Fractions in Ratio Problems Revision Notes | GCSE AQA Higher Maths

1. Understanding Fractions in Ratios

Imagine you're sharing a pizza between friends, but instead of whole slices, you're dealing with fractional portions. This is where fractions meet ratios - and it's trickier than it first appears.

When ratios involve fractions, we're comparing parts of wholes rather than whole numbers. The key insight is that we can't just compare fractions directly in a ratio - we need to find a common way to express them first.

Consider the ratio $\frac{1}{2} : \frac{1}{3}$. This doesn't mean "1 to 3" - it means "half of something compared to a third of something". The relationship between these fractions determines the actual ratio.

The most reliable method for handling fraction ratios is the common denominator approach. By converting all fractions to equivalent fractions with the same denominator, we can compare their numerators directly.

2. The Common Denominator Method

The common denominator method transforms fraction ratios into whole number ratios. This makes calculations much more manageable and reduces the chance of errors.

Here's how it works: find the lowest common multiple (LCM) of all denominators in the ratio. Convert each fraction to an equivalent fraction using this common denominator. The ratio of the numerators gives you the simplified form.

Worked example: Simplifying a Fraction Ratio

Simplify the ratio $\frac{2}{3} : \frac{1}{4} : \frac{5}{6}$

1. Denominators: 3, 4, 6 LCM(3, 4, 6) = 12 — We need all fractions to have denominator 12
2. $\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}$ $\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}$ $\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}$ — Multiply top and bottom by the same number to get denominator 12
3. $\frac{8}{12} : \frac{3}{12} : \frac{10}{12} = 8 : 3 : 10$ — The denominators are all the same, so we just compare numerators
4. HCF(8, 3, 10) = 1 — Since HCF is 1, the ratio is already in simplest form

Answer: 8 : 3 : 10

Notice how the common denominator method converts a complex fraction ratio into a simple whole number ratio. This makes it much easier to work with in further calculations or real-world applications.

3. Working with Mixed Numbers

Mixed numbers in ratios add another layer of complexity. The key is to convert mixed numbers to improper fractions first, then apply the common denominator method.

Remember that a mixed number like $2\frac{1}{3}$ represents $2 + \frac{1}{3} = \frac{7}{3}$. Always make this conversion before attempting to simplify the ratio.

Worked example: Ratio with Mixed Numbers

Simplify the ratio $1\frac{1}{2} : 2\frac{1}{4}$

1. $1\frac{1}{2} = \frac{3}{2}$ $2\frac{1}{4} = \frac{9}{4}$ — $1\frac{1}{2} = \frac{2 \times 1 + 1}{2} = \frac{3}{2}$ and $2\frac{1}{4} = \frac{4 \times 2 + 1}{4} = \frac{9}{4}$
2. LCM(2, 4) = 4 — 4 is the smallest number that both 2 and 4 divide into
3. $\frac{3}{2} = \frac{6}{4}$ $\frac{9}{4} = \frac{9}{4}$ — Multiply $\frac{3}{2}$ by $\frac{2}{2}$ to get $\frac{6}{4}$
4. $\frac{6}{4} : \frac{9}{4} = 6 : 9$ — Compare the numerators since denominators are equal
5. $6 : 9 = 2 : 3$ — Divide both parts by HCF(6, 9) = 3

Answer: 2 : 3

The checking step is vital in sharing problems. If your shares don't add up to the original total, you've made an error somewhere in the calculation.

4. Solving Sharing Problems

Real-world problems often involve sharing quantities according to fractional ratios. The key is to find the total number of parts and then calculate each person's share.

When sharing £120 in the ratio $\frac{1}{3} : \frac{1}{4}$, we first simplify the ratio to whole numbers, then use the standard sharing method. This approach prevents calculation errors with fractions.

Worked example: Sharing Money in Fractional Ratios

Share £84 between Amy and Ben in the ratio $\frac{2}{3} : \frac{1}{2}$

1. Common denominator of 3 and 2 is 6 $\frac{2}{3} = \frac{4}{6}$ and $\frac{1}{2} = \frac{3}{6}$ Ratio = 4 : 3 — Convert fractions to have common denominator, then compare numerators
2. Total parts = 4 + 3 = 7 — Add the numbers in the simplified ratio
3. Value per part = £84 ÷ 7 = £12 — Divide the total amount by the total number of parts
4. Amy gets: 4 × £12 = £48 Ben gets: 3 × £12 = £36 — Multiply each person's parts by the value per part
5. £48 + £36 = £84 ✓ — The shares should add up to the original total

Answer: Amy gets £48, Ben gets £36

The checking step is vital in sharing problems. If your shares don't add up to the original total, you've made an error somewhere in the calculation.

5. Advanced Techniques and Exam Strategy

Exam questions often combine fraction ratios with other mathematical concepts. You might need to work backwards from a given share to find the original ratio, or handle ratios involving decimals and fractions together.

When working backwards, use the fact that if you know one person's share and their ratio part, you can find the value per part. From there, you can calculate the other shares or the total amount.

Worked example: Working Backwards from a Share

Three friends share money in the ratio $\frac{1}{2} : \frac{1}{3} : \frac{1}{4}$. If the first person receives £30, how much money was shared in total?

1. LCM(2, 3, 4) = 12 $\frac{1}{2} : \frac{1}{3} : \frac{1}{4} = \frac{6}{12} : \frac{4}{12} : \frac{3}{12} = 6 : 4 : 3$ — Convert to common denominator and compare numerators
2. First person has 6 parts and receives £30 Value per part = £30 ÷ 6 = £5 — Divide the known share by the number of parts for that person
3. Total parts = 6 + 4 + 3 = 13 — Add all parts in the simplified ratio
4. Total money = 13 × £5 = £65 — Multiply total parts by value per part
5. Check: $\frac{6}{13} \times £65 = £30$ ✓ — The first person's share should equal the given amount

Answer: £65

Another common exam trick is mixing fractions with decimals in the same ratio. Convert everything to fractions first - it's much more reliable than trying to work with mixed formats.