Fractions without a Calculator

These pages show you how to cope with fraction calculations without your beloved calculator.

1) Cancelling down

• Understand how to identify common factors in the numerator and denominator to simplify the fraction before performing any calculations.

EXAMPLE
Simplify 18/24.

To simplify 18/24, we can find the greatest common factor, which is 6. Dividing both the numerator and denominator by 6, we get 3/4.

2) Mixed numbers

A mixed number consists of a whole number and a proper fraction. To convert an improper fraction to a mixed number, divide the numerator by the denominator. The quotient gives the whole number, and the remainder becomes the new numerator. For example, to convert 7/4 into a mixed number, dividing 7 by 4 gives 1 with a remainder of 3, so the mixed number would be 1 3/4.

EXAMPLES
1. Write 4 2/3 as an improper fraction.

To convert the mixed number 4 2/3 into an improper fraction, multiply the whole number (4) by the denominator (3) to get 12, then add the numerator (2) to get 14. Thus, the improper fraction is 14/3.

2. Write 31/4 as a mixed number.

   To convert the improper fraction 31/4 into a mixed number, divide the numerator (31) by the denominator (4) to find how many whole times 4 fits into 31, which is 7, with a remainder of 3. Thus, the mixed number is 7 3/4.

3) Multiplying

To multiply fractions, multiply the numerators together and the denominators together. For example, to calculate 2/3 multiplied by 4/5, you would perform the operation as follows:

• Multiply the numerators: 2 x 4 = 8

• Multiply the denominators: 3 x 5 = 15

The result is 8/15.

EXAMPLE
Find 8/5 × 7/12.

Multiply the numerators: 8 x 7 = 56
Multiply the denominators: 5 x 12 = 60
The result is 56/60, which simplifies to 14/15.

4) Dividing

To divide fractions, multiply by the reciprocal of the second fraction. For example, to divide ( \frac{3}{4} ) by ( \frac{2}{5} ), you would multiply ( \frac{3}{4} ) by ( \frac{5}{2} ). This results in ( \frac{3 \times 5}{4 \times 2} = \frac{15}{8} ), which can be left as is or converted to a mixed number: 1 7/8.

EXAMPLE

Find 2 1/3 divided by 3 1/2.

• First, convert the mixed numbers to improper fractions: 2 1/3 becomes ( \frac{7}{3} ) and 3 1/2 becomes ( \frac{7}{2} ).

• Next, multiply by the reciprocal of the second fraction: ( \frac{7}{3} ) multiplied by ( \frac{2}{7} ).

• This results in ( \frac{7 \times 2}{3 \times 7} = \frac{14}{21} ), which simplifies to ( \frac{2}{3} ).

5) Common denominators

are necessary when adding or subtracting fractions. To find a common denominator, identify the least common multiple (LCM) of the denominators involved. For example, to add ( rac{1}{4} ) and ( rac{1}{6} ), the LCM of 4 and 6 is 12, allowing us to convert the fractions to ( rac{3}{12} ) and ( rac{2}{12} ), respectively, resulting in a sum of ( rac{5}{12} ).

EXAMPLE
Put these fractions in ascending order of size:

8/3 5/4 12/5

To compare these fractions, we first need to find a common denominator, which in this case can be 60. This allows us to convert the fractions as follows: ( 8/3 = 160/60 ), ( 5/4 = 75/60 ), and ( 12/5 = 144/60 ). Putting these fractions in ascending order results in ( 5/4 ), ( 12/5 ), and then ( 8/3 ).

6) Adding, subtracting - sort the denominators first.

EXAMPLE
Calculate 2 1/5 - 1 1/2.

To solve this, we first convert the mixed numbers to improper fractions: 2 1/5 becomes 11/5, and 1 1/2 becomes 3/2. Next, we find a common denominator, which is 10, allowing us to rewrite the fractions as (11/5 = 22/10) and (3/2 = 15/10). Now we can perform the subtraction: (22/10 - 15/10 = 7/10), leading us to the result of 2 1/5 - 1 1/2 = 7/10. After simplifying, the final answer can be converted back to a mixed number if needed. In this case, since 7/10 is already in its simplest form, we can conclude that the solution is complete.

7) Fractions of something

EXAMPLE
What is 9/20 of £360?

To find this, we multiply £360 by the fraction 9/20. This can be calculated as follows:

• First, simplify the multiplication:

  • £360 ÷ 20 = £18

• Then, multiply the result by 9:

  • £18 × 9 = £162

Thus, 9/20 of £360 equals £162.

Expressing as a fraction

EXAMPLE
Write 180 as a fraction of 80.

To express 180 as a fraction of 80, we can write it as ( \frac{180}{80} ). Simplifying this fraction, we divide both the numerator and denominator by their greatest common divisor, which is 20, resulting in ( \frac{9}{4} ) or 2.25.