Ratio notation and sharing

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Ratio Notation

A ratio compares two or more quantities of the same kind. It is written with a colon, e.g. 2 : 3.
The order matters: 2 : 3 means the first quantity is 2 parts and the second is 3 parts.
Ratios can be written as part-to-part (e.g. 2 : 3) or part-to-whole (e.g. 2 : 5, where 5 is the total parts).
A ratio can be simplified by dividing all parts by a common factor, e.g. 10 : 15 simplifies to 2 : 3.
To express a ratio in the form 1 : n, divide both parts by the first number. For example, 5 : 20 becomes 1 : 4.
Ratios with different units must be converted to the same unit before simplifying. For example, 2 m : 50 cm becomes 200 cm : 50 cm = 4 : 1.

Sharing a Quantity in a Given Ratio

To share an amount in a given ratio, first find the total number of parts by adding the parts of the ratio.
Find the value of one part by dividing the total amount by the total number of parts.
Multiply the value of one part by each part of the ratio to find each share.
Example: Share £60 in the ratio 2 : 3. Total parts = 2 + 3 = 5. One part = £60 ÷ 5 = £12. Shares are £24 and £36.
Always check that the shares add up to the original total.

Share £60 in the ratio 2:3

Part-to-Whole Ratios and Fractions

A part-to-whole ratio compares a share with the total. If A:B = 2:3 is a part-to-part ratio, then A:(A + B) = 2:5 and A is 2/5 of the whole.
To convert a ratio to a fraction of the whole, write the part over the total parts. For 2 : 3, the first part is 2/5 and the second is 3/5.
To convert a fraction of the whole back to a ratio, write the parts in the order given. For example, if one share is 3/8 of the total, the other share is 5/8, so the ratio is 3 : 5.
Be careful: a ratio like 2 : 3 does not mean the first part is 2/3 of the whole; it means 2 parts out of 5.

Part-to-part versus the whole

Working Backwards from a Share

If you know one share and the ratio, you can find the value of one part by dividing that share by its ratio number.
Then multiply by the other ratio numbers to find the other shares or the total.
Example: In a ratio 3 : 5, the smaller share is £18. One part = £18 ÷ 3 = £6. The larger share = £6 × 5 = £30. Total = £18 + £30 = £48.
If you know the total and one share, you can find the other share by subtraction, then set up the ratio.

Working Backwards from a Difference

If you are given the difference between two shares, find the difference in the ratio parts first.
Divide the given difference by the difference in ratio parts to find the value of one part.
Example: Two shares are in the ratio 4 : 7 and the difference is £15. Difference in parts = 7 - 4 = 3. One part = £15 ÷ 3 = £5. Shares are £20 and £35. Total = £55.
With more than two shares, use the difference between the two ratio entries specifically named by the question; they need not be the largest and smallest.

Combining Ratios

To combine two ratios that share a common element, make the common element have the same value in both ratios.
Find the lowest common multiple (LCM) of the two numbers representing the common element.
Multiply each ratio so that the common element becomes that LCM.
Then write the three-part ratio by combining the non-common parts.
Example: If a : b = 2 : 3 and b : c = 4 : 5, the common element is b. LCM of 3 and 4 is 12. So a : b = 8 : 12 and b : c = 12 : 15. Therefore a : b : c = 8 : 12 : 15.

Comparing Lengths, Areas and Volumes

If two shapes are similar with a length ratio of a : b, then their area ratio is a² : b².
Their volume ratio is a³ : b³.
Example: If the lengths of two similar cubes are in the ratio 2 : 3, their surface areas are in the ratio 4 : 9 and their volumes in the ratio 8 : 27.
When comparing areas or volumes, always square or cube the length ratio, not the other way round.

Common Misconceptions

Do not treat one part of a ratio as a fraction of the whole. For 2 : 3, the first part is 2/5, not 2/3.
Do not divide by the number of parts written; divide by the sum of the parts.
Do not use the total when the question gives only the difference between shares; use the difference in parts.
When combining ratios, make the common element match; otherwise the three-part ratio is wrong.
Always convert units before simplifying a ratio; otherwise the ratio is incorrect.

Typical Exam Questions

Share £120 in the ratio 3 : 5.
Given that one share is £45 in a ratio 3 : 4, find the other share and the total.
The difference between two shares is £24 and they are in the ratio 5 : 8. Find the shares.
Express 12 : 30 in the form 1 : n.
A recipe uses flour and sugar in the ratio 5 : 2. If 300 g of flour is used, how much sugar is needed?
Combine the ratios a : b = 2 : 5 and b : c = 3 : 4 to find a : b : c.

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1.What is the ratio 12:18 in its simplest form?
Easy
- A2:3
- B3:2
- C4:6
- D6:9
2.Express 5:8 in the form 1:n, giving n as a terminating decimal.
Easy
- A1:1.6
- B1:0.625
- C1:8/5
- D1:5/8
3.Ali and Ben share money in the ratio 3:5. Ali gets £36. How much does Ben get?
Medium
- A£60
- B£45
- C£54
- D£72
4.The ratio of boys to girls in a class is 3:2. What fraction of the class are girls?
Medium
- A2/5
- B2/3
- C3/5
- D3/2
5.A recipe for 6 people uses 300 g of flour. How much flour is needed for 8 people?
Medium
- A400 g
- B450 g
- C225 g
- D240 g
6.Which of the following are equivalent to the ratio 2:3? (select all that apply)
Medium
- A4:6
- B6:9
- C8:12
- D10:15
- E1:2
7.For a part-to-part comparison of two shares: The ratio 2:3 is the same as the fraction 2/3 of the whole.
Easy
True or false?
8.Match each ratio to its simplest form.
Easy
- 4:6
- 10:25
- 18:24
- 2:3
- 2:5
- 3:4

Plus de questions

9.Order the ratios from smallest to largest quotient of the first quantity divided by the second.
Medium
- 2:3
- 3:4
- 5:8
- 1:2
10.For a part-to-part comparison of two shares: The ratio 3:5 is equivalent to the fraction 3/5 of the whole.
Easy
True or false?
11.A sum of money is shared between two people in the ratio 5:7. The larger share is £84. What is the total amount shared?
Medium
- A£120
- B£144
- C£168
- D£192
12.The ratio of the volumes of two cubes is 8:27. What is the ratio of their side lengths?
Hard
- A2:3
- B4:9
- C8:27
- D√8:√27
13.Which of the following is the correct method to share £60 in the ratio 2:3?
Easy
- ADivide £60 by 2 and give one person that amount, then divide by 3 for the other.
- BDivide £60 by 5 to find one part, then multiply by 2 and by 3.
- CDivide £60 by 2 and multiply by 3 for the larger share.
- DMultiply £60 by 2 and by 3, then add the results.
14.Share £120 in the ratio 2:3. How much does the person with the smaller share receive?
Medium
- A£24
- B£48
- C£40
- D£60
15.For a part-to-part comparison of two shares: The ratio 5:7 can be written as the fraction 5/7 of the whole.
Easy
True or false?
16.Two people share an amount in the ratio 5:9. The difference between their shares is £24. What is the total amount shared?
Medium
- A£84
- B£96
- C£60
- D£120
17.Match each ratio to its equivalent ratio in the form 1:n.
Medium
- 3:6
- 5:20
- 2:10
- 1:2
- 1:4
- 1:5
18.Which of the following ratios are equivalent to 6:9? (select all that apply)
Medium
- A2:3
- B3:4
- C12:18
- D18:27
- E4:6
19.Order the ratios from smallest to largest quotient of the first quantity divided by the second.
Medium
- 2:3
- 3:4
- 5:6
- 1:2
20.A rectangle has length 12 cm and width 8 cm. A similar rectangle has length 18 cm. What is the ratio of the area of the smaller rectangle to the area of the larger rectangle?
Hard
- A4:9
- B2:3
- C8:18
- D16:81
21.To share £50 in the ratio 2:3, you should first divide £50 by 5.
Easy
True or false?
22.What is the ratio notation for the comparison of 4 to 5?
Easy
- A4:5
- B5:4
- C4/5
- D5/4
23.The ratio 3:4 means that for every 3 parts of one quantity, there are 4 parts of another.
Easy
True or false?
24.Which of the following is the correct first step to share £80 in the ratio 3:5?
Easy
- ADivide £80 by 3
- BDivide £80 by 5
- CDivide £80 by 8
- DDivide £80 by 2
25.A recipe for 12 cookies uses 240 g of flour. How much flour is needed for 30 cookies?
Medium
- A600 g
- B480 g
- C960 g
- D720 g
26.Which of the following ratios are equivalent to 4:6? (select all that apply)
Medium
- A2:3
- B8:12
- C6:9
- D1:1.5
- E3:2
27.The ratio of the lengths of two similar rectangles is 2:5. The area of the smaller rectangle is 24 cm². If the widths are in the same ratio, what is the area of the larger rectangle?
Medium
- A60 cm²
- B150 cm²
- C120 cm²
- D240 cm²
28.Match each ratio to its simplified form.
Medium
- 12:16
- 18:30
- 15:35
- 3:4
- 3:5
- 3:7
29.Order the ratios from smallest to largest quotient of the first quantity divided by the second.
Medium
- 3:4
- 5:8
- 2:3
30.Which of the following are equivalent to the ratio 4:10? (select all that apply)
Medium
- A2:5
- B8:20
- C1:2.5
- D3:8
31.Share £45 in the ratio 2:3. How much is the smaller share?
Easy
- A£18
- B£27
- C£20
- D£15
32.For a part-to-part comparison of two shares: The ratio 3:5 can be written as the fraction 3/5 of the whole.
Easy
True or false?
33.A recipe for 6 people uses 240 g of pasta. How much pasta is needed for 10 people?
Medium
- A400 g
- B300 g
- C360 g
- D480 g
34.The ratio of the volumes of two cubes is 27:64. What is the ratio of their side lengths?
Hard
- A3:4
- B9:16
- C27:64
- D1:2
35.The ratio of cats to dogs in a pet shop is 4:7. There are 12 cats. How many dogs are there?
Medium
- A21
- B19
- C28
- D33
36.Two people share an amount in the ratio 3:5. The difference between their shares is £16. What is the total amount shared?
Medium
- A£64
- B£40
- C£80
- D£48
37.For a part-to-part comparison of two shares: The ratio 2:3 means that the first quantity is 2/3 of the whole.
Easy
True or false?
38.Which of the following ratios are equivalent to 8:12? (select all that apply)
Medium
- A2:3
- B4:6
- C6:9
- D3:2
- E16:24
39.Order the ratios from smallest to largest quotient of the first quantity divided by the second.
Medium
- 3:5
- 2:3
- 4:7
40.A recipe for 8 people uses 500 g of flour. How much flour is needed for 12 people?
Medium
- A750 g
- B625 g
- C800 g
- D1000 g
41.Share £60 in the ratio 2:3. How much is the smaller share?
Medium
42.Two people share an amount in the ratio 4:7. The difference between their shares is £18. Find the total amount shared.
Medium
43.Complete the sentence.
Easy
To share a quantity in a given ratio, first find the value of ____ part by dividing the total by the ____ of the parts.
44.Write 3:5 in the form 1:n, with n rounded to 2 decimal places. Give the complete ratio.
Medium
45.The ratio of the lengths of two rectangles is 2:5. The area of the smaller rectangle is 24 cm². If the widths are the same, what is the area of the larger rectangle?
Medium
46.Write 4:6 as a complete ratio in the form 1:n. Give n as a simplest fraction or terminating decimal.
Medium
47.Complete the sentence about sharing in a ratio.
Medium
When sharing an amount in the ratio a:b, the total number of parts is ____, and each part is worth the total amount divided by ____. Fill in the blanks with the same expression.
48.Fill in the blanks to combine these ratios into a three-part ratio.
Medium
The ratio of apples to bananas is 2:3 and the ratio of bananas to cherries is 4:5. The combined ratio of apples:bananas:cherries is ____:____:____.
49.A rectangle has length 8 cm and width 6 cm. Its simplest length-to-width ratio is a:3. Enter the value of a.
Easy
50.Fill in the blank to express the ratio 4:7 in the form 1:n. Give your answer as a decimal.
Easy
4:7 = 1:____
51.A recipe for 4 people uses 200 g of flour. How much flour is needed for 10 people?
Medium
52.Write the ratio 15:25 in its simplest form.
Medium
53.If two shares have part-to-part ratio A:B = 2:3, explain why A is not 2/3 of the combined whole A + B.
Hard
54.Two shares have part-to-part ratio A:B = 2:3. Express each share as a fraction of their combined whole, explaining your reasoning.
Hard
55.A bag contains red and blue marbles in the ratio 3:5. There are 24 red marbles. Explain how to find the total number of marbles.
Hard
56.Explain how to share £120 in the ratio 3:5, and find the larger share.
Hard
57.Red, blue and green marbles are in the ratio 2:3:5. There are 42 more green than red marbles. Explain how to find the total, and calculate it.
Hard
58.Share £60 in the ratio 2:3. How much is the smaller share?
Medium
- A24 £
- B22 £
- C28 £
- D26 £
59.Two people share an amount in the ratio 4:7. The difference between their shares is £18. Find the total amount shared.
Medium
- A66 £
- B80 £
- C59 £
- D73 £
60.The ratio of the lengths of two rectangles is 2:5. The area of the smaller rectangle is 24 cm². If the widths are the same, what is the area of the larger rectangle?
Medium
- A180 cm²
- B135 cm²
- C165 cm²
- D60 cm²
61.A rectangle has length 8 cm and width 6 cm. Write the ratio of its length to its width in its simplest form.
Easy
- A4 :3
- B3 :3
- C6 :3
- D5 :3
62.A recipe for 4 people uses 200 g of flour. How much flour is needed for 10 people?
Medium
- A600 g
- B450 g
- C550 g
- D500 g

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