Chapter 5 of 7

All 7 chapters
  1. 1Percentage increase and decrease, readunderstood
  2. 2Finding the original amount, readunderstood
  3. 3Simple and compound interest, readunderstood
  4. 4Growth and decay, readunderstood
  5. 5Harder ratio problems, readunderstood
  6. 6Proportion as a formula, readunderstood
  7. 7Graphs of proportion, readunderstood

M4.2.5 Chapter 5

Harder ratio problems

Mei and Ana share beads in the ratio 2 : 3. Ana and Tariq share beads in the ratio 4 : 5. What is the ratio of all three?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Combine two ratios into one, like a : b : c
  • Solve a ratio problem that changes when some are added

Mei and Ana share beads in the ratio 2 : 3. Ana and Tariq share beads in the ratio 4 : 5. Ana appears in both ratios. We want one ratio for all three.

Ana is 3 parts in the first ratio and 4 parts in the second. Those are different sizes of part, so we cannot join the numbers yet. First we make Ana the same in both.

Draw it first

Make the middle the same

  1. Two small pairs of bars. The first pair is Mei: 2 parts and Ana: 3 parts, each part 4 pieces wide. The second pair is Ana: 4 parts and Tariq: 5 parts, each part 3 pieces wide. In step 3 four bars of the same lengths show the pieces: Mei: 8 pieces, Ana, first ratio: 12 pieces, Ana, second ratio: 12 pieces, Tariq: 15 pieces. Thick lines cut the bars into groups.Mei: 2 partsAna: 3 parts

    1 Mei and Ana: 2 parts to 3 parts. Ana has 3 parts here.

  2. Two small pairs of bars. The first pair is Mei: 2 parts and Ana: 3 parts, each part 4 pieces wide. The second pair is Ana: 4 parts and Tariq: 5 parts, each part 3 pieces wide. In step 3 four bars of the same lengths show the pieces: Mei: 8 pieces, Ana, first ratio: 12 pieces, Ana, second ratio: 12 pieces, Tariq: 15 pieces. Thick lines cut the bars into groups.Mei: 2 partsAna: 3 partsAna: 4 partsTariq: 5 parts

    2 Ana and Tariq: 4 parts to 5 parts. Ana has 4 parts here, a different size of part.

  3. Two small pairs of bars. The first pair is Mei: 2 parts and Ana: 3 parts, each part 4 pieces wide. The second pair is Ana: 4 parts and Tariq: 5 parts, each part 3 pieces wide. In step 3 four bars of the same lengths show the pieces: Mei: 8 pieces, Ana, first ratio: 12 pieces, Ana, second ratio: 12 pieces, Tariq: 15 pieces. Thick lines cut the bars into groups.Mei: 8 piecesAna, first ratio: 12 piecesAna, second ratio: 12 piecesTariq: 15 pieces

    3 Cut each part into smaller equal pieces. Ana's bar keeps its length. Make Ana 12 pieces in both. Mei is then 8 pieces and Tariq is 15 pieces.

Multiplying both numbers of a ratio by the same number gives an equal ratio. 2 : 3 = 8 : 12 when you multiply by 4. Pick the numbers so Ana matches: the lowest common multiple of 3 and 4, which is 12.

Worked example

Mei : Ana is 2 : 3 and Ana : Tariq is 4 : 5. Find Mei : Ana : Tariq.

  1. Ana has 3 parts in the first ratio and 4 parts in the second.
  2. The lowest common multiple of 3 and 4 is 12. Make Ana 12 in both.
  3. First ratio, multiply by 4: 2 : 3 = 8 : 12.
  4. Second ratio, multiply by 3: 4 : 5 = 12 : 15.
  5. Ana is 12 in both, so Mei : Ana : Tariq = 8 : 12 : 15.

Answer: 8 : 12 : 15

With a : b : c you can share an amount. Add the parts to find how many equal shares there are. Then find one part, and multiply.

Three bars of equal blocks with the number 2 in every block. Mei: 8 parts, Ana: 12 parts and Tariq: 15 parts. Each block stands for 2 beads.Mei: 8 parts22222222Ana: 12 parts222222222222Tariq: 15 parts222222222222222
70 beads in 35 parts. Each part is 2 beads.

Worked example

Mei, Ana and Tariq have 70 beads together in the ratio 8 : 12 : 15. How many beads does each have?

  1. Add the parts: 8 + 12 + 15 = 35.
  2. One part is 70 ÷ 35 = 2 beads.
  3. Mei has 8 × 2 = 16, Ana has 12 × 2 = 24, and Tariq has 15 × 2 = 30.
  4. Check: 16 + 24 + 30 = 70.

Answer: 16, 24 and 30 beads

Now a different kind of problem. Some are added, and the ratio changes. One group stays the same size. That group gives you the key.

Three bars of equal blocks. Goats, both times: 3 parts, Sheep before: 2 parts, and Sheep after: 5 parts where the last 3 blocks are dotted because they were added.Goats, both times: 3 partsSheep before: 2 partsSheep after: 5 parts
The goats stay 3 parts. The sheep go from 2 parts to 5 parts. The dotted parts are the 12 sheep that were added.

Worked example

Kofi has goats and sheep in the ratio 3 : 2. He gets 12 more sheep, and now goats : sheep is 3 : 5. How many of each did he have at the start?

  1. The goats did not change, so 3 parts of goats keep the same size.
  2. The sheep went from 2 parts to 5 parts. That is 5 − 2 = 3 more parts.
  3. The 3 more parts are 12 sheep, so one part is 12 ÷ 3 = 4.
  4. At the start: goats 3 × 4 = 12, and sheep 2 × 4 = 8.
  5. Check: after, 12 goats and 8 + 12 = 20 sheep, and 12 : 20 = 3 : 5.

Answer: 12 goats and 8 sheep

See it three ways

With things

Mei has 2 beads for every 3 of Ana's. Put 4 such sets together. Mei has 8 beads and Ana has 12.

As a picture

Two bars. Mei: 8 parts is cut into 4 groups of 2 blocks. Ana: 12 parts is cut into 4 groups of 3 blocks.Mei: 8 partsAna: 12 parts

In symbols

2 : 3 = 8 : 12.

Try it

Check with stones

You need: 35 stones or beans.

  1. Lay out 8 stones for Mei, 12 for Ana and 15 for Tariq, in three rows.
  2. Group Mei's and Ana's stones in fours. Count the groups. Then group Ana's and Tariq's stones in threes, and count again.

What you will see: In fours, Mei has 2 groups and Ana has 3, so Mei : Ana is 2 : 3. In threes, Ana has 4 groups and Tariq has 5, so Ana : Tariq is 4 : 5.

If you cannot try it, this is what happens.
Four bars. Mei: 8 parts, Ana, first ratio: 12 parts, Ana, second ratio: 12 parts, Tariq: 15 parts. The first two are cut into groups of 4 blocks and the last two into groups of 3 blocks.Mei: 8 partsAna, first ratio: 12 partsAna, second ratio: 12 partsTariq: 15 parts

A wrong idea many people have

Mei : Ana is 2 : 3 and Ana : Tariq is 4 : 5, so Mei : Ana : Tariq is 2 : 3 : 5.

Why it feels right: The names Mei, Ana and Tariq come in order, and so it looks as if the numbers can come in order too.

The test that breaks it: In 2 : 3 : 5, Ana : Tariq would be 3 : 5. The question says Ana : Tariq is 4 : 5. That does not match.

What is true: The shared person must have the same number in both ratios. Make Ana 12 in both first, then join.

A wrong idea many people have

Goats : sheep is 3 : 2. Adding 12 sheep makes it 3 : 14.

Why it feels right: The sheep number in the ratio is 2, and 12 sheep are added, so 2 + 12 looks right.

The test that breaks it: The 2 in the ratio is 2 parts, not 2 sheep. A part is not the same size as one sheep. You cannot add 12 sheep to 2 parts.

What is true: Find how big one part is first. Here one part is 4 sheep, so 12 sheep are 3 parts.

Key words

lowest common multiple
the smallest whole number above 0 that both numbers divide into. For 3 and 4 it is 12.

Back to your guess

The answer is C: Mei : Ana : Tariq is 8 : 12 : 15.

Ana has 3 parts in one ratio and 4 parts in the other, so the numbers cannot just be joined. Make Ana the same in both: 12. Then 2 : 3 becomes 8 : 12, and 4 : 5 becomes 12 : 15. So Mei : Ana : Tariq is 8 : 12 : 15.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. A : B is 3 : 4 and B : C is 2 : 5. What is A : B : C?

    Tap your answer.

    1. Not this one. You joined the numbers without matching B. B is 4 in the first ratio and 2 in the second, so they are not the same size of part. Go to M4.2.5. Try another.

    2. Yes, that is right. B is 4 and 2. The lowest common multiple is 4. The first ratio stays 3 : 4. The second, multiplied by 2, is 4 : 10. So A : B : C is 3 : 4 : 10.

    3. Not this one. You doubled the first ratio but not the second. B is 8 in the first, and 2 in the second, which is not 8. Scale both ratios. Go to M4.2.5. Try another.

    4. Not this one. You took A from the first ratio and B : C from the second. B is 4 in one and 2 in the other, so the parts are different sizes. Make B the same first. Go to M4.2.5. Try another.

    See the answer

    3 : 4 : 10

    B is 4 and 2. The lowest common multiple is 4. The first ratio stays 3 : 4. The second, multiplied by 2, is 4 : 10. So A : B : C is 3 : 4 : 10.

    • If you chose A: You joined the numbers without matching B. B is 4 in the first ratio and 2 in the second, so they are not the same size of part. Go to M4.2.5.
    • If you chose C: You doubled the first ratio but not the second. B is 8 in the first, and 2 in the second, which is not 8. Scale both ratios. Go to M4.2.5.
    • If you chose D: You took A from the first ratio and B : C from the second. B is 4 in one and 2 in the other, so the parts are different sizes. Make B the same first. Go to M4.2.5.

  2. A : B is 2 : 5 and B : C is 10 : 3. What is A : B : C?

    Tap your answer.

    1. Not this one. You joined the numbers without matching B. B is 5 in the first ratio and 10 in the second. Go to M4.2.5. Try another.

    2. Not this one. You took B from the second ratio but did not scale A with it. If B doubles from 5 to 10, A must double too. Go to M4.2.5. Try another.

    3. Not this one. You doubled A but kept B at 5. Doubling the first ratio makes 4 : 10, so B must be 10. Go to M4.2.5. Try another.

    4. Yes, that is right. B is 5 and 10. The lowest common multiple is 10. The first ratio, multiplied by 2, is 4 : 10. The second ratio stays 10 : 3. So A : B : C is 4 : 10 : 3.

    See the answer

    4 : 10 : 3

    B is 5 and 10. The lowest common multiple is 10. The first ratio, multiplied by 2, is 4 : 10. The second ratio stays 10 : 3. So A : B : C is 4 : 10 : 3.

    • If you chose A: You joined the numbers without matching B. B is 5 in the first ratio and 10 in the second. Go to M4.2.5.
    • If you chose B: You took B from the second ratio but did not scale A with it. If B doubles from 5 to 10, A must double too. Go to M4.2.5.
    • If you chose C: You doubled A but kept B at 5. Doubling the first ratio makes 4 : 10, so B must be 10. Go to M4.2.5.

  3. Share 90 beads in the ratio 2 : 3 : 4. How many beads is the largest share?

    Tap your answer.

    1. Yes, that is right. The parts add up to 2 + 3 + 4 = 9. One part is 90 ÷ 9 = 10. The largest share is 4 × 10 = 40.

    2. Not this one. This is the middle share, 3 parts. The largest share is the 4 parts. Go to M3.4.3. Try another.

    3. Not this one. This is the size of one part, 90 ÷ 9. The largest share is 4 of those parts. Go to M3.4.3. Try another.

    4. Not this one. You divided 90 by 4, the largest number in the ratio. Divide by the total of the parts, 2 + 3 + 4 = 9, then take 4 of them. Go to M3.4.3. Try another.

    See the answer

    40

    The parts add up to 2 + 3 + 4 = 9. One part is 90 ÷ 9 = 10. The largest share is 4 × 10 = 40.

    • If you chose B: This is the middle share, 3 parts. The largest share is the 4 parts. Go to M3.4.3.
    • If you chose C: This is the size of one part, 90 ÷ 9. The largest share is 4 of those parts. Go to M3.4.3.
    • If you chose D: You divided 90 by 4, the largest number in the ratio. Divide by the total of the parts, 2 + 3 + 4 = 9, then take 4 of them. Go to M3.4.3.

  4. Red and blue beads are in the ratio 2 : 3. After 10 more blue beads are added, the ratio is 2 : 5. How many red beads are there?

    Tap your answer.

    1. Not this one. This is the size of one part. There are 2 parts of red, so red is 2 × 5. Go to M4.2.5. Try another.

    2. Not this one. This is the number of blue beads at the start, 3 parts. The question asks for red. Go to M4.2.5. Try another.

    3. Yes, that is right. The red beads did not change. Blue went from 3 parts to 5 parts, which is 2 more parts. So 2 parts are 10 beads, and one part is 5. Red is 2 parts, so 2 × 5 = 10.

    4. Not this one. This is the number of blue beads at the end, 5 parts. The question asks for red. Go to M4.2.5. Try another.

    See the answer

    10

    The red beads did not change. Blue went from 3 parts to 5 parts, which is 2 more parts. So 2 parts are 10 beads, and one part is 5. Red is 2 parts, so 2 × 5 = 10.

    • If you chose A: This is the size of one part. There are 2 parts of red, so red is 2 × 5. Go to M4.2.5.
    • If you chose B: This is the number of blue beads at the start, 3 parts. The question asks for red. Go to M4.2.5.
    • If you chose D: This is the number of blue beads at the end, 5 parts. The question asks for red. Go to M4.2.5.

  5. The bars show A, B and C. What is the ratio A : B : C in its simplest form?

    Three bars of equal blocks. A: 4 blocks, B: 6 blocks, C: 10 blocks.A: 4 blocksB: 6 blocksC: 10 blocks
    Tap your answer.

    1. Not this one. This is a true ratio for the bars, but not the simplest. Divide each number by 2 to get the simplest form. Go to M3.4.1. Try another.

    2. Yes, that is right. The bars hold 4, 6 and 10 blocks. All three divide by 2, giving 2, 3 and 5. No bigger number divides all three, so 2 : 3 : 5 is simplest.

    3. Not this one. This is the right numbers in the wrong order. A is the first bar, so its number comes first. Go to M3.4.1. Try another.

    See the answer

    2 : 3 : 5

    The bars hold 4, 6 and 10 blocks. All three divide by 2, giving 2, 3 and 5. No bigger number divides all three, so 2 : 3 : 5 is simplest.

    • If you chose A: This is a true ratio for the bars, but not the simplest. Divide each number by 2 to get the simplest form. Go to M3.4.1.
    • If you chose C: This is the right numbers in the wrong order. A is the first bar, so its number comes first. Go to M3.4.1.

  6. From an earlier chapter

    After a 20% rise an amount is 180. What was it before? Type the number only.

    See the answer

    150

    A rise of 20% is a multiplier of 1.2. Divide to go back: 180 ÷ 1.2 = 150.

  7. Share 84 stones in the ratio 1 : 2 : 4. How many stones is the largest share?

    See the answer

    48

    The parts add up to 1 + 2 + 4 = 7. One part is 84 ÷ 7 = 12. The largest share is 4 × 12 = 48.

  8. A : B is 3 : 5 and B : C is 10 : 7. Write A : B : C. Type the three numbers.

    See the answer

    6 : 10 : 7

    B is 5 and 10. The lowest common multiple is 10. The first ratio, multiplied by 2, is 6 : 10. The second stays 10 : 7. So A : B : C is 6 : 10 : 7.

Keep it

Copy this into your exercise book.

To join a : b and b : c, make b the same in both with the lowest common multiple. When some are added, find what stayed the same, then the size of one part.

Draw: Two pairs of bars with the shared bar cut into 12 equal pieces, and the joined bars 8 : 12 : 15 beneath.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

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