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To express one quantity as a fraction of another, write the first as the numerator and the second as the denominator, then simplify. Free GCSE Maths notes.
Expressing One Quantity as a Fraction of Another is part of Ratio and Proportion Fundamentals in GCSE AQA Higher Maths.
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To express one quantity as a fraction of another, write the first as the numerator and the second as the denominator, then simplify. Free GCSE Maths notes.
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Use this lesson to stay inside Ratio and Proportion Fundamentals while connecting the detail back to the wider Ratio, proportion and rates of change topic.
Read the core explanations from Expressing One Quantity as a Fraction of Another before testing yourself from memory.
When you compare two quantities using fractions, you're asking: "How much of the second quantity makes up the first?" This might sound abstract, but you use this thinking every day.
If you score 15 marks out of 20 on a test, you've scored $\frac{15}{20}$ of the total marks. If your friend scores 25 marks out of 20 (perhaps with bonus questions), they've scored $\frac{25}{20}$ of the standard total.
The key insight is this: when the first quantity is smaller than the second, your fraction will be less than 1. When the first quantity is larger, your fraction will be greater than 1.
The general method is straightforward: to express quantity A as a fraction of quantity B, write $\frac{A}{B}$. The challenge comes in recognising which quantity goes where and simplifying your answer.
Every fraction comparison follows the same three-step process. Master this method and you'll handle any variation the exam throws at you.
The three steps are: identify the quantities, write the fraction, then simplify. Let's see this in action with a concrete example.
Worked example: Express 12 as a fraction of 16
Express 12 as a fraction of 16. Give your answer in its simplest form.
Answer: $\frac{3}{4}$
Notice that $\frac{3}{4}$ is less than 1 because 12 is smaller than 16. This makes sense: 12 is three-quarters of 16.
Here's where many students get confused. When the first quantity is larger than the second, your fraction will be greater than 1. This is perfectly normal and often appears in exam questions.
Think about it practically: if you eat 3 slices of pizza and your friend eats 2 slices, you've eaten $\frac{3}{2}$ times as much as your friend. That's 1.5 times as much, which makes sense.
Worked example: Express 20 as a fraction of 8
Express 20 as a fraction of 8. Give your answer in its simplest form.
Answer: $\frac{5}{2}$
The fraction $\frac{5}{2}$ tells us that 20 is two-and-a-half times as big as 8. You can check this: $8 \times 2.5 = 20$. The improper fraction is the exact answer the exam wants.
Some students want to convert improper fractions to mixed numbers (like $2\frac{1}{2}$), but unless the question specifically asks for this, leave your answer as an improper fraction in simplest form.
Real exam questions often involve quantities with units. The key principle is that both quantities must be in the same units before you can form a meaningful fraction.
If you're comparing 30 minutes to 2 hours, you need to convert to the same units first: either both in minutes or both in hours.
Worked example: Express 45 minutes as a fraction of 2 hours
Express 45 minutes as a fraction of 2 hours. Give your answer in its simplest form.
Answer: $\frac{3}{8}$
Notice that the final answer has no units. When you express one quantity as a fraction of another with the same units, the units cancel out. The fraction $\frac{3}{8}$ tells us that 45 minutes is three-eighths of 2 hours.
You could equally convert both to hours: 45 minutes = 0.75 hours, so the fraction becomes \(\frac{0.75}{2} = \frac{3}{8}\), which yields the same answer.
Fraction comparisons become clearer when you visualize them. Let's see how different fraction values relate to the number 1 on a number line.
This visual representation helps you check your answers. If you're expressing a smaller quantity as a fraction of a larger one, your answer should be less than 1 (to the left of 1 on the number line).
Conversely, if you're expressing a larger quantity as a fraction of a smaller one, your answer should be greater than 1 (to the right of 1 on the number line).
The quickest way to check if your fraction is in simplest form is to find the highest common factor (HCF) of the numerator and denominator. If the HCF is 1, your fraction is already simplified.
For finding HCFs quickly, try these shortcuts: if both numbers are even, they share a factor of 2. If both numbers end in 0 or 5 (i.e. both are multiples of 5), they share a factor of 5. For other cases, use the division method or prime factorisation.
Worked example: Quick HCF method for 24 and 36
Find the HCF of 24 and 36 to simplify $\frac{24}{36}$
Answer: $\frac{2}{3}$
Exam questions on this topic follow predictable patterns, but they're designed to catch common mistakes. The most frequent error is getting the fraction upside down.
When a question says "express A as a fraction of B", A always goes on top. Read the question twice to identify which quantity is which. Circle the key words if it helps.
Another trap is forgetting to simplify. Even if your fraction is correct, you'll lose marks if it's not in simplest form. Always check if the numerator and denominator share any common factors.
In calculator-allowed exam questions, you might be tempted to convert fractions to decimals. Resist this urge unless specifically asked. Exact fractional answers are usually preferred and avoid rounding errors.
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Move to the related lessons or back to the topic page when you need the wider sequence before exam-style practice.
24/36 simplifies to 2/3 by dividing both parts by their HCF of 12. The HCF is found by division: 36 ÷ 24 = 1 remainder 12, then 24 ÷ 12 = 2 remainder 0, so the last non-zero remainder, 12, is the HCF.
Write the first quantity as the numerator and the second as the denominator, then simplify by dividing by the HCF. For example, 12 as a fraction of 16 is 12/16, and dividing by 4 gives 3/4.
A fraction is greater than 1 when the first quantity is larger than the second, and it is called an improper fraction. For example, 20 as a fraction of 8 is 20/8 = 5/2, meaning 20 is two-and-a-half times as big as 8.
15 marks out of 20 is the fraction 15/20 of the total marks, which simplifies to 3/4 by dividing both parts by 5. A score of 25 out of 20 would be 25/20, which is greater than 1.
Convert to the same units first: 2 hours = 120 minutes. Then 45 minutes as a fraction of 2 hours is 45/120, and dividing by the HCF of 15 gives 3/8; the units cancel out, so the answer has no units.
Divide the larger number by the smaller, then divide the previous divisor by the remainder, until the remainder is 0; the last non-zero remainder is the HCF. For 24 and 36, 36 ÷ 24 = 1 remainder 12 and 24 ÷ 12 = 2 remainder 0, so the HCF is 12 and 24/36 = 2/3.
Leave the answer as an improper fraction in simplest form unless the question asks for a mixed number. For example, 20 as a fraction of 8 is best given as 5/2 rather than 2 1/2.
12 as a fraction of 16 is 3/4. Writing 12/16 and dividing both parts by the HCF of 4 gives 3/4, which is less than 1 because 12 is smaller than 16.
When expressing A as a fraction of B, A always goes on top and B on the bottom. Getting the fraction upside down is the most frequent error, so the question should be read twice to identify which quantity is which.
Both quantities must be in the same units before forming the fraction. For 45 minutes as a fraction of 2 hours, convert 2 hours to 120 minutes, giving 45/120 = 3/8, and the units cancel so the answer has no units.
Quick ways to simplify a fraction are to check common factors: if both numbers are even they share a factor of 2, and if both end in 0 or 5 they share a factor of 5. If the HCF of the numerator and denominator is 1, the fraction is already in simplest form.
A fraction comparison answer should be left as an exact fraction in simplest form unless a decimal is asked for. Converting to decimals can cause rounding errors, even on calculator-allowed questions.