GCSE Maths: One quantity as a fraction of another (Foundation)
Part of the GCSE Maths ratio and proportion chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
One quantity as a fraction of another
Suppose 18 of the 30 students in a class are girls. One way to describe this is with a fraction. A fraction compares a part with the whole: the part goes on the top (the numerator) and the whole goes on the bottom (the denominator). Here the part is 18 and the whole is 30, so the girls are 1830 of the class. Remember from Chapter 3 that we should always simplify: divide the top and the bottom by their highest common factor. Both 18 and 30 divide by 6, so 1830 = 35.
When a question says “write A as a fraction of B”, the quantity after the word “of” is the whole. It goes on the bottom. For example, “write 3 as a fraction of 8” gives 38. The picture shows why: if the bar is cut into 8 equal blocks and 3 of them are blue, then blue is 3 out of 8 blocks.
Units must match. To write 45 minutes as a fraction of 2 hours, we cannot write 452 because minutes and hours are different units. We change both to the same unit first: 2 hours = 120 minutes, and then the fraction is 45120 = 38. Always change to the smaller unit, because that gives whole numbers.
Fractions bigger than 1. The part is allowed to be bigger than the whole. If one plank is 5 units long and another is 4 units long, the first plank is 54 of the second. This is an improper fraction. It is the same as 114, which tells us the first plank is one whole plank and a quarter more.
Key ideas
- A fraction of a quantity is part ÷ whole: the part on top, the whole on the bottom.
- In “A as a fraction of B”, B (the quantity after “of”) is the whole.
- Both quantities must be in the same unit before you write the fraction. Change to the smaller unit.
- Simplify the fraction by dividing the top and bottom by the highest common factor.
- If the part is bigger than the whole, the fraction is bigger than 1 (an improper fraction).
Key words. A fraction of a quantity compares part of it with the whole. The numerator is the top (the part) and the denominator is the bottom (the whole).
Method. Write the part over the whole (in the same units), then simplify the fraction.
- Decide which quantity is the part and which is the whole. In “A as a fraction of B”, B is the whole (it comes after “of”) and A is the part. The fraction is part over whole.Part = 18
Whole = 30 - Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).1830
- List the factors of the top and the bottom. We want the biggest number that divides into both. This is the highest common factor (HCF).Factors of 18: 1, 2, 3, 6, 9, 18
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
The highest common factor is 6. - Divide the top and the bottom by 6. Dividing both by the same number does not change the value of the fraction, but it makes the numbers smaller.18 ÷ 6 = 3 and 30 ÷ 6 = 5
1830 = 35
Check. Take the fraction of the whole: 35 of 30 = 30 ÷ 5 × 3 = 18, which is the part 18. ✓
Watch out. Put the quantity that comes after “of” on the bottom, and make sure both quantities use the same unit first.
Key words. A fraction of a quantity compares part of it with the whole. The numerator is the top (the part) and the denominator is the bottom (the whole).
Method. Write the part over the whole (in the same units), then simplify the fraction.
- Decide which quantity is the part and which is the whole. In “A as a fraction of B”, B is the whole (it comes after “of”) and A is the part. The fraction is part over whole.Part = 45 minutes
Whole = 2 hours - Write both quantities in the same unit. A fraction compares two amounts, so they must be in the same unit. Change to the smaller unit to avoid decimals.1 hours = 60 minutes
2 hours = 2 × 60 = 120 minutes - Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).45120
- List the factors of the top and the bottom. We want the biggest number that divides into both. This is the highest common factor (HCF).Factors of 45: 1, 3, 5, 9, 15, 45
Factors of 120: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120
The highest common factor is 15. - Divide the top and the bottom by 15. Dividing both by the same number does not change the value of the fraction, but it makes the numbers smaller.45 ÷ 15 = 3 and 120 ÷ 15 = 8
45120 = 38
Check. Take the fraction of the whole: 38 of 120 = 120 ÷ 8 × 3 = 45, which is the part 45. ✓
Watch out. Put the quantity that comes after “of” on the bottom, and make sure both quantities use the same unit first.
Key words. A fraction of a quantity compares part of it with the whole. The numerator is the top (the part) and the denominator is the bottom (the whole).
Method. Write the part over the whole (in the same units), then simplify the fraction.
- Decide which quantity is the part and which is the whole. In “A as a fraction of B”, B is the whole (it comes after “of”) and A is the part. The fraction is part over whole.Part = 0.6 kg
Whole = 2.4 kg - Multiply both numbers by 5 to remove the decimals. Multiplying the top and the bottom of a fraction by the same number does not change its value.0.6 × 5 = 3
2.4 × 5 = 12 - Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).312
- List the factors of the top and the bottom. We want the biggest number that divides into both. This is the highest common factor (HCF).Factors of 3: 1, 3
Factors of 12: 1, 2, 3, 4, 6, 12
The highest common factor is 3. - Divide the top and the bottom by 3. Dividing both by the same number does not change the value of the fraction, but it makes the numbers smaller.3 ÷ 3 = 1 and 12 ÷ 3 = 4
312 = 14
Check. Take the fraction of the whole: 14 of 12 = 12 ÷ 4 × 1 = 3, which is the part 3. ✓
Watch out. Put the quantity that comes after “of” on the bottom, and make sure both quantities use the same unit first.
Key words. A fraction of a quantity compares part of it with the whole. The numerator is the top (the part) and the denominator is the bottom (the whole).
Method. Write the part over the whole (in the same units), then simplify the fraction.
- Decide which quantity is the part and which is the whole. In “A as a fraction of B”, B is the whole (it comes after “of”) and A is the part. The fraction is part over whole.Part = 1.5 m
Whole = 60 cm - Write both quantities in the same unit. A fraction compares two amounts, so they must be in the same unit. Change to the smaller unit to avoid decimals.1 m = 100 cm
1.5 m = 1.5 × 100 = 150 cm - Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).15060
- List the factors of the top and the bottom. We want the biggest number that divides into both. This is the highest common factor (HCF).Factors of 150: 1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, 150
Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
The highest common factor is 30. - Divide the top and the bottom by 30. Dividing both by the same number does not change the value of the fraction, but it makes the numbers smaller.150 ÷ 30 = 5 and 60 ÷ 30 = 2
15060 = 52 - Notice the size of the answer. The part is bigger than the whole, so the fraction is bigger than 1 (an improper fraction). That is allowed.52 = 212
Check. Take the fraction of the whole: 52 of 60 = 60 ÷ 2 × 5 = 150, which is the part 150. ✓
Watch out. Put the quantity that comes after “of” on the bottom, and make sure both quantities use the same unit first.
Watch out
Do not put the numbers the wrong way round. “12 as a fraction of 30” is 1230 = 25, not 3012. And never write a fraction until both quantities are in the same unit: 40p as a fraction of £2 is 40200, not 402.Exercise 15A One quantity as a fraction of another
Non-calculatorKey words. A fraction of a quantity compares part of it with the whole. The numerator is the top (the part) and the denominator is the bottom (the whole).
Method. Write the part over the whole (in the same units), then simplify the fraction.
- Decide which quantity is the part and which is the whole. In “A as a fraction of B”, B is the whole (it comes after “of”) and A is the part. The fraction is part over whole.Part = 7
Whole = 10 - Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).710
- Look for a common factor of the top and the bottom. A fraction is in its simplest form when the only number that divides into both is 1.Factors of 7: 1, 7
Factors of 10: 1, 2, 5, 10
The only common factor is 1, so 710 is already in its simplest form.
Check. Take the fraction of the whole: 710 of 10 = 10 ÷ 10 × 7 = 7, which is the part 7. ✓
Watch out. Put the quantity that comes after “of” on the bottom, and make sure both quantities use the same unit first.
Key words. A fraction of a quantity compares part of it with the whole. The numerator is the top (the part) and the denominator is the bottom (the whole).
Method. Write the part over the whole (in the same units), then simplify the fraction.
- Decide which quantity is the part and which is the whole. In “A as a fraction of B”, B is the whole (it comes after “of”) and A is the part. The fraction is part over whole.Part = 8
Whole = 20 - Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).820
- List the factors of the top and the bottom. We want the biggest number that divides into both. This is the highest common factor (HCF).Factors of 8: 1, 2, 4, 8
Factors of 20: 1, 2, 4, 5, 10, 20
The highest common factor is 4. - Divide the top and the bottom by 4. Dividing both by the same number does not change the value of the fraction, but it makes the numbers smaller.8 ÷ 4 = 2 and 20 ÷ 4 = 5
820 = 25
Check. Take the fraction of the whole: 25 of 20 = 20 ÷ 5 × 2 = 8, which is the part 8. ✓
Watch out. Put the quantity that comes after “of” on the bottom, and make sure both quantities use the same unit first.
Key words. A fraction of a quantity compares part of it with the whole. The numerator is the top (the part) and the denominator is the bottom (the whole).
Method. Write the part over the whole (in the same units), then simplify the fraction.
- Decide which quantity is the part and which is the whole. In “A as a fraction of B”, B is the whole (it comes after “of”) and A is the part. The fraction is part over whole.Part = 12
Whole = 30 - Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).1230
- List the factors of the top and the bottom. We want the biggest number that divides into both. This is the highest common factor (HCF).Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
The highest common factor is 6. - Divide the top and the bottom by 6. Dividing both by the same number does not change the value of the fraction, but it makes the numbers smaller.12 ÷ 6 = 2 and 30 ÷ 6 = 5
1230 = 25
Check. Take the fraction of the whole: 25 of 30 = 30 ÷ 5 × 2 = 12, which is the part 12. ✓
Watch out. Put the quantity that comes after “of” on the bottom, and make sure both quantities use the same unit first.
Key words. A fraction of a quantity compares part of it with the whole. The numerator is the top (the part) and the denominator is the bottom (the whole).
Method. Write the part over the whole (in the same units), then simplify the fraction.
- Decide which quantity is the part and which is the whole. In “A as a fraction of B”, B is the whole (it comes after “of”) and A is the part. The fraction is part over whole.Part = 9
Whole = 36 - Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).936
- List the factors of the top and the bottom. We want the biggest number that divides into both. This is the highest common factor (HCF).Factors of 9: 1, 3, 9
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
The highest common factor is 9. - Divide the top and the bottom by 9. Dividing both by the same number does not change the value of the fraction, but it makes the numbers smaller.9 ÷ 9 = 1 and 36 ÷ 9 = 4
936 = 14
Check. Take the fraction of the whole: 14 of 36 = 36 ÷ 4 × 1 = 9, which is the part 9. ✓
Watch out. Put the quantity that comes after “of” on the bottom, and make sure both quantities use the same unit first.
Key words. A fraction of a quantity compares part of it with the whole. The numerator is the top (the part) and the denominator is the bottom (the whole).
Method. Write the part over the whole (in the same units), then simplify the fraction.
- Decide which quantity is the part and which is the whole. In “A as a fraction of B”, B is the whole (it comes after “of”) and A is the part. The fraction is part over whole.Part = 14
Whole = 35 - Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).1435
- List the factors of the top and the bottom. We want the biggest number that divides into both. This is the highest common factor (HCF).Factors of 14: 1, 2, 7, 14
Factors of 35: 1, 5, 7, 35
The highest common factor is 7. - Divide the top and the bottom by 7. Dividing both by the same number does not change the value of the fraction, but it makes the numbers smaller.14 ÷ 7 = 2 and 35 ÷ 7 = 5
1435 = 25
Check. Take the fraction of the whole: 25 of 35 = 35 ÷ 5 × 2 = 14, which is the part 14. ✓
Watch out. Put the quantity that comes after “of” on the bottom, and make sure both quantities use the same unit first.
Key words. A fraction of a quantity compares part of it with the whole. The numerator is the top (the part) and the denominator is the bottom (the whole).
Method. Write the part over the whole (in the same units), then simplify the fraction.
- Decide which quantity is the part and which is the whole. In “A as a fraction of B”, B is the whole (it comes after “of”) and A is the part. The fraction is part over whole.Part = 16
Whole = 64 - Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).1664
- List the factors of the top and the bottom. We want the biggest number that divides into both. This is the highest common factor (HCF).Factors of 16: 1, 2, 4, 8, 16
Factors of 64: 1, 2, 4, 8, 16, 32, 64
The highest common factor is 16. - Divide the top and the bottom by 16. Dividing both by the same number does not change the value of the fraction, but it makes the numbers smaller.16 ÷ 16 = 1 and 64 ÷ 16 = 4
1664 = 14
Check. Take the fraction of the whole: 14 of 64 = 64 ÷ 4 × 1 = 16, which is the part 16. ✓
Watch out. Put the quantity that comes after “of” on the bottom, and make sure both quantities use the same unit first.
Key words. A fraction of a quantity compares part of it with the whole. The numerator is the top (the part) and the denominator is the bottom (the whole).
Method. Write the part over the whole (in the same units), then simplify the fraction.
- Decide which quantity is the part and which is the whole. In “A as a fraction of B”, B is the whole (it comes after “of”) and A is the part. The fraction is part over whole.Part = 4
Whole = 10 - Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).410
- List the factors of the top and the bottom. We want the biggest number that divides into both. This is the highest common factor (HCF).Factors of 4: 1, 2, 4
Factors of 10: 1, 2, 5, 10
The highest common factor is 2. - Divide the top and the bottom by 2. Dividing both by the same number does not change the value of the fraction, but it makes the numbers smaller.4 ÷ 2 = 2 and 10 ÷ 2 = 5
410 = 25
Check. Take the fraction of the whole: 25 of 10 = 10 ÷ 5 × 2 = 4, which is the part 4. ✓
Watch out. Put the quantity that comes after “of” on the bottom, and make sure both quantities use the same unit first.
Key words. A fraction of a quantity compares part of it with the whole. The numerator is the top (the part) and the denominator is the bottom (the whole).
Method. Write the part over the whole (in the same units), then simplify the fraction.
- Decide which quantity is the part and which is the whole. In “A as a fraction of B”, B is the whole (it comes after “of”) and A is the part. The fraction is part over whole.Part = 9
Whole = 24 - Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).924
- List the factors of the top and the bottom. We want the biggest number that divides into both. This is the highest common factor (HCF).Factors of 9: 1, 3, 9
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
The highest common factor is 3. - Divide the top and the bottom by 3. Dividing both by the same number does not change the value of the fraction, but it makes the numbers smaller.9 ÷ 3 = 3 and 24 ÷ 3 = 8
924 = 38
Check. Take the fraction of the whole: 38 of 24 = 24 ÷ 8 × 3 = 9, which is the part 9. ✓
Watch out. Put the quantity that comes after “of” on the bottom, and make sure both quantities use the same unit first.
Key words. A fraction of a quantity compares part of it with the whole. The numerator is the top (the part) and the denominator is the bottom (the whole).
Method. Write the part over the whole (in the same units), then simplify the fraction.
- Decide which quantity is the part and which is the whole. In “A as a fraction of B”, B is the whole (it comes after “of”) and A is the part. The fraction is part over whole.Part = 36
Whole = 45 - Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).3645
- List the factors of the top and the bottom. We want the biggest number that divides into both. This is the highest common factor (HCF).Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Factors of 45: 1, 3, 5, 9, 15, 45
The highest common factor is 9. - Divide the top and the bottom by 9. Dividing both by the same number does not change the value of the fraction, but it makes the numbers smaller.36 ÷ 9 = 4 and 45 ÷ 9 = 5
3645 = 45
Check. Take the fraction of the whole: 45 of 45 = 45 ÷ 5 × 4 = 36, which is the part 36. ✓
Watch out. Put the quantity that comes after “of” on the bottom, and make sure both quantities use the same unit first.
Key words. A fraction of a quantity compares part of it with the whole. The numerator is the top (the part) and the denominator is the bottom (the whole).
Method. Write the part over the whole (in the same units), then simplify the fraction.
- Decide which quantity is the part and which is the whole. In “A as a fraction of B”, B is the whole (it comes after “of”) and A is the part. The fraction is part over whole.Part = 6
Whole = 10 - Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).610
- List the factors of the top and the bottom. We want the biggest number that divides into both. This is the highest common factor (HCF).Factors of 6: 1, 2, 3, 6
Factors of 10: 1, 2, 5, 10
The highest common factor is 2. - Divide the top and the bottom by 2. Dividing both by the same number does not change the value of the fraction, but it makes the numbers smaller.6 ÷ 2 = 3 and 10 ÷ 2 = 5
610 = 35
Check. Take the fraction of the whole: 35 of 10 = 10 ÷ 5 × 3 = 6, which is the part 6. ✓
Watch out. Put the quantity that comes after “of” on the bottom, and make sure both quantities use the same unit first.
Key words. A fraction of a quantity compares part of it with the whole. The numerator is the top (the part) and the denominator is the bottom (the whole).
Method. Write the part over the whole (in the same units), then simplify the fraction.
- Decide which quantity is the part and which is the whole. In “A as a fraction of B”, B is the whole (it comes after “of”) and A is the part. The fraction is part over whole.Part = 4
Whole = 10 - Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).410
- List the factors of the top and the bottom. We want the biggest number that divides into both. This is the highest common factor (HCF).Factors of 4: 1, 2, 4
Factors of 10: 1, 2, 5, 10
The highest common factor is 2. - Divide the top and the bottom by 2. Dividing both by the same number does not change the value of the fraction, but it makes the numbers smaller.4 ÷ 2 = 2 and 10 ÷ 2 = 5
410 = 25
Check. Take the fraction of the whole: 25 of 10 = 10 ÷ 5 × 2 = 4, which is the part 4. ✓
Watch out. Put the quantity that comes after “of” on the bottom, and make sure both quantities use the same unit first.
Key words. A fraction of a quantity compares part of it with the whole. The numerator is the top (the part) and the denominator is the bottom (the whole).
Method. Write the part over the whole (in the same units), then simplify the fraction.
- Decide which quantity is the part and which is the whole. In “A as a fraction of B”, B is the whole (it comes after “of”) and A is the part. The fraction is part over whole.Part = 20 cm
Whole = 1 m - Write both quantities in the same unit. A fraction compares two amounts, so they must be in the same unit. Change to the smaller unit to avoid decimals.1 m = 100 cm
1 m = 1 × 100 = 100 cm - Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).20100
- List the factors of the top and the bottom. We want the biggest number that divides into both. This is the highest common factor (HCF).Factors of 20: 1, 2, 4, 5, 10, 20
Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100
The highest common factor is 20. - Divide the top and the bottom by 20. Dividing both by the same number does not change the value of the fraction, but it makes the numbers smaller.20 ÷ 20 = 1 and 100 ÷ 20 = 5
20100 = 15
Check. Take the fraction of the whole: 15 of 100 = 100 ÷ 5 × 1 = 20, which is the part 20. ✓
Watch out. Put the quantity that comes after “of” on the bottom, and make sure both quantities use the same unit first.
Key words. A fraction of a quantity compares part of it with the whole. The numerator is the top (the part) and the denominator is the bottom (the whole).
Method. Write the part over the whole (in the same units), then simplify the fraction.
- Decide which quantity is the part and which is the whole. In “A as a fraction of B”, B is the whole (it comes after “of”) and A is the part. The fraction is part over whole.Part = 300 g
Whole = 2 kg - Write both quantities in the same unit. A fraction compares two amounts, so they must be in the same unit. Change to the smaller unit to avoid decimals.1 kg = 1000 g
2 kg = 2 × 1000 = 2000 g - Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).3002000
- List the factors of the top and the bottom. We want the biggest number that divides into both. This is the highest common factor (HCF).Factors of 300: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 25, 30, 50, 60, 75, 100, 150, 300
Factors of 2000: 1, 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, 125, 200, 250, 400, 500, 1000, 2000
The highest common factor is 100. - Divide the top and the bottom by 100. Dividing both by the same number does not change the value of the fraction, but it makes the numbers smaller.300 ÷ 100 = 3 and 2000 ÷ 100 = 20
3002000 = 320
Check. Take the fraction of the whole: 320 of 2000 = 2000 ÷ 20 × 3 = 300, which is the part 300. ✓
Watch out. Put the quantity that comes after “of” on the bottom, and make sure both quantities use the same unit first.
Key words. A fraction of a quantity compares part of it with the whole. The numerator is the top (the part) and the denominator is the bottom (the whole).
Method. Write the part over the whole (in the same units), then simplify the fraction.
- Decide which quantity is the part and which is the whole. In “A as a fraction of B”, B is the whole (it comes after “of”) and A is the part. The fraction is part over whole.Part = 45 minutes
Whole = 3 hours - Write both quantities in the same unit. A fraction compares two amounts, so they must be in the same unit. Change to the smaller unit to avoid decimals.1 hours = 60 minutes
3 hours = 3 × 60 = 180 minutes - Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).45180
- List the factors of the top and the bottom. We want the biggest number that divides into both. This is the highest common factor (HCF).Factors of 45: 1, 3, 5, 9, 15, 45
Factors of 180: 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180
The highest common factor is 45. - Divide the top and the bottom by 45. Dividing both by the same number does not change the value of the fraction, but it makes the numbers smaller.45 ÷ 45 = 1 and 180 ÷ 45 = 4
45180 = 14
Check. Take the fraction of the whole: 14 of 180 = 180 ÷ 4 × 1 = 45, which is the part 45. ✓
Watch out. Put the quantity that comes after “of” on the bottom, and make sure both quantities use the same unit first.
Key words. A fraction of a quantity compares part of it with the whole. The numerator is the top (the part) and the denominator is the bottom (the whole).
Method. Write the part over the whole (in the same units), then simplify the fraction.
- Decide which quantity is the part and which is the whole. In “A as a fraction of B”, B is the whole (it comes after “of”) and A is the part. The fraction is part over whole.Part = 40p
Whole = £2 - Write both quantities in the same unit. A fraction compares two amounts, so they must be in the same unit. Change to the smaller unit to avoid decimals.£1 = 100p
£2 = 2 × 100 = 200p - Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).40200
- List the factors of the top and the bottom. We want the biggest number that divides into both. This is the highest common factor (HCF).Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40
Factors of 200: 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 200
The highest common factor is 40. - Divide the top and the bottom by 40. Dividing both by the same number does not change the value of the fraction, but it makes the numbers smaller.40 ÷ 40 = 1 and 200 ÷ 40 = 5
40200 = 15
Check. Take the fraction of the whole: 15 of 200 = 200 ÷ 5 × 1 = 40, which is the part 40. ✓
Watch out. Put the quantity that comes after “of” on the bottom, and make sure both quantities use the same unit first.
Key words. A fraction of a quantity compares part of it with the whole. The numerator is the top (the part) and the denominator is the bottom (the whole).
Method. Write the part over the whole (in the same units), then simplify the fraction.
- Decide which quantity is the part and which is the whole. In “A as a fraction of B”, B is the whole (it comes after “of”) and A is the part. The fraction is part over whole.Part = 250 m
Whole = 2 km - Write both quantities in the same unit. A fraction compares two amounts, so they must be in the same unit. Change to the smaller unit to avoid decimals.1 km = 1000 m
2 km = 2 × 1000 = 2000 m - Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).2502000
- List the factors of the top and the bottom. We want the biggest number that divides into both. This is the highest common factor (HCF).Factors of 250: 1, 2, 5, 10, 25, 50, 125, 250
Factors of 2000: 1, 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, 125, 200, 250, 400, 500, 1000, 2000
The highest common factor is 250. - Divide the top and the bottom by 250. Dividing both by the same number does not change the value of the fraction, but it makes the numbers smaller.250 ÷ 250 = 1 and 2000 ÷ 250 = 8
2502000 = 18
Check. Take the fraction of the whole: 18 of 2000 = 2000 ÷ 8 × 1 = 250, which is the part 250. ✓
Watch out. Put the quantity that comes after “of” on the bottom, and make sure both quantities use the same unit first.
Key words. A fraction of a quantity compares part of it with the whole. The numerator is the top (the part) and the denominator is the bottom (the whole).
Method. Write the part over the whole (in the same units), then simplify the fraction.
- Decide which quantity is the part and which is the whole. In “A as a fraction of B”, B is the whole (it comes after “of”) and A is the part. The fraction is part over whole.Part = 500 ml
Whole = 3 litres - Write both quantities in the same unit. A fraction compares two amounts, so they must be in the same unit. Change to the smaller unit to avoid decimals.1 litres = 1000 ml
3 litres = 3 × 1000 = 3000 ml - Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).5003000
- List the factors of the top and the bottom. We want the biggest number that divides into both. This is the highest common factor (HCF).Factors of 500: 1, 2, 4, 5, 10, 20, 25, 50, 100, 125, 250, 500
Factors of 3000: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 25, 30, 40, 50, 60, 75, 100, 120, 125, 150, 200, 250, 300, 375, 500, 600, 750, 1000, 1500, 3000
The highest common factor is 500. - Divide the top and the bottom by 500. Dividing both by the same number does not change the value of the fraction, but it makes the numbers smaller.500 ÷ 500 = 1 and 3000 ÷ 500 = 6
5003000 = 16
Check. Take the fraction of the whole: 16 of 3000 = 3000 ÷ 6 × 1 = 500, which is the part 500. ✓
Watch out. Put the quantity that comes after “of” on the bottom, and make sure both quantities use the same unit first.
Key words. A fraction of a quantity compares part of it with the whole. The numerator is the top (the part) and the denominator is the bottom (the whole).
Method. Write the part over the whole (in the same units), then simplify the fraction.
- Decide which quantity is the part and which is the whole. In “A as a fraction of B”, B is the whole (it comes after “of”) and A is the part. The fraction is part over whole.Part = 9
Whole = 6 - Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).96
- List the factors of the top and the bottom. We want the biggest number that divides into both. This is the highest common factor (HCF).Factors of 9: 1, 3, 9
Factors of 6: 1, 2, 3, 6
The highest common factor is 3. - Divide the top and the bottom by 3. Dividing both by the same number does not change the value of the fraction, but it makes the numbers smaller.9 ÷ 3 = 3 and 6 ÷ 3 = 2
96 = 32 - Notice the size of the answer. The part is bigger than the whole, so the fraction is bigger than 1 (an improper fraction). That is allowed.32 = 112
Check. Take the fraction of the whole: 32 of 6 = 6 ÷ 2 × 3 = 9, which is the part 9. ✓
Watch out. Put the quantity that comes after “of” on the bottom, and make sure both quantities use the same unit first.
Key words. A fraction of a quantity compares part of it with the whole. The numerator is the top (the part) and the denominator is the bottom (the whole).
Method. Write the part over the whole (in the same units), then simplify the fraction.
- Decide which quantity is the part and which is the whole. In “A as a fraction of B”, B is the whole (it comes after “of”) and A is the part. The fraction is part over whole.Part = 15
Whole = 10 - Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).1510
- List the factors of the top and the bottom. We want the biggest number that divides into both. This is the highest common factor (HCF).Factors of 15: 1, 3, 5, 15
Factors of 10: 1, 2, 5, 10
The highest common factor is 5. - Divide the top and the bottom by 5. Dividing both by the same number does not change the value of the fraction, but it makes the numbers smaller.15 ÷ 5 = 3 and 10 ÷ 5 = 2
1510 = 32 - Notice the size of the answer. The part is bigger than the whole, so the fraction is bigger than 1 (an improper fraction). That is allowed.32 = 112
Check. Take the fraction of the whole: 32 of 10 = 10 ÷ 2 × 3 = 15, which is the part 15. ✓
Watch out. Put the quantity that comes after “of” on the bottom, and make sure both quantities use the same unit first.
Key words. A fraction of a quantity compares part of it with the whole. The numerator is the top (the part) and the denominator is the bottom (the whole).
Method. Write the part over the whole (in the same units), then simplify the fraction.
- Decide which quantity is the part and which is the whole. In “A as a fraction of B”, B is the whole (it comes after “of”) and A is the part. The fraction is part over whole.Part = 7
Whole = 4 - Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).74
- Look for a common factor of the top and the bottom. A fraction is in its simplest form when the only number that divides into both is 1.Factors of 7: 1, 7
Factors of 4: 1, 2, 4
The only common factor is 1, so 74 is already in its simplest form. - Notice the size of the answer. The part is bigger than the whole, so the fraction is bigger than 1 (an improper fraction). That is allowed.74 = 134
Check. Take the fraction of the whole: 74 of 4 = 4 ÷ 4 × 7 = 7, which is the part 7. ✓
Watch out. Put the quantity that comes after “of” on the bottom, and make sure both quantities use the same unit first.
Key words. A fraction of a quantity compares part of it with the whole. The numerator is the top (the part) and the denominator is the bottom (the whole).
Method. Write the part over the whole (in the same units), then simplify the fraction.
- Decide which quantity is the part and which is the whole. In “A as a fraction of B”, B is the whole (it comes after “of”) and A is the part. The fraction is part over whole.Part = 90 minutes
Whole = 1 hours - Write both quantities in the same unit. A fraction compares two amounts, so they must be in the same unit. Change to the smaller unit to avoid decimals.1 hours = 60 minutes
1 hours = 1 × 60 = 60 minutes - Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).9060
- List the factors of the top and the bottom. We want the biggest number that divides into both. This is the highest common factor (HCF).Factors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90
Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
The highest common factor is 30. - Divide the top and the bottom by 30. Dividing both by the same number does not change the value of the fraction, but it makes the numbers smaller.90 ÷ 30 = 3 and 60 ÷ 30 = 2
9060 = 32 - Notice the size of the answer. The part is bigger than the whole, so the fraction is bigger than 1 (an improper fraction). That is allowed.32 = 112
Check. Take the fraction of the whole: 32 of 60 = 60 ÷ 2 × 3 = 90, which is the part 90. ✓
Watch out. Put the quantity that comes after “of” on the bottom, and make sure both quantities use the same unit first.
Key words. A fraction of a quantity compares part of it with the whole. The numerator is the top (the part) and the denominator is the bottom (the whole).
Method. Write the part over the whole (in the same units), then simplify the fraction.
- Decide which quantity is the part and which is the whole. In “A as a fraction of B”, B is the whole (it comes after “of”) and A is the part. The fraction is part over whole.Part = £5
Whole = 80p - Write both quantities in the same unit. A fraction compares two amounts, so they must be in the same unit. Change to the smaller unit to avoid decimals.£1 = 100p
£5 = 5 × 100 = 500p - Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).50080
- List the factors of the top and the bottom. We want the biggest number that divides into both. This is the highest common factor (HCF).Factors of 500: 1, 2, 4, 5, 10, 20, 25, 50, 100, 125, 250, 500
Factors of 80: 1, 2, 4, 5, 8, 10, 16, 20, 40, 80
The highest common factor is 20. - Divide the top and the bottom by 20. Dividing both by the same number does not change the value of the fraction, but it makes the numbers smaller.500 ÷ 20 = 25 and 80 ÷ 20 = 4
50080 = 254 - Notice the size of the answer. The part is bigger than the whole, so the fraction is bigger than 1 (an improper fraction). That is allowed.254 = 614
Check. Take the fraction of the whole: 254 of 80 = 80 ÷ 4 × 25 = 500, which is the part 500. ✓
Watch out. Put the quantity that comes after “of” on the bottom, and make sure both quantities use the same unit first.
Key words. A fraction of a quantity compares part of it with the whole. The numerator is the top (the part) and the denominator is the bottom (the whole).
Method. Write the part over the whole (in the same units), then simplify the fraction.
- Decide which quantity is the part and which is the whole. In “A as a fraction of B”, B is the whole (it comes after “of”) and A is the part. The fraction is part over whole.Part = 0.6 kg
Whole = 2.4 kg - Multiply both numbers by 5 to remove the decimals. Multiplying the top and the bottom of a fraction by the same number does not change its value.0.6 × 5 = 3
2.4 × 5 = 12 - Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).312
- List the factors of the top and the bottom. We want the biggest number that divides into both. This is the highest common factor (HCF).Factors of 3: 1, 3
Factors of 12: 1, 2, 3, 4, 6, 12
The highest common factor is 3. - Divide the top and the bottom by 3. Dividing both by the same number does not change the value of the fraction, but it makes the numbers smaller.3 ÷ 3 = 1 and 12 ÷ 3 = 4
312 = 14
Check. Take the fraction of the whole: 14 of 12 = 12 ÷ 4 × 1 = 3, which is the part 3. ✓
Watch out. Put the quantity that comes after “of” on the bottom, and make sure both quantities use the same unit first.
Key words. A fraction of a quantity compares part of it with the whole. The numerator is the top (the part) and the denominator is the bottom (the whole).
Method. Write the part over the whole (in the same units), then simplify the fraction.
- Decide which quantity is the part and which is the whole. In “A as a fraction of B”, B is the whole (it comes after “of”) and A is the part. The fraction is part over whole.Part = 0.9 m
Whole = 1.5 m - Multiply both numbers by 10 to remove the decimals. Multiplying the top and the bottom of a fraction by the same number does not change its value.0.9 × 10 = 9
1.5 × 10 = 15 - Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).915
- List the factors of the top and the bottom. We want the biggest number that divides into both. This is the highest common factor (HCF).Factors of 9: 1, 3, 9
Factors of 15: 1, 3, 5, 15
The highest common factor is 3. - Divide the top and the bottom by 3. Dividing both by the same number does not change the value of the fraction, but it makes the numbers smaller.9 ÷ 3 = 3 and 15 ÷ 3 = 5
915 = 35
Check. Take the fraction of the whole: 35 of 15 = 15 ÷ 5 × 3 = 9, which is the part 9. ✓
Watch out. Put the quantity that comes after “of” on the bottom, and make sure both quantities use the same unit first.
Key words. A fraction of a quantity compares part of it with the whole. The numerator is the top (the part) and the denominator is the bottom (the whole).
Method. Write the part over the whole (in the same units), then simplify the fraction.
- Decide which quantity is the part and which is the whole. In “A as a fraction of B”, B is the whole (it comes after “of”) and A is the part. The fraction is part over whole.Part = £3.50
Whole = £5.60 - Multiply both numbers by 10 to remove the decimals. Multiplying the top and the bottom of a fraction by the same number does not change its value.3.5 × 10 = 35
5.6 × 10 = 56 - Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).3556
- List the factors of the top and the bottom. We want the biggest number that divides into both. This is the highest common factor (HCF).Factors of 35: 1, 5, 7, 35
Factors of 56: 1, 2, 4, 7, 8, 14, 28, 56
The highest common factor is 7. - Divide the top and the bottom by 7. Dividing both by the same number does not change the value of the fraction, but it makes the numbers smaller.35 ÷ 7 = 5 and 56 ÷ 7 = 8
3556 = 58
Check. Take the fraction of the whole: 58 of 56 = 56 ÷ 8 × 5 = 35, which is the part 35. ✓
Watch out. Put the quantity that comes after “of” on the bottom, and make sure both quantities use the same unit first.
Key words. A fraction of a quantity compares part of it with the whole. The numerator is the top (the part) and the denominator is the bottom (the whole).
Method. Write the part over the whole (in the same units), then simplify the fraction.
- Decide which quantity is the part and which is the whole. In “A as a fraction of B”, B is the whole (it comes after “of”) and A is the part. The fraction is part over whole.Part = 14
Whole = 38 - Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).1438
- List the factors of the top and the bottom. We want the biggest number that divides into both. This is the highest common factor (HCF).Factors of 14: 1, 2, 7, 14
Factors of 38: 1, 2, 19, 38
The highest common factor is 2. - Divide the top and the bottom by 2. Dividing both by the same number does not change the value of the fraction, but it makes the numbers smaller.14 ÷ 2 = 7 and 38 ÷ 2 = 19
1438 = 719
Check. Take the fraction of the whole: 719 of 38 = 38 ÷ 19 × 7 = 14, which is the part 14. ✓
Watch out. Put the quantity that comes after “of” on the bottom, and make sure both quantities use the same unit first.
Key words. A fraction of a quantity compares part of it with the whole. The numerator is the top (the part) and the denominator is the bottom (the whole).
Method. Write the part over the whole (in the same units), then simplify the fraction.
- Decide which quantity is the part and which is the whole. In “A as a fraction of B”, B is the whole (it comes after “of”) and A is the part. The fraction is part over whole.Part = 9
Whole = 38 - Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).938
- Look for a common factor of the top and the bottom. A fraction is in its simplest form when the only number that divides into both is 1.Factors of 9: 1, 3, 9
Factors of 38: 1, 2, 19, 38
The only common factor is 1, so 938 is already in its simplest form.
Check. Take the fraction of the whole: 938 of 38 = 38 ÷ 38 × 9 = 9, which is the part 9. ✓
Watch out. Put the quantity that comes after “of” on the bottom, and make sure both quantities use the same unit first.
Method. First find how many matches were lost, then write that number over the total.
- Find the number lost. Wins, draws and losses add up to all the matches, so take the wins and draws away from the total.38 − 14 − 9 = 15
- Write the losses over the total matches. Part over whole.1538
- Simplify. 15 and 38 have no common factor above 1 (they share only 1).1538 = 1538
Check. The three fractions should add to 1: 1438 + 938 + 1538 = 3838 = 1. ✓
Watch out. The total in the denominator is the number of matches played (38), not the number of wins.
Method. Find the number of pages still to read, then write that part over the whole book.
- Find the pages not yet read. The unread pages are the whole book take away the pages already read.270 − 45 = 225
- Write them over the total. Part over whole. The whole is the whole book.225270
- Simplify. Divide top and bottom by their HCF, 45.225270 = 56
Check. The read fraction is 45270 = 16. Then 16 + 56 = 1. ✓
Watch out. The question asks for the part not read. Do not give the fraction that has been read.
Method. The race is in km and the distance run is in metres, so first use the same unit. Then work out the distance left.
- Change 2 km to metres. 1 km = 1000 m, and 2 km is 2 lots of 1000 m.2 × 1000 = 2000 m
- Find the distance left. Take away the distance already run.2000 − 750 = 1250 m
- Write the distance left over the whole race. Part over whole, both in metres.12502000
- Simplify. Divide top and bottom by 250.12502000 = 58
Check. Kofi has run 7502000 = 38 of the race, and 38 + 58 = 1. ✓
Watch out. Do not mix km and m in one fraction.
Key words. A fraction of a quantity compares part of it with the whole. The numerator is the top (the part) and the denominator is the bottom (the whole).
Method. Write the part over the whole (in the same units), then simplify the fraction.
- Decide which quantity is the part and which is the whole. In “A as a fraction of B”, B is the whole (it comes after “of”) and A is the part. The fraction is part over whole.Part = 12
Whole = 28 - Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).1228
- List the factors of the top and the bottom. We want the biggest number that divides into both. This is the highest common factor (HCF).Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 28: 1, 2, 4, 7, 14, 28
The highest common factor is 4. - Divide the top and the bottom by 4. Dividing both by the same number does not change the value of the fraction, but it makes the numbers smaller.12 ÷ 4 = 3 and 28 ÷ 4 = 7
1228 = 37
Check. Take the fraction of the whole: 37 of 28 = 28 ÷ 7 × 3 = 12, which is the part 12. ✓
Watch out. Put the quantity that comes after “of” on the bottom, and make sure both quantities use the same unit first.
Method. Work out how many come by car, then write that number as a fraction of the class.
- Find how many come by car. Walkers, bus users and car users make up the whole class.28 − 7 − 12 = 9
- Write over the class size. Part over whole.928
- Simplify. 9 and 28 share only the factor 1, so the fraction is already in its simplest form.928
Check. Walk 728 + bus 1228 + car 928 = 2828 = 1. ✓
Watch out. Check that your fractions add up to 1 whole.
Key words. A fraction of a quantity compares part of it with the whole. The numerator is the top (the part) and the denominator is the bottom (the whole).
Method. Write the part over the whole (in the same units), then simplify the fraction.
- Decide which quantity is the part and which is the whole. In “A as a fraction of B”, B is the whole (it comes after “of”) and A is the part. The fraction is part over whole.Part = 90p
Whole = £2.40 - Write both quantities in the same unit. A fraction compares two amounts, so they must be in the same unit. Change to the smaller unit to avoid decimals.£1 = 100p
£2.40 = 2.4 × 100 = 240p - Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).90240
- List the factors of the top and the bottom. We want the biggest number that divides into both. This is the highest common factor (HCF).Factors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90
Factors of 240: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240
The highest common factor is 30. - Divide the top and the bottom by 30. Dividing both by the same number does not change the value of the fraction, but it makes the numbers smaller.90 ÷ 30 = 3 and 240 ÷ 30 = 8
90240 = 38
Check. Take the fraction of the whole: 38 of 240 = 240 ÷ 8 × 3 = 90, which is the part 90. ✓
Watch out. Put the quantity that comes after “of” on the bottom, and make sure both quantities use the same unit first.
Method. Write each score as a fraction of the total, then compare the fractions by giving them the same denominator.
- Write Amy's score as a fraction. Score over total marks.1720
- Write Ben's score as a fraction. Score over total marks.2125
- Give both fractions the denominator 100. Fractions with the same denominator can be compared by looking at the numerators. 20 × 5 = 100 and 25 × 4 = 100.1720 = 85100 and 2125 = 84100
- Compare. 85 is bigger than 84, so Amy's fraction is bigger.85100 > 84100. Amy did better.
Check. As decimals: 17 ÷ 20 = 0.85 and 21 ÷ 25 = 0.84. 0.85 > 0.84. ✓
Watch out. A bigger score does not always mean better: compare the fractions of the marks available.
Method. Check which number is the part and which is the whole.
- Find the whole. The whole is the number after the word “of”. Here it is 30.Part = 12, whole = 30
- Explain the mistake. The part must be on the top and the whole on the bottom. Sara has put them the wrong way round, which gives a fraction bigger than 1 although 12 is smaller than 30.Sara: 3012 (wrong)
- Write the correct fraction and simplify. Part over whole, then divide by the HCF, which is 6.1230 = 25
Check. 2 ÷ 5 = 0.4 and 12 ÷ 30 = 0.4. ✓
Watch out. A part that is smaller than the whole must give a fraction smaller than 1.
Method. Choose a convenient number for B that divides by 5, find A, then write B as a fraction of A.
- Choose B = 5. Any number would do, but a multiple of 5 makes 35 of B easy to find.B = 5
- Find A. A is three fifths of B. Divide 5 by 5, then multiply by 3.A = 35 of 5 = 5 ÷ 5 × 3 = 3
- Write B as a fraction of A. B is the part and A is the whole.B as a fraction of A = 53
- Conclude. This matches Ivan's fraction, so he is right. In general, flipping the fraction swaps the roles of part and whole.Ivan is right.
Check. Try B = 10: A = 35 of 10 = 6, and 10 as a fraction of 6 is 106 = 53. ✓
Watch out. A single example shows a rule works for that example; here the argument works for every number because the roles of part and whole are simply swapped.
Method. Put every mass in grams, find the total mass, then write the sugar over the total.
- Change 0.3 kg to grams. 1 kg = 1000 g, so multiply by 1000.0.3 × 1000 = 300 g
- Find the total mass. Add all three ingredients, all in grams now.450 + 300 + 150 = 900 g
- Write the sugar over the total. Part over whole.300900
- Simplify. Divide top and bottom by 300, their HCF.300900 = 13
Check. One third of 900 g is 900 ÷ 3 = 300 g, which is the mass of sugar. ✓
Watch out. The whole here is the total of all three ingredients (900 g), not the flour alone.