This worksheet guides students through combining two ratios, such as A:B and B:C, into a single ratio A:B:C using lowest common multiples.

Let’s solve each problem step by step. We’re combining two ratios into one big ratio (like A:B:C or P:Q:R) by making the middle term (B or Q) the same in both ratios using the lowest common multiple (LCM).

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Section A

1) A:B = 2:1 and B:C = 3:5

We need to make the “B” part the same in both ratios.

- In A:B, B is 1.
- In B:C, B is 3.
→ LCM of 1 and 3 is 3.

So multiply A:B by 3 → A:B = 6:3
Keep B:C as is → B:C = 3:5

Now combine: A:B:C = 6:3:5

✔ Final Answer for 1: 6:3:5

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2) A:B = 5:4 and B:C = 2:7

Make B the same.

- A:B has B=4
- B:C has B=2
→ LCM of 4 and 2 is 4.

Multiply B:C by 2 → B:C = 4:14
A:B stays 5:4

Combine: A:B:C = 5:4:14

✔ Final Answer for 2: 5:4:14

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3) A:B = 1:2 and B:C = 3:4

Make B the same.

- A:B has B=2
- B:C has B=3
→ LCM of 2 and 3 is 6.

Multiply A:B by 3 → A:B = 3:6
Multiply B:C by 2 → B:C = 6:8

Combine: A:B:C = 3:6:8

✔ Final Answer for 3: 3:6:8

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4) A:B = 3:5 and B:C = 2:1

Make B the same.

- A:B has B=5
- B:C has B=2
→ LCM of 5 and 2 is 10.

Multiply A:B by 2 → A:B = 6:10
Multiply B:C by 5 → B:C = 10:5

Combine: A:B:C = 6:10:5

✔ Final Answer for 4: 6:10:5

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5) A:B = 5:3 and B:C = 2:7

Make B the same.

- A:B has B=3
- B:C has B=2
→ LCM of 3 and 2 is 6.

Multiply A:B by 2 → A:B = 10:6
Multiply B:C by 3 → B:C = 6:21

Combine: A:B:C = 10:6:21

✔ Final Answer for 5: 10:6:21

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6) A:B = 5:4 and B:C = 4:3

B is already 4 in both!

So just combine: A:B:C = 5:4:3

✔ Final Answer for 6: 5:4:3

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Section B

1) P:Q = 3:2 and Q:R = 4:5

Make Q the same.

- P:Q has Q=2
- Q:R has Q=4
→ LCM of 2 and 4 is 4.

Multiply P:Q by 2 → P:Q = 6:4
Q:R stays 4:5

Combine: P:Q:R = 6:4:5

✔ Final Answer for 1: 6:4:5

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2) P:Q = 2:3 and Q:R = 4:1

Make Q the same.

- P:Q has Q=3
- Q:R has Q=4
→ LCM of 3 and 4 is 12.

Multiply P:Q by 4 → P:Q = 8:12
Multiply Q:R by 3 → Q:R = 12:3

Combine: P:Q:R = 8:12:3

✔ Final Answer for 2: 8:12:3

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3) P:Q = 1:4 and Q:R = 6:1

Make Q the same.

- P:Q has Q=4
- Q:R has Q=6
→ LCM of 4 and 6 is 12.

Multiply P:Q by 3 → P:Q = 3:12
Multiply Q:R by 2 → Q:R = 12:2

Combine: P:Q:R = 3:12:2

✔ Final Answer for 3: 3:12:2

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4) P:R = 3:2 and Q:R = 5:7

This one is tricky — we have P:R and Q:R. So R is the common term.

We want P:Q:R.

First, make R the same in both.

- P:R has R=2
- Q:R has R=7
→ LCM of 2 and 7 is 14.

Multiply P:R by 7 → P:R = 21:14
Multiply Q:R by 2 → Q:R = 10:14

Now write all three: P:Q:R = 21:10:14

✔ Final Answer for 4: 21:10:14

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5) P:Q = 8:3 and P:R = 6:7

Here, P is common.

We want P:Q:R.

Make P the same.

- P:Q has P=8
- P:R has P=6
→ LCM of 8 and 6 is 24.

Multiply P:Q by 3 → P:Q = 24:9
Multiply P:R by 4 → P:R = 24:28

Now combine: P:Q:R = 24:9:28

✔ Final Answer for 5: 24:9:28

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6) Q:P = 9:2 and R:Q = 5:6

We need P:Q:R. But the given ratios are Q:P and R:Q.

First, flip Q:P to get P:Q.

Q:P = 9:2 → P:Q = 2:9

R:Q = 5:6 → This means Q:R = 6:5 (flip it)

Now we have:

P:Q = 2:9
Q:R = 6:5

Make Q the same.

- P:Q has Q=9
- Q:R has Q=6
→ LCM of 9 and 6 is 18.

Multiply P:Q by 2 → P:Q = 4:18
Multiply Q:R by 3 → Q:R = 18:15

Combine: P:Q:R = 4:18:15

✔ Final Answer for 6: 4:18:15

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Final Answer:
Section A:
1) 6:3:5
2) 5:4:14
3) 3:6:8
4) 6:10:5
5) 10:6:21
6) 5:4:3

Section B:
1) 6:4:5
2) 8:12:3
3) 3:12:2
4) 21:10:14
5) 24:9:28
6) 4:18:15