Hard GCSE Ratio notation and simplifying Questions

Challenging, exam-style GCSE Ratio notation and simplifying questions with worked solutions. Stretch yourself on the hardest chaining two ratios, lowest common multiple, three-part ratios, ratios with units problems.

chaining two ratioslowest common multiplethree-part ratiosratios with unitsunit conversionhighest common factor
GCSE Foundation34 questionsStep-by-step solutions
Question 1
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?
Show worked solution

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
Question 2
6 markschallenging
The ratio of pp to qq is 2:32 : 3 and the ratio of qq to rr is 5:45 : 4. Write the ratio p:rp : r in the form 1:n1 : n.
Show worked solution

Worked solution

  1. Write down the two ratios

    p:q=2:3,q:r=5:4\text{p} : \text{q} = 2 : 3, \quad \text{q} : \text{r} = 5 : 4

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

  2. Spot the linking quantity

    q=3 in the first ratio, q=5 in the second\text{q} = 3 \text{ in the first ratio, } \text{q} = 5 \text{ in the second}

    q 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(3,5)=15\text{LCM}(3, 5) = 15

    15 is the smallest number that both 3 and 5 divide into, so make q worth 15 parts in BOTH ratios.

  4. Scale the first ratio

    2:3=2×5:3×5=10:152 : 3 = 2 \times 5 : 3 \times 5 = 10 : 15

    Multiply both parts of the first ratio by 5 so that q becomes 15.

  5. Scale the second ratio

    5:4=5×3:4×3=15:125 : 4 = 5 \times 3 : 4 \times 3 = 15 : 12

    Multiply both parts of the second ratio by 3 so that q becomes 15 here too.

  6. Check the linking quantity now matches

    15=15=1515 = 15 = 15

    q is worth 15 parts in both ratios, so the two ratios can now be joined.

  7. Combine into a three-part ratio

    p:q:r=10:15:12\text{p} : \text{q} : \text{r} = 10 : 15 : 12

    Take p from the first ratio, the shared q, and r from the second.

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

    HCF(10,15,12)=1\text{HCF}(10, 15, 12) = 1

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

  9. Check the first ratio is still there

    10:15=2:310 : 15 = 2 : 3

    Cancelling the first two parts of the answer returns the original p to q ratio, as it must.

  10. Check the second ratio is still there

    15:12=5:415 : 12 = 5 : 4

    Cancelling the last two parts returns the original q to r ratio, so the combined ratio agrees with both starting ratios.

  11. Warn against the classic mistake

    2:3:4 is wrong2 : 3 : 4 \text{ is wrong}

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

  12. Pick out the two parts that are wanted

    p:r=10:12\text{p} : \text{r} = 10 : 12

    The three-part ratio already contains the p to r ratio: just read off those two parts.

  13. Simplify that pair

    10:12=5:610 : 12 = 5 : 6

    Both parts divide by 2.

  14. Divide both parts by the first part

    5÷5:6÷5=1:1.25 \div 5 : 6 \div 5 = 1 : 1.2

    To reach the form 1 : n the first part must be 1, so divide BOTH parts by 5.

  15. Check by scaling back up

    1×5:1.2×5=5:61 \times 5 : 1.2 \times 5 = 5 : 6

    Scaling 1 : 1.2 back up returns 5 : 6, so the answer is consistent.

Answer
1:1.21 : 1.2
Question 3
6 markschallenging
Write the ratio 34:56:12\frac{3}{4} : \frac{5}{6} : \frac{1}{2} in its simplest form.
Show worked solution

Worked solution

  1. Read the ratio of fractions

    34:56:12\frac{3}{4} : \frac{5}{6} : \frac{1}{2}

    Three fractions are being compared. A ratio is only in an acceptable form when its parts are whole numbers with no common factor.

  2. Write down the denominators

    4,6,24, 6, 2

    The denominators are what stand in the way of whole numbers.

  3. Find the lowest common multiple of the denominators

    LCM(4,6,2)=12\text{LCM}(4, 6, 2) = 12

    12 is the smallest number that all the denominators divide into exactly, so multiplying by 12 clears every fraction at once.

  4. Multiply part 1 by the lowest common multiple

    34×12=3×124=9\frac{3}{4} \times 12 = \frac{3 \times 12}{4} = 9

    12 divided by 4 is 3, and 3 times 3 is 9.

  5. Multiply part 2 by the lowest common multiple

    56×12=5×126=10\frac{5}{6} \times 12 = \frac{5 \times 12}{6} = 10

    12 divided by 6 is 2, and 2 times 5 is 10.

  6. Multiply part 3 by the lowest common multiple

    12×12=1×122=6\frac{1}{2} \times 12 = \frac{1 \times 12}{2} = 6

    12 divided by 2 is 6, and 6 times 1 is 6.

  7. Write the ratio with whole numbers

    34:56:12=9:10:6\frac{3}{4} : \frac{5}{6} : \frac{1}{2} = 9 : 10 : 6

    Every part was multiplied by the same number, so the new ratio is equivalent to the old one.

  8. Look for a common factor

    HCF(9,10,6)=1\text{HCF}(9, 10, 6) = 1

    The three parts share no common factor bigger than 1.

  9. Write the ratio in its simplest form

    9:10:6=9:10:69 : 10 : 6 = 9 : 10 : 6

    This is the simplified whole-number ratio.

  10. Check the answer is in its simplest form

    HCF(9,10,6)=1\text{HCF}(9, 10, 6) = 1

    The parts share no common factor bigger than 1, so the ratio cannot be cancelled any further.

  11. Check the first two parts against the original fractions

    34÷56=910=910\frac{3}{4} \div \frac{5}{6} = \frac{9}{10} = \frac{9}{10}

    The first fraction divided by the second gives the same value as the first part of the answer divided by the second, so those two parts are in the right proportion.

  12. Check the last two parts as well

    56÷12=53=106\frac{5}{6} \div \frac{1}{2} = \frac{5}{3} = \frac{10}{6}

    The same check works on the second and third parts, so the whole three-part ratio is correct.

  13. Find the total number of parts

    9+10+6=259 + 10 + 6 = 25

    If these parts made up a whole, the whole would be 25 parts.

  14. Express the middle part as a fraction of the whole

    1025=25\frac{10}{25} = \frac{2}{5}

    The middle part is 10 of the 25 parts, which is 2/5 of the total. A part is always taken out of the SUM of the parts.

  15. Avoid the common mistake

    3:5:1≠9:10:63 : 5 : 1 \ne 9 : 10 : 6

    Comparing only the numerators and ignoring the denominators gives a completely different, wrong ratio. The fractions must be cleared using the LCM.

Answer
9:10:69 : 10 : 6
Question 4
6 markschallenging
In a survey, 25%25\% of people chose tea and 35%35\% chose coffee. Everyone else chose water. Write the ratio of tea to coffee to water in its simplest form.
Show worked solution

Worked solution

  1. Read the percentages given

    Tea=25%,Coffee=35%\text{Tea} = 25\%, \quad \text{Coffee} = 35\%

    Two of the three percentages are given directly in the question.

  2. Find the missing percentage

    100%−25%−35%=40%100\% - 25\% - 35\% = 40\%

    Everyone else is in the third group, so subtract the two known percentages from 100. That gives 40 percent.

  3. Check the three percentages total 100

    25%+35%+40%=100%25\% + 35\% + 40\% = 100\%

    Each person is counted exactly once, so the three percentages must add to 100.

  4. Turn the percentages into parts

    25:35:4025 : 35 : 40

    A percentage is already a number out of 100, so the three percentages can be written straight down as a ratio.

  5. Explain why that works

    25100:35100:40100\frac{25}{100} : \frac{35}{100} : \frac{40}{100}

    Each percentage is a fraction with denominator 100, and multiplying all three parts by 100 clears that denominator, leaving the percentages themselves.

  6. Look for a common factor

    HCF(25,35,40)=5\text{HCF}(25, 35, 40) = 5

    5 divides into all three parts.

  7. Divide every part by the highest common factor

    25÷5:35÷5:40÷5=5:7:825 \div 5 : 35 \div 5 : 40 \div 5 = 5 : 7 : 8

    Dividing all three parts by the same number gives an equivalent ratio.

  8. Check the answer is in its simplest form

    HCF(5,7,8)=1\text{HCF}(5, 7, 8) = 1

    The parts share no common factor bigger than 1, so the ratio cannot be cancelled any further.

  9. Check the parts still describe the whole

    5+7+8=205 + 7 + 8 = 20

    The whole is now 20 parts instead of 100, but the proportions are unchanged.

  10. Check the first group as a percentage

    520×100%=25%\frac{5}{20} \times 100\% = 25\%

    Turning the first part back into a percentage returns 25 percent, exactly what the question gave.

  11. Check the second group as a percentage

    720×100%=35%\frac{7}{20} \times 100\% = 35\%

    This returns 35 percent, which also matches.

  12. Check the third group as a percentage

    820×100%=40%\frac{8}{20} \times 100\% = 40\%

    This returns 40 percent, the group that was left over.

  13. Test the ratio with real numbers

    200:50:70:80=5:7:8200: \quad 50 : 70 : 80 = 5 : 7 : 8

    With 200 people the three groups would have 50, 70, 80 members, and cancelling those numbers gives the same ratio.

  14. Avoid the common mistake

    25:35≠5:7:825 : 35 \ne 5 : 7 : 8

    Forgetting the leftover group and writing only a two-part ratio loses the third group entirely.

  15. State the ratio in its simplest form

    5:7:85 : 7 : 8

    The three groups are in the ratio 5 : 7 : 8.

Answer
5:7:85 : 7 : 8
Question 5
6 markschallenging
The ratio a:ba : b is 3:43 : 4 and the ratio b:cb : c is 2:72 : 7. Which one of these is the ratio a:b:ca : b : c in its simplest form?
Show worked solution

Worked solution

  1. Write down the two ratios

    a:b=3:4,b:c=2:7a : b = 3 : 4, \quad b : c = 2 : 7

    The two ratios share the letter b, which is the link between them.

  2. Spot the problem with the link

    b=4 then b=2b = 4 \text{ then } b = 2

    b is worth 4 parts in the first ratio but only 2 parts in the second, so the ratios cannot be written side by side until b matches.

  3. Find the lowest common multiple of the two values of b

    LCM(4,2)=4\text{LCM}(4, 2) = 4

    4 is the smallest number that both 4 and 2 divide into, so make b worth 4 parts in both ratios.

  4. Scale the first ratio

    3:4=3×1:4×1=3:43 : 4 = 3 \times 1 : 4 \times 1 = 3 : 4

    b is already 4 here, so the first ratio does not need changing.

  5. Scale the second ratio

    2:7=2×2:7×2=4:142 : 7 = 2 \times 2 : 7 \times 2 = 4 : 14

    Multiplying both parts of the second ratio by 2 makes b worth 4 parts here too.

  6. Check the link now matches

    4=44 = 4

    b is worth 4 parts in both scaled ratios, so they can be joined together.

  7. Combine the ratios

    a:b:c=3:4:14a : b : c = 3 : 4 : 14

    Take a from the first ratio, the shared b, and c from the second ratio.

  8. Check the combined ratio is in its simplest form

    HCF(3,4,14)=1\text{HCF}(3, 4, 14) = 1

    No number bigger than 1 divides all three parts, so this is fully simplified.

  9. Check the first ratio survives

    3:43 : 4

    Reading off the first two parts gives a : b=3b = 3 : 4, exactly as the question stated.

  10. Check the second ratio survives

    4:14=4÷2:14÷2=2:74 : 14 = 4 \div 2 : 14 \div 2 = 2 : 7

    Reading off the last two parts and cancelling gives b : c=2c = 2 : 7, exactly as the question stated.

  11. Reject the naive answer

    3:4:73 : 4 : 7

    Writing the numbers straight down would give b : c=4c = 4 : 7, but the question says b : c=2c = 2 : 7. So this option is wrong.

  12. Reject the next option

    6:8:76 : 8 : 7

    Here a:b=6:8=3:4a : b = 6 : 8 = 3 : 4, which is right, but b:c=8:7b : c = 8 : 7, which is not 2:72 : 7. So c has not been scaled with b.

  13. Reject the next option

    3:2:73 : 2 : 7

    Here a : b=3b = 3 : 2, but the question says a : b=3b = 3 : 4, so this option contradicts the first ratio.

  14. Reject the last option

    3:8:143 : 8 : 14

    Here a : b=3b = 3 : 8, not 3 : 4, so b has been scaled without scaling a. The first ratio has been broken.

  15. State the answer

    3:4:143 : 4 : 14

    This is the only option that agrees with BOTH given ratios and is in its simplest form.

Answer
3:4:143 : 4 : 14

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