GCSE Ratio notation and simplifying Practice Questions

Free GCSE Ratio notation and simplifying practice questions with full step-by-step worked solutions. Covers simplifying ratios, common factors, highest common factor, writing a ratio. Practise exam-style problems and check your method.

simplifying ratioscommon factorshighest common factorwriting a ratioratio from a contextequivalent ratios
GCSE Foundation70 questionsStep-by-step solutions
Question 1
1 markeasy
Write the ratio 4:64 : 6 in its simplest form.
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Worked solution

  1. Look for a common factor

    4:64 : 6

    Common factors of 4 and 6: 1, 2. The highest is 2.

  2. Divide both parts by the highest common factor

    4÷2=2,6÷2=34 \div 2 = 2, \quad 6 \div 2 = 3

    Dividing both parts of a ratio by the same number does not change the ratio.

  3. Write the simplified ratio

    4:6=2:34 : 6 = 2 : 3

    2 and 3 have no common factor left, so this is the simplest form.

Answer
2:32 : 3
Question 2
2 markseasy
Write the ratio 2:4:62 : 4 : 6 in its simplest form.
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Worked solution

  1. Look for a factor common to all three parts

    2:4:62 : 4 : 6

    2 divides into 2, 4 and 6 exactly, and it is the highest number that does.

  2. Divide every part by the highest common factor

    2÷2:4÷2:6÷22 \div 2 : 4 \div 2 : 6 \div 2

    All three parts must be divided by the same number, or the ratio changes.

  3. Write the simplified ratio

    2:4:6=1:2:32 : 4 : 6 = 1 : 2 : 3

    The three parts now share no common factor, so this is simplest form.

Answer
1:2:31 : 2 : 3
Question 3
2 marksintermediate
A student writes: "20 cm:1 m=20:120\text{ cm} : 1\text{ m} = 20 : 1." Explain what is wrong with this statement and give the correct ratio in its simplest form.
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Worked solution

  1. Find the flaw in the statement

    20 cm:1 m20\text{ cm} : 1\text{ m}

    The two amounts are in different units, so the numbers 20 and 1 cannot be compared directly. The student has ignored the units.

  2. State the conversion fact

    1 m=100 cm1\text{ m} = 100\text{ cm}

    Centimetres are the smaller unit, so change the metre into centimetres.

  3. Convert both amounts to the same unit

    20 cm:100 cm20\text{ cm} : 100\text{ cm}

    The second amount is 100 cm, and now both parts share the unit centimetres.

  4. Drop the units and write the ratio

    20:10020 : 100

    Once both parts are in the same unit, the ratio is just a comparison of the two numbers.

  5. Simplify the ratio

    20÷20:100÷20=1:520 \div 20 : 100 \div 20 = 1 : 5

    The highest common factor of 20 and 100 is 20, and dividing gives 1 : 5.

  6. Sense check the student answer

    20:1≠1:520 : 1 \ne 1 : 5

    The student answer of 20 : 1 would mean the first amount is twenty times the second, but 20 cm is much SMALLER than 1 m. The correct ratio, 1 : 5, correctly shows the second amount is the bigger one.

Answer
The units are different, so they must be made the same first. Since 1 m=100m = 100 cm, the ratio is 20 cm:100 cm=20:100=1:520\text{ cm} : 100\text{ cm} = 20 : 100 = 1 : 5.
Question 4
4 markshard
Three prizes AA, BB and CC are shared in the ratio 2:3:52 : 3 : 5. What percentage of the total does prize BB receive?
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Worked solution

  1. Read the ratio

    Prize A:Prize B:Prize C=2:3:5\text{Prize A} : \text{Prize B} : \text{Prize C} = 2 : 3 : 5

    The numbers in a ratio are parts. They are not percentages and not totals.

  2. Add the parts to find the whole

    2+3+5=102 + 3 + 5 = 10

    The whole group is made of 10 equal parts.

  3. Write the wanted group as a fraction of the whole

    310\frac{3}{10}

    The prize b are 3 of the 10 parts.

  4. Simplify the fraction

    310=310\frac{3}{10} = \frac{3}{10}

    The fraction is already in its simplest form.

  5. Convert the fraction to a percentage

    310×100%=30%\frac{3}{10} \times 100\% = 30\%

    A percentage is a number out of 100, so multiply the fraction of the whole by 100.

  6. Find the scale factor that makes the total 100

    100÷10=10100 \div 10 = 10

    Scaling the ratio so that the parts total 100 turns each part directly into a percentage.

  7. Scale the whole ratio to a total of 100

    20:30:5020 : 30 : 50

    Every part is now written out of 100, so each number is that group as a percentage.

  8. Read off the percentage

    30%30\%

    The prize b make up 30 percent of the total.

  9. Check the percentages add to 100

    20%+30%+50%=100%20\% + 30\% + 50\% = 100\%

    Every member of the group is counted exactly once, so the percentages must total 100.

  10. Avoid the classic mistake

    37≠310\frac{3}{7} \ne \frac{3}{10}

    Comparing the wanted part with the OTHER part instead of with the total is the usual error here. A percentage of the whole needs the TOTAL on the bottom.

Answer
30%30\%
Question 5
6 markschallenging
In a school the ratio of Year 10 students to Year 11 students is 4:54 : 5, and the ratio of Year 11 students to Year 12 students is 10:310 : 3. There are 4040 more Year 11 students than Year 10 students. How many students are there in Years 10, 11 and 12 altogether?
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Worked solution

  1. Write down the two ratios

    Year 10:Year 11=4:5,Year 11:Year 12=10:3\text{Year 10} : \text{Year 11} = 4 : 5, \quad \text{Year 11} : \text{Year 12} = 10 : 3

    Two separate ratios share the quantity Year 11, which is the link between them.

  2. Spot the linking quantity

    Year 11=5 in the first ratio, Year 11=10 in the second\text{Year 11} = 5 \text{ in the first ratio, } \text{Year 11} = 10 \text{ in the second}

    Year 11 appears in both ratios but as a different number of parts, so the two ratios cannot simply be written side by side yet.

  3. Find the lowest common multiple of the two linking numbers

    LCM(5,10)=10\text{LCM}(5, 10) = 10

    10 is the smallest number that both 5 and 10 divide into, so make Year 11 worth 10 parts in BOTH ratios.

  4. Scale the first ratio

    4:5=4×2:5×2=8:104 : 5 = 4 \times 2 : 5 \times 2 = 8 : 10

    Multiply both parts of the first ratio by 2 so that Year 11 becomes 10.

  5. Scale the second ratio

    10:3=10×1:3×1=10:310 : 3 = 10 \times 1 : 3 \times 1 = 10 : 3

    Year 11 is already worth 10 parts here, so the second ratio does not need scaling.

  6. Check the linking quantity now matches

    10=10=1010 = 10 = 10

    Year 11 is worth 10 parts in both ratios, so the two ratios can now be joined.

  7. Combine into a three-part ratio

    Year 10:Year 11:Year 12=8:10:3\text{Year 10} : \text{Year 11} : \text{Year 12} = 8 : 10 : 3

    Take Year 10 from the first ratio, the shared Year 11, and Year 12 from the second.

  8. Check the three-part ratio is in simplest form

    HCF(8,10,3)=1\text{HCF}(8, 10, 3) = 1

    The three parts share no common factor, so no cancelling is needed.

  9. Check the first ratio is still there

    8:10=4:58 : 10 = 4 : 5

    Cancelling the first two parts of the answer returns the original Year 10 to Year 11 ratio, as it must.

  10. Check the second ratio is still there

    10:3=10:310 : 3 = 10 : 3

    Cancelling the last two parts returns the original Year 11 to Year 12 ratio, so the combined ratio agrees with both starting ratios.

  11. Warn against the classic mistake

    4:5:3 is wrong4 : 5 : 3 \text{ is wrong}

    Writing the given numbers side by side ignores that Year 11 counts differently in each ratio. The linking part must be scaled to match first.

  12. Find the difference in parts

    10−8=2 parts10 - 8 = 2 \text{ parts}

    In the combined ratio there are 2 more parts of Year 11 than of Year 10.

  13. Match the difference in parts to the difference given

    2 parts=402 \text{ parts} = 40

    The question says there are 40 more Year 11 than Year 10, and that gap is worth 2 parts.

  14. Work out the value of one part

    1 part=40÷2=201 \text{ part} = 40 \div 2 = 20

    Dividing the difference by the number of parts in the difference gives the size of a single part.

  15. Multiply by the total number of parts

    21×20=42021 \times 20 = 420

    The whole group is 21 parts and each part is worth 20.

Answer
420420

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