Fractions in Ratio

Fractions in Ratio Problems

A ratio describes how a total is divided into parts. Every ratio can be expressed using fractions, which then allow the full toolkit of fraction arithmetic to be applied. The central connection is straightforward: if a total is divided in the ratio \( a : b \), the two parts are \( \frac{a}{a+b} \) and \( \frac{b}{a+b} \) of the total respectively.

Ratio as Fractions

In the ratio \( 3 : 5 \), the total number of parts is \( 3 + 5 = 8 \). The first part is \( \frac{3}{8} \) of the whole; the second part is \( \frac{5}{8} \) of the whole.

3 parts 5 parts ⅓¹ of total = 3/8 of total = 5/8 Ratio 3 : 5 — total 8 parts

This fraction representation connects ratio work directly to all other fraction skills: finding a fraction of a quantity, setting up equations, comparing ratios and working backwards from one part to the total.

Sharing a Quantity in a Ratio

To share a quantity in a ratio, find the value of one part and then multiply by each number in the ratio.

Example: share £200 in the ratio 3 : 7.
Total parts: \( 3 + 7 = 10 \). One part = \( \frac{£200}{10} = £20 \).
First share: \( 3 \times £20 = £60 \). Second share: \( 7 \times £20 = £140 \). Check: \( £60 + £140 = £200 \checkmark \)

Given One Part, Find the Total or Other Parts

If one part is known, use its fraction of the total to work backwards.

Example: in ratio 2 : 5, the smaller part is 14. Find the larger part.
The smaller part is \( \frac{2}{7} \) of the total: \( \frac{2}{7} \times T = 14 \implies T = 49 \).
Larger part: \( 49 - 14 = 35 \). Check: \( 14 : 35 = 2 : 5 \checkmark \)

Worked Examples

A bag of flour and sugar is mixed in the ratio 4 : 1. What fraction of the mixture is flour? What fraction is sugar?

Total parts: \( 4 + 1 = 5 \).

\[ \text{Flour} = \frac{4}{5} \text{ of the mixture}, \qquad \text{Sugar} = \frac{1}{5} \text{ of the mixture} \]
Share £360 in the ratio 5 : 4. Then verify the answer.

Total parts: \( 5 + 4 = 9 \). One part: \( \frac{360}{9} = £40 \).

\[ \text{First share} = 5 \times £40 = £200, \qquad \text{Second share} = 4 \times £40 = £160 \]

Verify: \( £200 + £160 = £360 \checkmark \) and \( 200 : 160 = 5 : 4 \checkmark \)

The ratio of orange juice to lemonade in a drink is 3 : 8. There is 420 ml of lemonade. How much orange juice is there, and what fraction of the drink is orange juice?

Lemonade is \( \frac{8}{11} \) of the total. Total drink:

\[ \frac{8}{11} \times T = 420 \implies T = \frac{420 \times 11}{8} = 577.5 \text{ ml} \]

Orange juice: \( 577.5 - 420 = 157.5 \text{ ml} \). Alternatively: \( \frac{3}{8} \times 420 = 157.5 \text{ ml} \) (since OJ : Lemonade = 3 : 8, OJ = \( \frac{3}{8} \) of lemonade).

Fraction of total that is orange juice: \( \frac{3}{11} \).

 Key Takeaways

  • In ratio \( a : b \), each part as a fraction of the total is \( \frac{a}{a+b} \) and \( \frac{b}{a+b} \).
  • To share a quantity: find one part by dividing the total by the sum of the ratio, then multiply by each ratio value.
  • Always verify by checking the parts sum to the original total and that the ratio is maintained.
  • To find the total given one part: set up the equation \( \frac{a}{a+b} \times T = \text{known part} \) and solve for \( T \).
  • Three-part ratios work the same way — add all three values to find the total number of parts.