11+ Maths: Fractions and Decimals

Fractions and decimals are two fundamental concepts in mathematics that express parts of a whole. Understanding how to work with fractions and decimals is essential for solving a variety of mathematical problems, particularly in the 11+ exam.

Key Concepts

1. Fractions

A fraction represents a part of a whole and consists of two numbers:

• Numerator: The top part, indicating how many parts we have.
• Denominator: The bottom part, indicating how many equal parts the whole is divided into.

For example, in the fraction $$\frac{3}{4}$$, 3 is the numerator and 4 is the denominator.

2. Types of Fractions

• Proper Fractions: The numerator is less than the denominator (e.g., $$\frac{3}{4}$$).
• Improper Fractions: The numerator is greater than or equal to the denominator (e.g., $$\frac{5}{4}$$).
• Mixed Numbers: A whole number combined with a proper fraction (e.g., $$1\frac{1}{4}$$).

3. Operations with Fractions

• Addition: To add fractions with the same denominator, add the numerators. For different denominators, find a common denominator first.
• Subtraction: Similar to addition, ensure the denominators are the same.
• Multiplication: Multiply the numerators together and the denominators together (e.g., $$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$).
• Division: Multiply by the reciprocal of the second fraction (e.g., $$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$$).

4. Decimals

Decimals are another way to represent fractions, particularly those with denominators of 10, 100, etc. The decimal point separates the whole number part from the fractional part.

For example:

• The fraction $$\frac{1}{10}$$ can be expressed as 0.1 in decimal form.
• The fraction $$\frac{3}{4}$$ can be expressed as 0.75.

5. Converting Between Fractions and Decimals

To convert a fraction to a decimal, divide the numerator by the denominator. To convert a decimal to a fraction, write the decimal as a fraction with a power of 10 in the denominator and simplify if possible.

Practice Questions on Fractions and Decimals

Easy Level

1. What is $$\frac{1}{2} + \frac{1}{2}$$?
2. Convert $$\frac{3}{4}$$ to a decimal.
3. What is $$\frac{2}{5} – \frac{1}{5}$$?
4. If you have 0.25 of a cake, what fraction of the cake is that?
5. Add $$\frac{1}{3} + \frac{1}{3}$$.
6. Convert 0.5 to a fraction.
7. What is $$\frac{1}{4} + \frac{2}{4}$$?
8. Convert $$\frac{7}{10}$$ to a decimal.
9. If you have $$0.75$$ of a pizza, what fraction of the pizza is that?
10. What is $$\frac{3}{8} + \frac{1}{8}$$?
11. Convert 0.2 to a fraction.
12. What is $$\frac{5}{10}$$ as a decimal?
13. What is $$\frac{2}{3} – \frac{1}{3}$$?
14. Convert $$0.6$$ to a fraction.
15. If you have $$0.125$$ of a pound, what fraction of a pound is that?
16. Add $$\frac{2}{5} + \frac{1}{5}$$.
17. Convert $$\frac{4}{10}$$ to a decimal.
18. What is $$\frac{1}{2} – \frac{1}{4}$$?
19. Convert 0.75 to a fraction.
20. What is $$\frac{6}{8}$$ as a decimal?

Medium Level

1. What is $$\frac{3}{5} + \frac{2}{5}$$?
2. Convert $$\frac{5}{6}$$ to a decimal.
3. If a recipe requires $$\frac{3}{4}$$ of a cup of sugar, how much is that in decimal form?
4. What is $$\frac{3}{10} – \frac{1}{10}$$?
5. Convert 0.2 to a fraction in simplest form.
6. What is $$\frac{7}{8} + \frac{1}{8}$$?
7. If you have $$0.3$$ of a cake, what fraction of the cake is that?
8. Add $$\frac{2}{3} + \frac{1}{3}$$.
9. Convert $$0.125$$ to a fraction.
10. What is $$\frac{4}{5} – \frac{1}{5}$$?
11. Convert 0.4 to a fraction.
12. If a drink is 0.5 litres, how many millilitres is that?
13. What is $$\frac{2}{6}$$ in simplest form as a fraction?
14. If you have $$0.9$$ of a pizza, what fraction of the pizza is that?
15. Convert $$\frac{9}{10}$$ to a decimal.
16. What is $$\frac{1}{5} + \frac{2}{5}$$?
17. Convert 0.8 to a fraction in simplest form.
18. If a piece of chocolate weighs 0.2 kg, what is that in grams?
19. What is $$\frac{1}{2} + \frac{2}{4}$$?
20. Convert $$\frac{5}{12}$$ to a decimal.

Hard Level

1. If a piece of fabric is 2.5 metres long, what is that in centimetres?
2. What is $$\frac{4}{5} + \frac{3}{10}$$?
3. Convert $$0.625$$ to a fraction.
4. If a recipe requires $$\frac{5}{8}$$ of a cup of flour, how much is that in decimal form?
5. What is $$\frac{7}{12} – \frac{1}{3}$$?
6. Convert 1.75 to a fraction.
7. If a jar holds 2.2 litres of jam, how many millilitres is that?
8. What is $$\frac{5}{9} + \frac{1}{3}$$?
9. Convert $$\frac{11}{20}$$ to a decimal.
10. If a runner completed $$0.9$$ of a race, what fraction of the race did they complete?
11. What is $$\frac{7}{15} + \frac{1}{5}$$?
12. If you have 0.125 of a cake, what fraction of the cake is that in simplest form?
13. Convert $$\frac{13}{25}$$ to a decimal.
14. What is $$\frac{2}{7} – \frac{1}{14}$$?
15. If a car travels $$0.6$$ of a mile, what is that in fractions?
16. What is $$\frac{3}{4} + \frac{1}{8}$$?
17. Convert $$0.03$$ to a fraction in simplest form.
18. If a book is 1.5 kg, what is that in grams?
19. What is $$\frac{5}{16} + \frac{3}{8}$$?
20. Convert $$\frac{2}{3}$$ to a decimal.

Answers and Explanations

Easy Level

1. $$ \frac{1}{2} + \frac{1}{2} = 1 $$
2. $$ \frac{3}{4} = 0.75 $$
3. $$ \frac{2}{5} – \frac{1}{5} = \frac{1}{5} $$
4. $$ 0.25 = \frac{1}{4} $$
5. $$ \frac{1}{3} + \frac{1}{3} = \frac{2}{3} $$
6. $$ 0.5 = \frac{1}{2} $$
7. $$ \frac{1}{4} + \frac{2}{4} = \frac{3}{4} $$
8. $$ \frac{7}{10} = 0.7 $$
9. $$ 0.75 = \frac{3}{4} $$
10. $$ \frac{3}{8} + \frac{1}{8} = \frac{4}{8} = \frac{1}{2} $$
11. $$ 0.2 = \frac{1}{5} $$
12. $$ \frac{5}{10} = 0.5 $$
13. $$ \frac{2}{3} – \frac{1}{3} = \frac{1}{3} $$
14. $$ 0.6 = \frac{3}{5} $

$

15. $$ 0.125 = \frac{1}{8} $$
16. $$ \frac{2}{5} + \frac{1}{5} = \frac{3}{5} $$
17. $$ \frac{4}{10} = 0.4 $$
18. $$ \frac{1}{2} – \frac{1}{4} = \frac{1}{4} $$
19. $$ 0.75 = \frac{3}{4} $$
20. $$ \frac{6}{8} = 0.75 $$

Medium Level

1. $$ \frac{3}{5} + \frac{2}{5} = \frac{5}{5} = 1 $$
2. $$ \frac{5}{6} = 0.8333… $$
3. $$ \frac{3}{4} = 0.75 $$
4. $$ \frac{3}{10} – \frac{1}{10} = \frac{2}{10} = \frac{1}{5} $$
5. $$ 0.2 = \frac{1}{5} $$
6. $$ \frac{7}{8} + \frac{1}{8} = 1 $$
7. $$ 0.3 = \frac{3}{10} $$
8. $$ \frac{2}{3} + \frac{1}{3} = 1 $$
9. $$ 0.125 = \frac{1}{8} $$
10. $$ \frac{4}{5} – \frac{1}{5} = \frac{3}{5} $$
11. $$ 0.4 = \frac{2}{5} $$
12. $$ \frac{90}{100} = 0.9 $$
13. $$ \frac{3}{4} + \frac{1}{8} = \frac{6}{8} + \frac{1}{8} = \frac{7}{8} $$
14. $$ 0.3 = \frac{3}{10} $$
15. $$ \frac{1}{2} – \frac{1}{4} = \frac{1}{4} $$
16. $$ \frac{5}{9} + \frac{1}{3} = \frac{5}{9} + \frac{3}{9} = \frac{8}{9} $$
17. $$ 0.4 = \frac{2}{5} $$
18. $$ \frac{4}{5} + \frac{1}{5} = 1 $$
19. $$ 0.125 = \frac{1}{8} $$
20. $$ \frac{1}{4} = 0.25 $$

Hard Level

1. $$ \text{Volume} = 5 \times 3 \times 2 = 30 \text{ m}^3 $$
2. $$ 0.5 \text{ litres} = 500 \text{ ml} $$
3. $$ 1.8 \text{ l} = 1800 \text{ ml} $$
4. $$ \text{Area} = 9 \times 6 = 54 \text{ m}^2 $$
5. $$ \text{Volume} = 8 \times 4 \times 3 = 96 \text{ cm}^3 $$
6. $$ 150 \text{ km} \div 1.5 \text{ hours} = 100 \text{ km/h} $$
7. $$ 5000 \text{ ml} = 5 \text{ litres} $$
8. $$ 300 \text{ g} \text{ for 5 servings} \Rightarrow 300 \div 5 = 60 \text{ g per serving} \Rightarrow 8 \text{ servings} = 480 \text{ g} $$
9. $$ 250 \text{ cm} = 2.5 \text{ m} $$
10. $$ 750 \text{ grams} = 0.75 \text{ kg} $$
11. $$ 1.2 \text{ litres} = 1200 \text{ ml} $$
12. $$ \text{Perimeter} = 2 \times (10 + 4) = 28 \text{ m} $$
13. $$ 300 – 1 = 299 \text{ grams} $$
14. $$ 750 \text{ grams} = 0.75 \text{ kg} $$
15. $$ 3.14 \times 2^2 \times 10 \approx 125.6 \text{ cm}^3 $$
16. $$ 150 \text{ cm} = 1.5 \text{ m} $$
17. $$ 10 \text{ m}^2 = 200 \text{ cm}^3 $$
18. $$ 25 \text{ grams} $$
19. $$ 5 \text{ kg} = 5000 \text{ grams} $$
20. $$ 150 \text{ grams} = 0.15 \text{ kg} $$

This set of questions and answers provides a comprehensive overview of fractions and decimals relevant to the 11+ exam, covering various difficulty levels and encouraging students to develop their understanding and problem-solving skills.