Expressing Quantities as Fractions and Ratios
From following a recipe to sharing money fairly, fractions and ratios are essential tools for comparing quantities. This guide will show you how to use them, simplify them, and understand the crucial link between them.
Fraction vs. Ratio: What’s the Difference?
Fractions: Part of a Whole
A fraction represents a part of a whole amount. The key is to put the ‘part’ you’re interested in over the ‘total’.
Example: In a box of 40 chocolates, 15 are milk. The fraction is $\frac{15}{40}$, which simplifies to $\frac{3}{8}$.
Ratios: Comparing Parts
A ratio compares two or more quantities to each other. The order of the numbers is important!
$3 : 2$
Example: If there are 10 boys and 15 girls, the ratio of boys to girls is $10:15$, which simplifies to $2:3$.
The Link Between Ratios and Fractions
To turn a ratio into fractions, you first need to find the total number of parts by adding the numbers in the ratio together.
Worked Examples
Simplifying a Ratio
Simplify the ratio $24:36$.
- Find the Highest Common Factor (HCF) of 24 and 36. The HCF is 12.
- Divide each number in the ratio by the HCF.
- $24 \div 12 = 2$.
- $36 \div 12 = 3$.
Answer: $2:3$
Converting a Fraction to a Ratio
$\frac{3}{7}$ of students in a class are boys. What is the ratio of boys to girls?
- The fraction means 3 parts are boys out of a total of 7 parts.
- Calculate the parts for girls: Total parts – Boy parts = $7 – 3 = 4$ parts.
- Write the ratio in the correct order (boys to girls).
Answer: $3:4$
Tutor Insights
🤔 Common Misunderstandings
- Confusing denominators and ratios. For a ratio of $2:3$, students often think the fraction is $\frac{2}{3}$. You must add the parts ($2+3=5$) to find the total for the denominator.
- Ratio order. Forgetting that the order of a ratio is important. ‘Apples to oranges’ is different from ‘oranges to apples’.
📝 Common Exam Mistakes
- Not simplifying fully. Always check if your fraction or ratio can be simplified further.
- Incorrect units. Forgetting to convert quantities to the same unit (e.g., cm and m) before forming a fraction or ratio.
- Not finding the *Highest* Common Factor, leading to an unsimplified answer.
Practice Questions
- In a class of 25 students, 10 wear glasses. What fraction of the students wear glasses?
- Simplify the ratio $18:45$.
- A fruit bowl contains 8 apples and 12 bananas. What is the ratio of apples to bananas in its simplest form?
- A bag contains red and blue marbles in the ratio $2:5$. What fraction of the marbles are blue?
Show Answers
- Working: Fraction = $\frac{10}{25}$. Simplify by dividing by 5.
Answer: $\frac{2}{5}$. - Working: The HCF of 18 and 45 is 9. $18 \div 9 = 2$, $45 \div 9 = 5$.
Answer: $2:5$. - Working: Ratio is $8:12$. The HCF is 4. $8 \div 4 = 2$, $12 \div 4 = 3$.
Answer: $2:3$. - Working: Total parts = $2+5=7$. Blue parts = 5.
Answer: $\frac{5}{7}$.
FAQs
What’s the main difference between a ratio and a fraction?
A ratio compares parts to each other (e.g., 2 red sweets to 3 blue sweets, written $2:3$). A fraction compares a part to the whole (e.g., 2 red sweets out of 5 total sweets, written $\frac{2}{5}$).
Do I always need to simplify?
Yes, unless a question specifically tells you not to, you should always give fractions and ratios in their simplest form. It’s often worth a mark in exams.
