Working With Ratios

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선생님을 위해: Working With Ratios(Maths [CIE], Extended)을(를) 위한 바로 쓸 수 있는 수업 슬라이드, 복습 노트, 다이어그램 — 수업에 사용하거나, 학생들이 실시간 게임으로 즐기는 인터랙티브 클래스 활동으로 진행하세요.

수업 노트

Understanding Ratios

A ratio compares one part of a whole to another part (e.g., 2 : 5).
The order matters: the first quantity mentioned corresponds to the first number in the ratio.
The total number of parts is the sum of all numbers in the ratio (e.g., 4 : 3 gives 7 parts).
Ratios are different from fractions: a ratio compares parts to parts, a fraction compares a part to the whole.

Equivalent Ratios

Equivalent ratios represent the same proportion (e.g., 5 : 10 is equivalent to 20 : 40).
Multiply or divide each part of the ratio by the same number to find an equivalent ratio.
Scaling up (multiplying) gives larger numbers that may be more realistic in context.
Scaling down (dividing) leads to simplification.

Simplifying Ratios

A ratio is in simplest form when all numbers are integers with no common factor greater than 1.
Divide each part by the highest common factor (HCF) to simplify in one step.
If the HCF is not used, repeat the division process until the ratio is fully simplified.
Example: 30 : 18 simplifies to 5 : 3 (HCF = 6).

Sharing an Amount in a Given Ratio

Add the parts to find the total number of parts.
Divide the total amount by the total number of parts to find the value of one part.
Multiply the value of one part by each part of the ratio to find each share.
Check that the shares add up to the original total.

Problem Solving: Difference Given

Find the difference in the number of parts between the two quantities.
Equate this difference to the given actual difference to find the value of one part.
Then multiply to find each quantity or the total.
Example: If ratio is 7 : 3 and Alfred eats 12 more than Bob, then 4 parts = 12, so 1 part = 3.

Problem Solving: One Quantity Given

Compare the given quantity to its corresponding number of parts in the ratio.
Divide to find the value of one part.
Multiply by the other part(s) to find the unknown quantity(ies).
Example: If red : white = 3 : 2 and Mark has 36 L red, then 3 parts = 36 L, so 1 part = 12 L, white = 24 L.

Combining Two Ratios into a Three-Part Ratio

Identify the common quantity that appears in both ratios (the link).
Find equivalent ratios so that the link has the same value in both.
Then write the three-part ratio combining the two.
Example: B:S = 5:2 and S:W = 6:7 → make S = 6 in both: B:S = 15:6, S:W = 6:7 → B:S:W = 15:6:7.

Direct Proportion

Direct proportion: as one quantity increases, the other increases by the same factor.
The ratio between the two quantities remains constant.
To solve, find the factor (new ÷ old) and multiply the other quantity by that factor.
The unitary method (find 1 unit first) can also be used.

Inverse Proportion

Inverse proportion: as one quantity increases, the other decreases by the same factor.
To solve, find the factor (new ÷ old) and divide the other quantity by that factor.
The unitary method: find the value for 1 unit (opposite operation), then scale.
Example: 3 pumps take 12 hours; 9 pumps take 12 ÷ (9/3) = 4 hours.

Best Value Problems

Compare prices per unit (e.g., per kg, per litre) to find the best value.
Divide the cost by the quantity to get the unit price.
The lowest unit price is the best value for money.
Sometimes rounding up is necessary (e.g., number of tins of paint).

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연습 문제

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1.A plane has 14 First Class seats, 70 Premium seats and 168 Economy seats. Find the ratio First Class seats : Premium seats : Economy seats in its simplest form.
Easy
- A1 : 5 : 12
- B2 : 10 : 24
- C14 : 70 : 168
- D7 : 35 : 84
2.Alex and Chris share sweets in the ratio Alex : Chris = 7 : 3. Alex receives 20 more sweets than Chris. Work out the number of sweets Chris receives.
Medium
- A15
- B20
- C12
- D18
3.Divide $24 in the ratio 7 : 5.
Medium
- A$14 and $10
- B$12 and $12
- C$7 and $5
- D$16.80 and $7.20
4.Kristian and Stephanie share some money in the ratio 3 : 2. Kristian receives $72. Work out how much Stephanie receives.
Medium
- A$48
- B$36
- C$24
- D$60
5.One day, the newspaper had 60 pages of news and advertisements. The ratio number of pages of news : number of pages of advertisements = 5:7. Calculate the number of pages of advertisements.
Medium
- A35
- B25
- C42
- D30
6.Marianne sells photos. The selling price of each photo is $6. The selling price for each photo is made up of two parts, printing cost and profit. For each photo, the ratio printing cost : profit = 5 : 3. Calculate the profit she makes on each photo.
Medium
- A$2.25
- B$3.75
- C$2.50
- D$3.00
7.The Muller family are on holiday in New Zealand. The family visit two waterfalls, the Humboldt Falls and the Bridal Veil Falls. The ratio of the heights Humboldt Falls : Bridal Veil Falls = 5 : 1. The Humboldt Falls are 220 m higher than the Bridal Veil Falls. Calculate the height of the Humboldt Falls.
Medium
- A275 m
- B220 m
- C330 m
- D440 m
8.Adele, Barbara and Collette share $680 in the ratio 9 : 7 : 4. Show that Adele receives $306. Calculate the amount that Barbara and Collette each receives.
Medium
- ABarbara $238, Collette $136
- BBarbara $272, Collette $102
- CBarbara $224, Collette $150
- DBarbara $252, Collette $122

더 많은 문제

9.Some visitors to a campsite are in the ratio men : women = 5 : 4 and women : children = 3 : 7. Calculate the ratio men : women : children in its simplest form.
Hard
- A15 : 12 : 28
- B5 : 4 : 7
- C15 : 12 : 7
- D5 : 3 : 7
10.Ali and Mo share a sum of money in the ratio Ali : Mo = 9 : 7. Ali receives $600 more than Mo. Calculate how much each receives.
Medium
- AAli $2700, Mo $2100
- BAli $2400, Mo $1800
- CAli $3000, Mo $2400
- DAli $3600, Mo $3000
11.For a morning flight, the costs of tickets are in the ratio First Class : Premium : Economy = 14 : 6 : 5. The cost of a Premium ticket is $114. Calculate the cost of a First Class ticket and the cost of an Economy ticket.
Medium
- AFirst Class $266, Economy $95
- BFirst Class $228, Economy $76
- CFirst Class $304, Economy $114
- DFirst Class $342, Economy $95
12.In a cycling club, the number of members are in the ratio males : females = 8 : 3. The club has 342 females. Find the total number of members.
Medium
- A1254
- B912
- C1026
- D1140
13.The fares for a train journey are shown: Standard fare adult $84, child $60. For the standard fare, write the ratio adult fare : child fare in its simplest form.
Easy
- A7 : 5
- B84 : 60
- C42 : 30
- D14 : 10
14.Mohsin has 600 pear trees and 720 apple trees on his farm. Write the ratio pear trees : apple trees in its simplest form.
Easy
- A5 : 6
- B600 : 720
- C10 : 12
- D3 : 4
15.One week, the total mass of pears picked was 18 540 kg. For this week, the ratio mass of apples : mass of pears = 13 : 9. Find the mass of apples picked that week.
Medium
- A26 780 kg
- B22 660 kg
- C18 540 kg
- D30 420 kg
16.A recipe for pancake mix uses flour and milk in the ratio 3 : 2. How much flour is needed if 400 ml of milk is used?
Easy
- A600 ml
- B400 ml
- C200 ml
- D800 ml
17.In a bag, the ratio of red marbles to blue marbles is 4 : 7. If there are 28 blue marbles, how many red marbles are there?
Easy
- A16
- B12
- C20
- D24
18.A fruit stall sells apples and oranges in the ratio 5 : 3. The stall has 120 apples. How many oranges are there?
Easy
- A72
- B60
- C48
- D84
19.The ratio of boys to girls in a school is 7 : 5. There are 240 more boys than girls. How many girls are there?
Medium
- A600
- B840
- C400
- D500
20.A map has a scale of 1 : 50 000. What is the actual distance, in km, represented by 8 cm on the map?
Medium
- A4 km
- B0.4 km
- C40 km
- D0.04 km
21.A sum of money is shared between Alice, Bob, and Carol in the ratio 2 : 3 : 5. Carol receives $60 more than Bob. How much does Alice receive?
Hard
- A$30
- B$40
- C$20
- D$50
22.A triangle has sides in the ratio 3 : 4 : 5. The shortest side is 12 cm. What is the perimeter of the triangle?
Medium
- A48 cm
- B36 cm
- C60 cm
- D24 cm
23.Divide 80 cm in the ratio 3 : 5. What is the length of the longer part?
Easy
- A50 cm
- B30 cm
- C40 cm
- D60 cm
24.In a school, the ratio of students who take music to those who take art is 7 : 4. The number taking music is 63. How many students take art?
Medium
- A36
- B28
- C45
- D56
25.The ratio of men to women in a club is 5 : 3. The ratio of women to children is 4 : 7. What is the ratio of men to children?
Hard
- A20 : 21
- B5 : 7
- C4 : 3
- D15 : 28
26.A paint mixture contains red, blue, and yellow in the ratio 2 : 3 : 5. If there is 6 litres of red, how many litres of blue are needed?
Medium
- A9
- B15
- C10
- D12
27.A cake recipe uses butter and sugar in the ratio 4 : 3. If 200 g of butter is used, how much sugar is needed?
Easy
- A150 g
- B200 g
- C100 g
- D250 g
28.A rectangle has length and width in the ratio 5 : 2. Its perimeter is 70 cm. What is the length?
Medium
- A25 cm
- B20 cm
- C15 cm
- D30 cm
29.The ratio of the ages of a father and son is 7 : 2. The father is 40 years older than the son. How old is the son?
Hard
- A16
- B14
- C12
- D18
30.A recipe for 20 biscuits uses 500 g of flour. How much flour is needed for 35 biscuits?
Medium
- A875 g
- B700 g
- C1000 g
- D750 g
31.A map scale is 1 : 200 000. Two towns are 5 cm apart on the map. What is the actual distance in km?
Medium
- A10 km
- B1 km
- C100 km
- D0.1 km
32.A ratio is a way of comparing one part of a whole to another. Which of the following best describes a ratio?
Easy
- AA comparison of a part to the whole
- BA comparison of one part to another part
- CA comparison of a part to a percentage
- DA comparison of a part to a decimal
33.In the ratio 4 : 3, how many parts make up the whole?
Easy
- A4
- B3
- C7
- D12
34.Which of the following is equivalent to the ratio 5 : 10?
Easy
- A1 : 2
- B2 : 1
- C10 : 5
- D5 : 5
35.A pizza is cut into 8 slices. If the ratio of slices eaten by person A to person B is 5 : 3, what fraction of the pizza does person A eat?
Medium
- A5/8
- B3/8
- C5/3
- D8/5
36.Simplify the ratio 30 : 66 : 12.
Easy
- A5 : 11 : 2
- B15 : 33 : 6
- C10 : 22 : 4
- D6 : 12 : 2
37.Share $200 in the ratio 5 : 3. How much does the person with the larger share receive?
Medium
- A$75
- B$100
- C$125
- D$150
38.The ratio of red paint to white paint is 3 : 2. If Mark has 36 litres of red paint, how many litres of white paint does he need to use all the red paint?
Medium
- A12 litres
- B24 litres
- C36 litres
- D60 litres
39.Alfred and Bob eat cabbage leaves in the ratio 7 : 3. Alfred eats 12 more leaves than Bob. How many leaves does Bob eat?
Hard
- A9
- B12
- C21
- D30
40.What does a ratio compare?
Easy
- AA part to the whole
- BOne part to another part
- CA part to a percentage
- DA whole to a part
41.Match each ratio to its equivalent ratio.
Easy
- 2:3
- 4:5
- 1:4
- 8:12
- 12:15
- 3:12
42.Put the following steps in the correct order to share $120 in the ratio 2:3.
Medium
- Add the parts: 2+3=5
- Divide $120 by 5 to get $24 per part
- Multiply $24 by 2 to get $48 and by 3 to get $72
43.Which of the following ratios are equivalent to 3:4? (Select all that apply.)
Medium
- A6:8
- B9:12
- C12:16
- D15:20
- E3:5
44.In the ratio 5:3, the whole is made up of 8 parts.
Easy
True or false?
45.A sum of money is shared between Alice and Bob in the ratio 7:3. Alice receives $84. How much does Bob receive?
Medium
- A$36
- B$48
- C$24
- D$60
46.The ratio of men to women in a club is 3:5. The ratio of women to children is 4:7. What is the ratio of men to children?
Hard
- A12:35
- B3:7
- C12:28
- D3:35
47.If it takes 4 workers 6 hours to paint a fence, how long would it take 3 workers? (Assume workers work at the same rate.)
Medium
- A4.5 hours
- B8 hours
- C2 hours
- D7.5 hours
48.Complete the sentence.
Medium
A pizza is cut into 8 slices. If the ratio of slices eaten by person A to person B is 5 : 3, the fraction of the pizza that person A eats is ____.
49.Simplify the ratio 30 : 66 : 12. Write your answer as three numbers separated by colons (e.g., 5:11:2).
Easy
50.A fruit drink is made by mixing orange juice and pineapple juice in the ratio 5:2. If 350 ml of orange juice is used, how many ml of pineapple juice is needed?
Easy
51.Simplify the ratio 24:36 to its simplest form.
Easy
The ratio in simplest form is ____ : ____
52.Write down the ratio of 30 cm to 2 m in its simplest form.
Medium
53.The ratio of the ages of a father and son is 7:2. The father is 40 years older than the son. How old is the son?
Medium
54.Divide $200 in the ratio 5:3. How much does the person with the larger share receive?
Medium
55.The ratio of men to women in a club is 3:5. The ratio of women to children is 4:7. What is the ratio of men to children? Write your answer in the form m:n, e.g., 12:35.
Hard
56.Explain how to divide $200 in the ratio 5:3. Include the amounts each person receives.
Medium
57.A fruit drink is made by mixing orange juice and pineapple juice in the ratio 5:2. If 350 ml of orange juice is used, how many ml of pineapple juice is needed?
Easy
- A154 ml
- B126 ml
- C168 ml
- D140 ml

복습 노트

## Ratios - A **ratio** compares one part to another part (e.g., 2:5). - The order matters: the first item mentioned corresponds to the first number. - **Equivalent ratios** are found by multiplying/dividing all parts by the same number. - **Simplifying** a ratio: divide all parts by their highest common factor (HCF). ### Sharing an amount in a given ratio 1. Add the parts to find total parts. 2. Divide the total amount by total parts → value of 1 part. 3. Multiply each part by that value. ### Difference given - Find the difference in parts; equate to the actual difference → find 1 part. ### One quantity given - Compare given quantity to its parts → find 1 part; then find the other. ### Combining two ratios (three-part ratio) - Make the common quantity (e.g., women) the same in both ratios by scaling. - Then combine into a single three-part ratio. ## Proportion ### Direct proportion - Both quantities increase/decrease by the same factor. - Use unitary method: find 1 unit, then multiply. - Or find factor = new/old, then multiply the other quantity. ### Inverse proportion - As one quantity increases, the other decreases by the same factor. - Use unitary method: find 1 unit (multiply), then divide for the required quantity. - Or find factor = new/old, then divide the other quantity. ### Best value - Find the price per unit (e.g., per kg) for each option; the lowest is best. ## Key Formulas - **Ratio to fraction**: part/whole (e.g., in ratio 5:3, first part = 5/8 of whole). - **Total parts** = sum of ratio numbers. - **Value of 1 part** = total amount ÷ total parts. ## Tips - Label your ratios (e.g., A:B = 3:4). - For inverse proportion, think: more workers → less time. - Round up when buying items (e.g., tins of paint).

Ratio: A way of comparing one part of a whole to another, written with colons (e.g., 2:5).
Equivalent ratio: Two ratios that represent the same proportion, found by multiplying or dividing all parts by the same number.
Simplest form: A ratio where all values are integers with no common factors (except 1).
Direct proportion: As one quantity increases/decreases, the other does so by the same factor; the ratio is constant.
Inverse proportion: As one quantity increases, the other decreases by the same factor (and vice versa).
Unitary method: Finding the value of one unit (e.g., 1 part) then scaling up/down to find the required amount.

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