11+ Maths: Ratios and Proportions

Ratios and proportions are fundamental concepts in mathematics that express relationships between quantities. Understanding these concepts is essential for solving various problems in everyday life and in the 11+ exam.

Key Concepts

1. Ratio

A ratio compares two or more quantities, showing the relative sizes of each quantity. Ratios can be expressed in different ways:

• As a fraction (e.g., $$\frac{a}{b}$$ ),
• With a colon (e.g., $$a:b$$ ),
• In words (e.g., “a to b”).

For example, if there are 4 apples and 2 oranges, the ratio of apples to oranges can be expressed as:

2. Proportion

A proportion states that two ratios are equal. It can be represented as:

For example, if 1 out of 3 students passed a test, and 2 out of 6 students passed a different test, we can say:

3. Finding Ratios

To find a ratio, divide the quantities of interest by their greatest common factor (GCF). For example:

• For the quantities 10 and 15, the GCF is 5.
• Therefore, the ratio can be expressed as:
  $$\frac{10 \div 5}{15 \div 5} = \frac{2}{3}$$ or $$2:3$$ .

4. Working with Proportions

To solve proportion problems, cross-multiply and simplify. For example, given:

Practice Questions on Ratios and Proportions

Easy Level

1. What is the ratio of 8 apples to 4 oranges?
2. If there are 3 cats and 9 dogs, what is the ratio of cats to dogs?
3. Simplify the ratio 10:5.
4. If a recipe calls for 2 cups of flour to 3 cups of sugar, what is the ratio of flour to sugar?
5. A bag contains 5 red balls and 10 blue balls. What is the ratio of red to blue balls?
6. There are 6 boys and 4 girls in a class. What is the ratio of boys to girls?
7. If there are 20 students in total and 5 are girls, what is the ratio of boys to girls?
8. A recipe needs 1 cup of milk for every 2 cups of flour. What is the ratio of milk to flour?
9. If a car travels 50 km in 2 hours, what is the ratio of distance to time?
10. There are 12 cookies and 8 brownies. What is the ratio of cookies to brownies?
11. A basket has 6 apples and 3 bananas. What is the ratio of apples to fruits?
12. If a class has 12 students, and 3 are wearing glasses, what is the ratio of students wearing glasses to total students?
13. What is the ratio of 15:45?
14. If 2 pizzas are shared among 8 friends, what is the ratio of pizzas to friends?
15. If a garden has 4 roses and 6 tulips, what is the ratio of roses to flowers?
16. There are 10 men and 5 women at a party. What is the ratio of men to women?
17. If a car uses 8 litres of petrol for 100 km, what is the ratio of litres to distance?
18. A team has 5 players and 2 substitutes. What is the ratio of players to substitutes?
19. If a cake recipe requires 2 eggs for 3 cups of flour, what is the ratio of eggs to flour?
20. A class has 10 boys and 15 girls. What is the ratio of boys to girls?

Medium Level

1. A recipe requires 3 cups of milk for 4 cups of sugar. What is the ratio of milk to sugar?
2. If 5 out of 20 students passed an exam, what is the ratio of students who passed to those who failed?
3. There are 50 students in a class, and 30 of them are boys. What is the ratio of girls to boys?
4. A fruit basket contains 12 apples, 6 bananas, and 9 oranges. What is the ratio of apples to total fruits?
5. If the ratio of red to blue marbles is 3:2 and there are 30 marbles in total, how many red marbles are there?
6. A bag contains 24 candies. If the ratio of chocolate to fruity candies is 2:1, how many chocolate candies are there?
7. If a school has 150 students and 60 of them are girls, what is the ratio of boys to girls?
8. A pizza is cut into 8 slices. If 3 slices are eaten, what is the ratio of slices eaten to remaining slices?
9. If the ratio of cats to dogs is 4:5 and there are 45 dogs, how many cats are there?
10. A car can travel 240 km on 10 litres of petrol. What is the ratio of distance to petrol used?
11. A class has 16 boys and 12 girls. What is the ratio of boys to girls in simplest form?
12. If a garden has 10 roses and 5 tulips, what is the ratio of roses to tulips?
13. A recipe uses 2 cups of rice for every 5 cups of water. What is the ratio of rice to water?
14. In a pack of 30 cards, if 10 are red, what is the ratio of red cards to total cards?
15. If 60% of a class are girls and there are 20 students, how many boys are there?
16. A car travels 90 km in 1.5 hours. What is the ratio of distance to time in simplest form?
17. The ratio of pencils to pens in a box is 7:3. If there are 30 items in total, how many pens are there?
18. A bag contains 15 green balls and 10 yellow balls. What is the ratio of green balls to yellow balls?
19. If a student scored 45 marks out of 60, what is the ratio of marks obtained to total marks?
20. The ratio of the ages of two siblings is 3:5. If the younger sibling is 9 years old, how old is the older sibling?

Hard Level

1. If the ratio of boys to girls in a school is 4:5 and there are 360 students, how many boys are there?
2. A recipe calls for 4 eggs for every 3 cups of flour. If you use 12 eggs, how many cups of flour do you need?
3. The ratio of students in two classes is 2:3. If the first class has 40 students, how many students are in the second class?
4. If the ratio of red to blue marbles is 5:2 and there are 70 marbles in total, how many are red?
5. A container holds 150 litres of liquid. If the ratio of water to juice is 2:3, how much juice is there?
6. If the ratio of the length to the width of a rectangle is 5:2 and the perimeter is 70 cm, what is the length?
7. A recipe for cookies requires a ratio of 2:3 for butter to sugar. If you use 300g of sugar, how much butter do you need?
8. If 45% of a group of people are women and there are 120 people, how many are men?
9. The ratio of the ages of two friends is 4:5. If the older friend is 25 years old, how old is the younger friend?
10. A box contains 60 apples and 90 oranges. What is the ratio of apples to oranges in simplest form?
11. If the ratio of boys to girls in a sports team is 3:4 and there are 28 girls, how many boys are there?
12. A car travels 180 km on 15 litres of petrol. What is the ratio of distance travelled to petrol used?
13. If the ratio of the sides of a triangle is 3:4:5 and the perimeter is 36 cm, what is the length of the longest side?
14. The ratio of students who like football to those who like basketball is 7:3. If there are 200 students in total, how many like basketball?
15. If a mixture contains 3 parts water to 2 parts juice, how much juice is needed for 20 litres of mixture?
16. A school has a ratio of 5 teachers for every 30 students. How many teachers are there if there are 120 students?
17. If the ratio of cats to dogs is 4:3 and there are 42 dogs, how many cats are there?
18. A recipe calls for a ratio of 1:4 for salt to flour. If you have 200g of salt, how much flour do you need?
19. If the ratio of red to green apples is 3:5 and there are 40 apples in total, how many are red?
20. The ratio of girls to boys in a class is 2:3. If there are 15 boys, how many girls are there?

Answers and

Explanations

Easy Level

1. $$8 \text{ apples}$$
2. $$6 \text{ marbles left}$$
3. $$2 \times 7 = 14 \text{ pounds}$$
4. $$12 – 3 = 9 \text{ cookies left}$$
5. $$4 + 6 = 10 \text{ balloons}$$
6. $$5 \times 2 = 10 \text{ pounds}$$
7. $$20 – 5 = 15 \text{ students}$$
8. $$3 + 2 = 5 \text{ pets}$$
9. $$8 – 3 = 5 \text{ pieces left}$$
10. $$15 – 5 = 10 \text{ pounds left}$$
11. $$52 – 5 = 47 \text{ cards left}$$
12. $$6 + 2 = 8 \text{ chairs}$$
13. $$2 – 1 = 1 \text{ banana left}$$
14. $$2 \times 10 = 20 \text{ dollars}$$
15. $$5 + 2 = 7 \text{ items left}$$
16. $$4 \times 6 = 24 \text{ students}$$
17. $$10 + 6 = 16 \text{ books}$$
18. $$10 + 2 = 12 \text{ bananas left}$$
19. $$2 \times 3 = 6 \text{ apples}$$
20. $$10 + 10 = 20 \text{ dollars}$$

Medium Level

1. $$\frac{3}{4} \text{ of } 20 = 15$$
2. $$\frac{20}{80} = \frac{1}{4}$$
3. $$4 \times 4 = 16 \text{ parts}$$
4. $$\frac{5}{3} = 1.67 \text{ parts}$$
5. $$\frac{4}{10} = 0.4 \text{ litres}$$
6. $$\frac{200 \text{ grams}}{300 \text{ grams}} = 0.67 \text{ parts}$$
7. $$3 + 4 + 5 = 12 \text{ parts}$$
8. $$\frac{10}{30} = \frac{1}{3}$$
9. $$20 + 30 + 20 + 40 + 40 = 150 \text{ grams}$$
10. $$15 + 4 + 5 + 6 + 1 = 31 \text{ students}$$

Hard Level

1. $$\text{Number of boys} = 360 \times \frac{4}{9} = 160$$
2. $$12 \text{ cups of flour}$$
3. $$\frac{40}{2} \times 3 = 60 \text{ students}$$
4. $$\text{Red marbles} = 5 \times \frac{70}{7} = 50$$
5. $$\text{Juice} = \frac{3}{5} \times 150 = 90 \text{ litres}$$
6. $$\text{Length} = 70 \text{ cm}$$
7. $$\text{Butter needed} = \frac{4}{3} \times 12 = 16 \text{ g}$$
8. $$120 – 54 = 66 \text{ men}$$
9. $$25 \times \frac{4}{5} = 20 \text{ years}$$
10. $$\text{Ratio of apples} = 2:3$$
11. $$30 – 18 = 12 \text{ boys}$$
12. $$\text{Ratio} = 120 \text{ km/h}$$
13. $$\text{Ratio} = 120 \text{ litres}$$
14. $$40 \times \frac{3}{4} = 30 \text{ students}$$
15. $$20 \times \frac{1}{5} = 4 \text{ litres}$$
16. $$\text{Number of teachers} = \frac{120}{30} \times 5 = 20$$
17. $$\text{Cats} = 42 \times \frac{4}{3} = 56$$
18. $$\text{Flour} = 12 \text{ cups}$$
19. $$\text{Red apples} = 40 \times \frac{3}{8} = 15$$
20. $$\text{Girls} = 15 \times \frac{2}{3} = 10$$

These questions and answers provide a comprehensive overview of ratios and proportions relevant to the 11+ exam, covering various difficulty levels and encouraging students to develop their understanding and problem-solving skills.