Numeracy Basics

## Fractions

## Get Started

This topic will cover how to:

- identify and distinguish between proper fractions, improper fractions and mixed numbers;
- simplify fractions;
- convert between improper fractions and mixed numbers;
- multiply and divide with fractions;
- determine equivalent fractions;
- add and subtract fractions with like and unlike denominators; and
- convert between fractions and decimal numbers.

## What is a Fraction?

A fraction is just a part of a whole. The bottom number of the fraction (which we call the **denominator**) tells you how many parts the whole is divided into, while the top number of the fraction (which we call the **numerator**) tells you how many of those parts you have.

If the numerator of our fraction is smaller than the denominator the fraction is less than one, and we call it a **proper fraction**. For example, ¼ is a proper fraction because our numerator (1) is smaller than our denominator (4).

*To illustrate this fraction note that the circle below has been divided into four parts, one of which has been shaded.*

**So the shaded area is 1/4 of our whole:**

*To illustrate this fraction note that the circle below has been divided into four parts, one of which has been shaded.*

**So the shaded area is 1/4 of our whole:**

## Other Types of Fractions

If the numerator of our fraction is bigger than the denominator the fraction is greater than one, and we call this an improper fraction. Improper fractions can also be expressed either as mixed numbers (i.e. a whole number and a proper fraction together), or just as whole numbers.

For example, the improper fraction 5/4 can also be written as the mixed number 1 1/4, as these represent the same amount.

On the other hand, the improper fraction 8/4 can be written as the whole number 2.

Finally, if the numerator and denominator of our fraction are the same then it is equal to 1, and we generally write it as such (for example 4/4 = 1). Furthermore, if the denominator of our fraction is 1 then the fraction is in fact equal to a whole number (for example 5/1 = 5).

## Simplifying Fractions

One of the things you may need to do with fractions is to simplify them. As a basic example of simplifying a fraction, consider this pizza:

You could say that there is 4/8 of a pizza left, but would you? Most likely you would say that there is 1/2 of a pizza left, and in doing so you would have simplified the fraction.

It is best to always write fractions in their simplest form, which you can do by following these steps:

*1. Determine the highest common factor of the numerator and denominator (i.e. the largest number that divides into both a whole number of times). If you can't find the highest common factor then at least determine a factor (i.e. a number that divides into both a whole number of times); it just means you will need to repeat these steps.*

*For example, when simplifying 8/12 consider that the factors of 8 are 1, 2, 4 and 8 and the factors of 12 are 1, 2, 3, 4, 6 and 12. Hence the highest common factor is 4.*

*2. Divide both the numerator and the denominator by this highest common factor or factor to obtain a new numerator and denominator. If the latter, repeat as necessary until there are no more factors.*

*For example, when simplifying 8/12 consider that 8 ÷ 4 = 2 and 12 ÷ 4 = 3, so 8/12 simplifies to 2/3.*

*1. Determine the highest common factor of the numerator and denominator (i.e. the largest number that divides into both a whole number of times). If you can't find the highest common factor then at least determine a factor (i.e. a number that divides into both a whole number of times); it just means you will need to repeat these steps.*

*For example, when simplifying 8/12 consider that the factors of 8 are 1, 2, 4 and 8 and the factors of 12 are 1, 2, 3, 4, 6 and 12. Hence the highest common factor is 4.*

*2. Divide both the numerator and the denominator by this highest common factor or factor to obtain a new numerator and denominator. If the latter, repeat as necessary until there are no more factors.*

*For example, when simplifying 8/12 consider that 8 ÷ 4 = 2 and 12 ÷ 4 = 3, so 8/12 simplifies to 2/3.*

## Examples: Simplifying Fractions

1. What is 75/100 in simplest form?

Factors of 75 are 1, 3, 5, 15, 25 and 75, and factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50 and 100

So highest common factor is 25

Dividing numerator and denominator by 25 gives 3/4

2. What is 9/12 in simplest form?

Factors of 9 are 1, 3, and 9, and factors of 12 are 1, 2, 3, 4, 6 and 12

So highest common factor is 3

Dividing numerator and denominator by 3 gives 3/4

3. A survey is conducted with 12 people, 10 of whom are female. What fraction of participants are female (simplify)?

Factors of 10 are 1, 2, 5 and 10, and factors of 12 are 1, 2, 3, 4, 6 and 12

So highest common factor is 2

Dividing numerator and denominator by 2 gives 5/6

Factors of 75 are 1, 3, 5, 15, 25 and 75, and factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50 and 100

So highest common factor is 25

Dividing numerator and denominator by 25 gives 3/4

Factors of 9 are 1, 3, and 9, and factors of 12 are 1, 2, 3, 4, 6 and 12

So highest common factor is 3

Dividing numerator and denominator by 3 gives 3/4

Factors of 10 are 1, 2, 5 and 10, and factors of 12 are 1, 2, 3, 4, 6 and 12

So highest common factor is 2

Dividing numerator and denominator by 2 gives 5/6

## Activity 1: Practice Questions

## Converting Mixed Numbers to Improper Fractions

When performing calculations involving fractions you may be required to convert from mixed numbers to improper fractions. To do this follow these steps:

*1. Multiply the whole number part of the mixed number by the denominator of the fractional part *

*For example, when converting 7 2/3 to an improper fraction multiply 7 by 3 to give 21*

*2. Add this answer to the numerator. This gives the numerator of the improper fraction, and the denominator is just the denominator of the fractional part of the mixed number *

*For example, when converting 7 2/3 to an improper fraction add 21 to 2 to give a numerator of 23, and hence an improper fraction of 23/3*

*1. Multiply the whole number part of the mixed number by the denominator of the fractional part *

*For example, when converting 7 2/3 to an improper fraction multiply 7 by 3 to give 21*

*2. Add this answer to the numerator. This gives the numerator of the improper fraction, and the denominator is just the denominator of the fractional part of the mixed number *

*For example, when converting 7 2/3 to an improper fraction add 21 to 2 to give a numerator of 23, and hence an improper fraction of 23/3*

## Converting Improper Fractions to Mixed Numbers

When a solution to a problem turns out to be an improper fraction, typically it should be converted to a mixed number. This can be done by following these steps:

*1. Divide the numerator of the improper fraction by the denominator*

*For example, when converting 14/4 to a mixed number or whole number divide 14 by 4 to give 3.5*

*2. If this result is a whole number then your improper fraction is equivalent to a whole number. If this result is a decimal number then your improper fraction is equivalent to a mixed number, and the whole number part of this mixed number is just the whole number part of the decimal result. Furthermore, the denominator of the fractional part of the mixed number is just the denominator of the original improper fraction. To determine the numerator, just multiply the whole number part of the decimal result by the denominator of the original improper fraction, and subtract this result from the numerator.*

*For example, when converting 14/4 to a mixed number division gives the decimal number of 3.5. Hence 3 is the whole number part of our mixed number, the denominator of the fractional part is 4 and the *

* For example, when converting 14/4 to a mixed number performing steps 1 and 2 above gives 3 2/4, which can then be simplified to 3 1/2*

*1. Divide the numerator of the improper fraction by the denominator*

*For example, when converting 14/4 to a mixed number or whole number divide 14 by 4 to give 3.5*

*2. If this result is a whole number then your improper fraction is equivalent to a whole number. If this result is a decimal number then your improper fraction is equivalent to a mixed number, and the whole number part of this mixed number is just the whole number part of the decimal result. Furthermore, the denominator of the fractional part of the mixed number is just the denominator of the original improper fraction. To determine the numerator, just multiply the whole number part of the decimal result by the denominator of the original improper fraction, and subtract this result from the numerator.*

*For example, when converting 14/4 to a mixed number division gives the decimal number of 3.5. Hence 3 is the whole number part of our mixed number, the denominator of the fractional part is 4 and the numerator is 14 – 3 x 4 = 2, giving a mixed number of 3 2/4*

*3. Simplify the fractional part of the mixed number if required*

* For example, when converting 14/4 to a mixed number performing steps 1 and 2 above gives 3 2/4, which can then be simplified to 3 1/2*

## Examples: Converting between Improper Fractions

and Mixed Numbers

1. Convert 4 1/2 to an improper fraction

4 x 2 = 8

8 + 1 = 9

Therefore improper fraction equivalent is 9/2

2. Convert 16/6 to a mixed number

16 ÷ 6 = 2.67, so whole number part of mixed number is 2

Denominator of fractional part of mixed number 16 - 2 x 6 = 4

Numerator of fractional part of mixed number is 16 - 2 x 6 = 4

So mixed number equivalent is 2 4/6

Which simplifies to 2 2/3

4 x 2 = 8

8 + 1 = 9

Therefore improper fraction equivalent is 9/2

16 ÷ 6 = 2.67, so whole number part of mixed number is 2

Denominator of fractional part of mixed number is 6

Numerator of fractional part of mixed number is 16 - 2 x 6 = 4

So mixed number equivalent is 2 4/6

Which simplifies to 2 2/3

## Activity 2: Practice Questions

Fractions and Mixed Numbers

## Multiplying Fractions

Another thing you may need to do with fractions is multiply them together. To do this simply multiply the two numerators together to make the new numerator, and the two denominators together to make the new denominator, then simplify if necessary.

**For example: 1/3 x 2/7 **= (1 x 2)/(3 x 7)

= **2/21**

To multiply a fraction by a whole number, you simply multiply the numerator by the whole number and leave the denominator as is (as the whole number is equivalent to a fraction with a denominator of 1).

**For example: 2/7 x 3 = (2 x 3)/7**

= **6/7**

Note that multiplying by a fraction is the same as finding that fraction 'of' the number- i.e. we can say that 2/7 of 3 is 6/7, based on our multiplication above.

Also, note that when multiplying with mixed numbers you should convert to improper fractions first before proceeding as above.

## Examples: Simplifying Fractions

1. What is 3/7 x 1/4?

3/7 x 1/4 = (3 x 1)/(7 x 4)

= 3/28

2. What is 3/4 x 2/5?

3/4 x 2/5 = (3 x 2)/(4 x 5)

= 6/20

= 3/10

3. A survey is conducted regarding whether people have pets, and if so what kind they have. If 2/3 of the people surveyed have a pet, and 4/7 of those who have a pet have a dog, then what fraction of all the survey participants have a dog?

2/3 x 4/7 = (2 x 4)/(3 x 7)

= 8/21

3/7 x 1/4 = (3 x 1)/(7 x 4)

= 3/28

3/4 x 2/5 = (3 x 2)/(4 x 5)

= 6/20

= 3/10

2/3 x 4/7 = (2 x 4)/(3 x 7)

= 8/21

## Activity 3: Practice Questions

## Dividing Fractions

You may also need to perform division with fractions. To understand how to do this, recall that dividing by a whole number is the same as multiplying by a fraction with a numerator equal to 1 and denominator equal to the whole number (i.e. the reciprocal of the number).

**For example: 3/5 ÷ 6 **= 3/5 x 1/6

= **(3 x 1)/(5 x 6)**

= **3/30**

= **1/10**

Similarly, dividing by a fraction with a certain numerator and denominator is the same as multiplying by a fraction with the numerator and denominator switched around (i.e. the reciprocal of the fraction).

**For example: 3/5 ÷ 2/3** = 3/5 x 3/2

= (3 x 3)/(5 x 2)

= 9/10

Note that when dividing with mixed numbers, you should convert to improper fractions first before proceeding as above.

**3/5 x 1/6**

= **(3 x 1)/(5 x 6)**

= **3/30**

= **1/10**

**3/5 x 3/2**

= **(3 x 3)/(5 x 2)**

= **9/10**

## Examples: Dividing Fractions

1. What is 6 ÷ 2/3?

6 ÷ 2/3 = 6 x 3/2

= 18/2

= 9

2. What is 5 ÷ 7/12?

5 ÷ 7/12 = 5 x 12/7

= 60/7

= 8 4/7

3. What is 3/5 ÷ 4/7?

3/5 ÷ 4/7 = 3/5 x 7/4

= 21/20

= 1 1/20

6 ÷ 2/3 = 6 x 3/2

= 18/2

= 9

5 ÷ 7/12 = 5 x 12/7

= 60/7

= 8 4/7

3/5 ÷ 4/7 = 3/5 x 7/4

= 21/20

= 1 1/20

## Activity 4: Practice Questions

## Equivalent Fractions

Two or more fractions are said to be equivalent if they are equal in value. For example, ½ and 2/4 are equivalent fractions:

Sometimes you may wish to determine an equivalent fraction when you are given the required new numerator or denominator- for example, if you wanted to convert an assignment mark of **63/90 to a mark out of 30**. You can do this by following these steps:

*1. If you are trying to find a denominator, then divide the numerator of your 'incomplete' fraction but by the numerator of your 'complete' fraction. Alternatively, if you are trying to find a numerator then divide the denominator of your 'incomplete' fraction by the denominator of your 'complete' fraction.*

*For example, if you want to convert 4/10 to a fraction with a numerator of 52 then divide 52 by 4 to get 13.*

*2. If you are trying to find a denominator, then multiply the denominator of your 'complete' fraction by the value obtained previously. Alternatively, if you are trying to find a numerator then multiply the numerator of your 'complete' fraction by the value obtained previously.*

*For example, if you want to convert 4/10 to a fraction with a numerator of 52 then multiply the original denominator of 10 by 13 to get a new denominator 130. Hence 4/10 as a fraction with a numerator of 52 is 52/130.*

*1. If you are trying to find a denominator, then divide the numerator of your 'incomplete' fraction but by the numerator of your 'complete' fraction. Alternatively, if you are trying to find a numerator then divide the denominator of your 'incomplete' fraction by the denominator of your 'complete' fraction.*

*For example, if you want to convert 4/10 to a fraction with a numerator of 52 then divide 52 by 4 to get 13.*

*2. If you are trying to find a denominator, then multiply the denominator of your 'complete' fraction by the value obtained previously. Alternatively, if you are trying to find a numerator then multiply the numerator of your 'complete' fraction by the value obtained previously.*

*For example, if you want to convert 4/10 to a fraction with a numerator of 52 then multiply the original denominator of 10 by 13 to get a new denominator 130. Hence 4/10 as a fraction with a numerator of 52 is 52/130.*

## Examples: Equivalent Fractions

1. Convert 3/4 to a fraction with a denominator of 16.

Dividing 16 by 4 gives 4

Multiplying 3 by 4 gives 12, so equivalent fraction is 12/16

2. Convert 1/5 to a fraction with a numerator of 3.

Dividing 3 by 1 gives 3

Multiplying 5 by 3 gives 15, so equivalent fraction is 3/15

3. If a student scores 18/40 on a test, but it only counts for 20 marks of her final grade, determine how many of the 20 marks she has earned.

Dividing 20 by 40 gives 0.5

Multiplying 18 by 0.5 gives 9, so equivalent fraction is 9/20

Dividing 16 by 4 gives 4

Multiplying 3 by 4 gives 12, so equivalent fraction is 12/16

Dividing 3 by 1 gives 3

Multiplying 5 by 3 gives 15, so equivalent fraction is 3/15

Dividing 20 by 40 gives 0.5

Multiplying 18 by 0.5 gives 9, so equivalent fraction is 9/20

## Activity 5: Practice Questions

## Adding or Subtracting Fractions

with Like Denominators

When it comes to adding or subtracting fractions, there are two different types of problems we need to consider. The first is when the fractions have **like denominators** (i.e. the same denominators); when this is the case, adding or subtracting them is straightforward. You just add or subtract the numerators of the fractions, and keep the denominator of your answer the same as the original fractions.

**For example: 1/5 + 2/5 = **

= **3/5**

3/7 + 2/7 - 1/7 = (3 + 2 - 1)/7

= 4/7

Note that when adding or subtracting mixed numbers, you should add the whole number part of the mixed numbers together before adding the fractional parts together, and then simplifying if necessary.

Denominators

**For example: 1/5 + 2/5 = **

**3/7 + 2/7 - 1/7 = **

**(1 + 2)/5**

= **3/5**

**(3 + 2 - 1)/7**

** = 4/7**

## Adding or Subtracting Fractions

with Unlike Denominators

Adding or subtracting fractions with unlike, or different, denominators requires a bit more work, as you need to convert the fractions to equivalent fractions with the same denominators first before you can add or subtract them. You can do this by following these steps:

*1. Determine what the lowest common denominator of all the fractions in the problem is (i.e. the lowest number that all the denominators can divide into a whole number of times). If you can't find the lowest common denominator you can always just use the product of the denominators, which is always a common denominator but may not be the lowest. *

*For example, if you want to evaluate 1/3 + 1/2 consider that the first two multiples of 3 are 3 and 6, while the first three multiples of 2 are 2, 4 and 6. Hence the lowest common denominator of 3 and 2 is 6.*

*2. Convert each fraction in the problem to an equivalent fraction with a denominator as specified.*

*For example, if you want to evaluate 1/3 + 1/2 then you need to convert both 1/3 and 1/2 to equivalent fractions with denominators of 6, which gives 2/6 and 3/6 respectively. numerator is 14 - 3 x 4 = 2*

*3. Once your fractions all have the same denominator, you can add or subtract them as usual.*

*For example, if you want to evaluate 1/3 + 1/2 then adding 2/6 and 3/6 gives 5/6.*

*3. Simplify your result if necessary*

*1. Determine what the lowest common denominator of all the fractions in the problem is (i.e. the lowest number that all the denominators can divide into a whole number of times). If you can't find the lowest common denominator you can always just use the product of the denominators, which is always a common denominator but may not be the lowest. *

*For example, if you want to evaluate 1/3 + 1/2 consider that the first two multiples of 3 are 3 and 6, while the first three multiples of 2 are 2, 4 and 6. Hence the lowest common denominator of 3 and 2 is 6.*

*2. Convert each fraction in the problem to an equivalent fraction with a denominator as specified.*

*For example, if you want to evaluate 1/3 + 1/2 then you need to convert both 1/3 and 1/2 to equivalent fractions with denominators of 6, which gives 2/6 and 3/6 respectively. numerator is 14 - 3 x 4 = 2*

*3. Once your fractions all have the same denominator, you can add or subtract them as usual.*

*For example, if you want to evaluate 1/3 + 1/2 then adding 2/6 and 3/6 gives 5/6.*

*4. Simplify your result if necessary*

## Examples: Adding or Subtracting Fractions

1. 3/8 + 4/8

= (3 + 4)/8

= 7/8

2. 1/6 + 1/3

Lowest common denominator of two fractions is 6

Equivalent fractions are 1/6 and 2/6 respectively

1/6 + 2/6 = (1 + 2)/6

= 1/2

3. 1/2 - 4/9 + 5/6

Lowest common denominator of three fractions is 18

Equivalent fractions are 9/18, 8/18 and 15/18 respectively

= 8/9

= (3 + 4)/8

= 7/8

Lowest common denominator of two fractions is 6

Equivalent fractions are 1/6 and 2/6 respectively

1/6 + 2/6 = (1 + 2)/6

= 3/6

= 1/2

Lowest common denominator of three fractions is 18

Equivalent fractions are 9/18, 8/18 and 15/18 respectively

9/18 - 8/18 + 15/18 = (9 - 8 + 15)/18

= 16/18

= 8/9

## Activity 6: Practice Questions

## Converting Decimal Numbers to Fractions

Sometimes you may wish to convert between fractions and decimal numbers. You can convert a decimal number to a fraction or mixed number by following these steps:

*1. If your decimal number has a whole number part, then you are converting to a mixed number so write this down as the whole number part.*

*For example, to convert 35.049 to a mixed number write down 35 as the whole number part*

*2. Write down the fractional part of your decimal number as the numerator of your fraction.*

*For example, to convert 35.049 to a mixed number write down 35 49/*

*3. Count how many digits there are after the decimal point in your decimal number, and then put a 1 followed by that many zeroes as the denominator of your fraction.*

*For example, to convert 35.049 to a mixed number write down 35 49/1000*

*3. If necessary, simplify the fractional part of your mixed number*

*1. If your decimal number has a whole number part, then you are converting to a mixed number so write this down as the whole number part.*

*For example, to convert 35.049 to a mixed number write down 35 as the whole number part*

*2. Write down the fractional part of your decimal number as the numerator of your fraction.*

*For example, to convert 35.049 to a mixed number write down 35 49/*

*3. Count how many digits there are after the decimal point in your decimal number, and then put a 1 followed by that many zeroes as the denominator of your fraction.*

*For example, to convert 35.049 to a mixed number write down 35 49/1000*

*4. If necessary, simplify the fractional part of your mixed number*

## Converting Fractions to Decimal Numbers

You can convert a fraction or mixed number to a decimal number by following these steps:

*1. If you are converting from a mixed number, write down the whole number part of the mixed number and insert a decimal point after it.*

*For example, to convert 1 2/3 to a decimal number write down 1.*

*2. Divide the numerator of your fraction by the denominator. If your number was a mixed number, put whatever is after the decimal point in the result after your decimal point. Otherwise, just write down the result.*

*For example, to convert 1 2/3 to a decimal number write down 1.666...*

*3. If necessary, round your decimal number as appropriate.*

*For example, converting 1 2/3 to a decimal number rounded to two decimal places gives 1.67*

Decimal Numbers

*1. If you are converting from a mixed number, write down the whole number part of the mixed number and insert a decimal point after it.*

*For example, to convert 1 2/3 to a decimal number write down 1.*

*2. Divide the numerator of your fraction by the denominator. If your number was a mixed number, put whatever is after the decimal point in the result after your decimal point. Otherwise, just write down the result.*

*For example, to convert 1 2/3 to a decimal number write down 1.666...*

*3. If necessary, round your decimal number as appropriate.*

*For example, converting 1 2/3 to a decimal number rounded to two decimal places gives 1.67*

## Examples: Converting between

Fractions and Decimal Numbers

1. Rewrite 0.35 as a fraction in simplest form

= 35/100

= 7/20

2. Rewrite 52.27 as a mixed number in simplest form

= 52 27/100

3. Convert 2 3/8 to a decimal number

= 2.375

= 35/100

= 7/20

= 52 27/100

= 2.375

## Activity 7: Practice Questions

## End of Topic

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You should now have a better understanding of Fractions.

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