Year 8 Maths: Addition of Fractions

Adding fractions is an essential mathematical concept that students need to master during Key Stage 3. Understanding how to add fractions allows students to solve more complex problems involving fractions, ratios, and real-world applications. Fractions can be added either when they have the same denominator or when they have different denominators, but each requires a slightly different approach.

Understanding Addition of Fractions

Adding Fractions with the Same Denominator

When fractions have the same denominator (also called the common denominator), adding them is simple. You only need to add the numerators while keeping the denominator the same.

Example

$$\frac{3}{8} + \frac{2}{8} = \frac{3 + 2}{8} = \frac{5}{8}$$

Adding Fractions with Different Denominators

When the fractions have different denominators, the process involves finding a common denominator first, which is usually the lowest common denominator (LCD) of the two denominators. Once the fractions are rewritten with the same denominator, you can add them as usual.

Steps to Add Fractions with Different Denominators

1. Find the Lowest Common Denominator (LCD) for the denominators.
2. Rewrite each fraction so that both fractions have the LCD as their denominator.
3. Add the numerators and keep the denominator the same.
4. Simplify the fraction, if possible.

Example

$$\frac{1}{4} + \frac{1}{6}$$

• The LCD of 4 and 6 is 12.
• Rewrite the fractions:
  $$\frac{1}{4} = \frac{3}{12}, \quad \frac{1}{6} = \frac{2}{12}$$
• Add the numerators:
  $$\frac{3}{12} + \frac{2}{12} = \frac{5}{12}$$

Practice Questions

Easy Level

1. $$\frac{1}{5} + \frac{2}{5}$$
2. $$\frac{1}{7} + \frac{3}{7}$$
3. $$\frac{2}{8} + \frac{3}{8}$$
4. $$\frac{1}{4} + \frac{2}{4}$$
5. $$\frac{2}{6} + \frac{1}{6}$$
6. $$\frac{3}{9} + \frac{2}{9}$$
7. $$\frac{4}{10} + \frac{1}{10}$$
8. $$\frac{1}{3} + \frac{2}{3}$$
9. $$\frac{5}{8} + \frac{2}{8}$$
10. $$\frac{1}{12} + \frac{5}{12}$$
11. $$\frac{3}{5} + \frac{1}{5}$$
12. $$\frac{4}{9} + \frac{3}{9}$$
13. $$\frac{1}{2} + \frac{1}{2}$$
14. $$\frac{2}{7} + \frac{4}{7}$$
15. $$\frac{3}{8} + \frac{5}{8}$$
16. $$\frac{1}{10} + \frac{7}{10}$$
17. $$\frac{2}{12} + \frac{4}{12}$$
18. $$\frac{3}{4} + \frac{1}{4}$$
19. $$\frac{6}{9} + \frac{2}{9}$$
20. $$\frac{1}{6} + \frac{3}{6}$$

Medium Level

1. $$\frac{1}{3} + \frac{1}{4}$$
2. $$\frac{2}{5} + \frac{1}{3}$$
3. $$\frac{5}{8} + \frac{1}{4}$$
4. $$\frac{2}{6} + \frac{3}{4}$$
5. $$\frac{1}{3} + \frac{2}{6}$$
6. $$\frac{2}{7} + \frac{3}{14}$$
7. $$\frac{1}{2} + \frac{1}{3}$$
8. $$\frac{4}{5} + \frac{1}{2}$$
9. $$\frac{5}{6} + \frac{1}{4}$$
10. $$\frac{1}{3} + \frac{1}{5}$$
11. $$\frac{3}{8} + \frac{1}{6}$$
12. $$\frac{5}{9} + \frac{1}{3}$$
13. $$\frac{4}{7} + \frac{3}{5}$$
14. $$\frac{2}{5} + \frac{3}{8}$$
15. $$\frac{3}{4} + \frac{2}{5}$$
16. $$\frac{7}{9} + \frac{1}{6}$$
17. $$\frac{3}{10} + \frac{4}{5}$$
18. $$\frac{2}{3} + \frac{3}{8}$$
19. $$\frac{5}{6} + \frac{1}{2}$$
20. $$\frac{3}{7} + \frac{5}{6}$$

Hard Level

1. $$\frac{7}{10} + \frac{5}{12}$$
2. $$\frac{9}{14} + \frac{3}{5}$$
3. $$\frac{3}{8} + \frac{7}{15}$$
4. $$\frac{11}{12} + \frac{3}{5}$$
5. $$\frac{5}{9} + \frac{8}{15}$$
6. $$\frac{13}{18} + \frac{7}{12}$$
7. $$\frac{9}{14} + \frac{11}{21}$$
8. $$\frac{4}{9} + \frac{7}{12}$$
9. $$\frac{5}{7} + \frac{11}{14}$$
10. $$\frac{6}{11} + \frac{4}{9}$$
11. $$\frac{8}{15} + \frac{9}{20}$$
12. $$\frac{11}{18} + \frac{13}{24}$$
13. $$\frac{7}{8} + \frac{5}{12}$$
14. $$\frac{9}{16} + \frac{11}{24}$$
15. $$\frac{13}{20} + \frac{5}{9}$$
16. $$\frac{4}{7} + \frac{8}{15}$$
17. $$\frac{7}{12} + \frac{11}{18}$$
18. $$\frac{9}{14} + \frac{4}{9}$$
19. $$\frac{10}{13} + \frac{9}{15}$$
20. $$\frac{13}{16} + \frac{7}{20}$$

Here are the answers and explanations for all the questions, organized by difficulty level. I’ll go step by step, explaining how to find the common denominator and perform the addition for each set.

Easy Level Answers and Explanations

1. $$\frac{1}{5} + \frac{2}{5} = \frac{3}{5}$$

• Both fractions already have the same denominator (5). Just add the numerators.

1. $$\frac{1}{7} + \frac{3}{7} = \frac{4}{7}$$

• Same denominator (7). Add the numerators.

1. $$\frac{2}{8} + \frac{3}{8} = \frac{5}{8}$$

• Same denominator (8). Add the numerators.

1. $$\frac{1}{4} + \frac{2}{4} = \frac{3}{4}$$

• Same denominator (4). Add the numerators.

1. $$\frac{2}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2}$$

• Same denominator (6). Add the numerators and simplify.

1. $$\frac{3}{9} + \frac{2}{9} = \frac{5}{9}$$

• Same denominator (9). Add the numerators.

1. $$\frac{4}{10} + \frac{1}{10} = \frac{5}{10} = \frac{1}{2}$$

• Same denominator (10). Add the numerators and simplify.

1. $$\frac{1}{3} + \frac{2}{3} = \frac{3}{3} = 1$$

• Same denominator (3). Add the numerators.

1. $$\frac{5}{8} + \frac{2}{8} = \frac{7}{8}$$

• Same denominator (8). Add the numerators.

1. $$\frac{1}{12} + \frac{5}{12} = \frac{6}{12} = \frac{1}{2}$$
   • Same denominator (12). Add the numerators and simplify.
2. $$\frac{3}{5} + \frac{1}{5} = \frac{4}{5}$$
   • Same denominator (5). Add the numerators.
3. $$\frac{4}{9} + \frac{3}{9} = \frac{7}{9}$$
   • Same denominator (9). Add the numerators.
4. $$\frac{1}{2} + \frac{1}{2} = 1$$
   • Same denominator (2). Add the numerators.
5. $$\frac{2}{7} + \frac{4}{7} = \frac{6}{7}$$
   • Same denominator (7). Add the numerators.
6. $$\frac{3}{8} + \frac{5}{8} = \frac{8}{8} = 1$$
   • Same denominator (8). Add the numerators.
7. $$\frac{1}{10} + \frac{7}{10} = \frac{8}{10} = \frac{4}{5}$$
   • Same denominator (10). Add the numerators and simplify.
8. $$\frac{2}{12} + \frac{4}{12} = \frac{6}{12} = \frac{1}{2}$$
   • Same denominator (12). Add the numerators and simplify.
9. $$\frac{3}{4} + \frac{1}{4} = \frac{4}{4} = 1$$
   • Same denominator (4). Add the numerators.
10. $$\frac{6}{9} + \frac{2}{9} = \frac{8}{9}$$
    • Same denominator (9). Add the numerators.
11. $$\frac{1}{6} + \frac{3}{6} = \frac{4}{6} = \frac{2}{3}$$
    • Same denominator (6). Add the numerators and simplify.

Medium Level Answers and Explanations

1. $$\frac{1}{3} + \frac{1}{4} = \frac{4}{12} + \frac{3}{12} = \frac{7}{12}$$

• LCD: 12. Convert both fractions to have the same denominator.

1. $$\frac{2}{5} + \frac{1}{3} = \frac{6}{15} + \frac{5}{15} = \frac{11}{15}$$

• LCD: 15. Convert both fractions and add the numerators.

1. $$\frac{5}{8} + \frac{1}{4} = \frac{5}{8} + \frac{2}{8} = \frac{7}{8}$$

• LCD: 8. Convert $$\frac{1}{4}$$ to $$\frac{2}{8}$$ .

1. $$\frac{2}{6} + \frac{3}{4} = \frac{4}{12} + \frac{9}{12} = \frac{13}{12} = 1 \frac{1}{12}$$

• LCD: 12. Convert and simplify to a mixed fraction.

1. $$\frac{1}{3} + \frac{2}{6} = \frac{2}{6} + \frac{2}{6} = \frac{4}{6} = \frac{2}{3}$$

• LCD: 6. Simplify after addition.

1. $$\frac{2}{7} + \frac{3}{14} = \frac{4}{14} + \frac{3}{14} = \frac{7}{14} = \frac{1}{2}$$

• LCD: 14. Simplify after adding.

1. $$\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}$$

• LCD: 6.

1. $$\frac{4}{5} + \frac{1}{2} = \frac{8}{10} + \frac{5}{10} = \frac{13}{10} = 1 \frac{3}{10}$$

• LCD: 10. Simplify after adding.

1. $$\frac{5}{6} + \frac{1}{4} = \frac{10}{12} + \frac{3}{12} = \frac{13}{12} = 1 \frac{1}{12}$$

• LCD: 12. Simplify after adding.

1. $$\frac{1}{3} + \frac{1}{5} = \frac{5}{15} + \frac{3}{15} = \frac{8}{15}$$
   • LCD: 15.
2. $$\frac{3}{8} + \frac{1}{6} = \frac{9}{24} + \frac{4}{24} = \frac{13}{24}$$
   • LCD: 24.
3. $$\frac{5}{9} + \frac{1}{3} = \frac{5}{9} + \frac{3}{9} = \frac{8}{9}$$
   • LCD: 9.
4. $$\frac{4}{7} + \frac{3}{5} = \frac{20}{35} + \frac{21}{35} = \frac{41}{35} = 1 \frac{6}{35}$$
   • LCD: 35. Simplify after adding.
5. $$\frac{2}{5} + \frac{3}{8} = \frac{16}{40} + \frac{15}{40} = \frac{31}{40}$$
   • LCD: 40.
6. $$\frac{3}{4} + \frac{2}{5} = \frac{15}{20} + \frac{8}{20} = \frac{23}{20} = 1 \frac{3}{20}$$
   • LCD: 20. Simplify after adding.
7. $$\frac{7}{9} + \frac{1}{6} = \frac{14}{18} + \frac{3}{18} = \frac{17}{18}$$
   • LCD: 18.
8. $$\frac{3}{10} + \frac{4}{5} = \frac{3}{10} + \frac{8}{10} = \frac{11}{10} = 1 \frac{1}{10}$$
   • LCD: 10. Simplify after adding.
9. $$\frac{2}{3} + \frac{3}{8} = \frac{16}{24} + \frac{9}{24} = \frac{25}{24} = 1 \frac{1}{24}$$
   • LCD: 24. Simplify after adding.
10. $$ \frac{5}{6} + \frac{1}{2} = \frac{5}{6} + \frac{

3}{6} = \frac{8}{6} = 1 \frac{2}{6} = 1 \frac{1}{3} $$
– LCD: 6. Simplify after adding.

20. $$\frac{3}{7} + \frac{5}{6} = \frac{18}{42} + \frac{35}{42} = \frac{53}{42} = 1 \frac{11}{42}$$
    • LCD: 42. Simplify after adding.

Hard Level Answers and Explanations

1. $$\frac{7}{10} + \frac{5}{12} = \frac{84}{120} + \frac{50}{120} = \frac{134}{120} = 1 \frac{14}{120} = 1 \frac{7}{60}$$

• LCD: 120. Simplify after adding.

1. $$\frac{9}{14} + \frac{3}{5} = \frac{45}{70} + \frac{42}{70} = \frac{87}{70} = 1 \frac{17}{70}$$

• LCD: 70. Simplify after adding.

1. $$\frac{3}{8} + \frac{7}{15} = \frac{45}{120} + \frac{56}{120} = \frac{101}{120}$$

• LCD: 120.

1. $$\frac{11}{12} + \frac{3}{5} = \frac{55}{60} + \frac{36}{60} = \frac{91}{60} = 1 \frac{31}{60}$$

• LCD: 60. Simplify after adding.

1. $$\frac{5}{9} + \frac{8}{15} = \frac{25}{45} + \frac{24}{45} = \frac{49}{45} = 1 \frac{4}{45}$$

• LCD: 45. Simplify after adding.

1. $$\frac{13}{18} + \frac{7}{12} = \frac{52}{72} + \frac{42}{72} = \frac{94}{72} = 1 \frac{22}{72} = 1 \frac{11}{36}$$

• LCD: 72. Simplify after adding.

1. $$\frac{9}{14} + \frac{11}{21} = \frac{27}{42} + \frac{22}{42} = \frac{49}{42} = 1 \frac{7}{42} = 1 \frac{1}{6}$$

• LCD: 42. Simplify after adding.

1. $$\frac{4}{9} + \frac{7}{12} = \frac{16}{36} + \frac{21}{36} = \frac{37}{36} = 1 \frac{1}{36}$$

• LCD: 36. Simplify after adding.

1. $$\frac{5}{7} + \frac{11}{14} = \frac{10}{14} + \frac{11}{14} = \frac{21}{14} = 1 \frac{7}{14} = 1 \frac{1}{2}$$

• LCD: 14. Simplify after adding.

1. $$\frac{6}{11} + \frac{4}{9} = \frac{54}{99} + \frac{44}{99} = \frac{98}{99}$$
   • LCD: 99.
2. $$\frac{8}{15} + \frac{9}{20} = \frac{160}{300} + \frac{135}{300} = \frac{295}{300}$$
   • LCD: 300.
3. $$\frac{11}{18} + \frac{13}{24} = \frac{88}{144} + \frac{78}{144} = \frac{166}{144} = 1 \frac{22}{144} = 1 \frac{11}{72}$$
   • LCD: 144. Simplify after adding.
4. $$\frac{7}{8} + \frac{5}{12} = \frac{21}{24} + \frac{10}{24} = \frac{31}{24} = 1 \frac{7}{24}$$
   • LCD: 24.
5. $$\frac{9}{16} + \frac{11}{24} = \frac{27}{48} + \frac{22}{48} = \frac{49}{48} = 1 \frac{1}{48}$$
   • LCD: 48.
6. $$\frac{13}{20} + \frac{5}{9} = \frac{117}{180} + \frac{100}{180} = \frac{217}{180} = 1 \frac{37}{180}$$
   • LCD: 180.
7. $$\frac{4}{7} + \frac{8}{15} = \frac{60}{105} + \frac{56}{105} = \frac{116}{105} = 1 \frac{11}{105}$$
   • LCD: 105.
8. $$\frac{7}{12} + \frac{11}{18} = \frac{21}{36} + \frac{22}{36} = \frac{43}{36} = 1 \frac{7}{36}$$
   • LCD: 36.
9. $$\frac{9}{14} + \frac{4}{9} = \frac{81}{126} + \frac{56}{126} = \frac{137}{126} = 1 \frac{11}{126}$$
   • LCD: 126.
10. $$\frac{10}{13} + \frac{9}{15} = \frac{150}{195} + \frac{117}{195} = \frac{267}{195} = 1 \frac{72}{195} = 1 \frac{24}{65}$$
    • LCD: 195. Simplify after adding.
11. $$\frac{13}{16} + \frac{7}{20} = \frac{65}{80} + \frac{28}{80} = \frac{93}{80} = 1 \frac{13}{80}$$
    • LCD: 80. Simplify after adding.

That completes the answers and explanations for all levels. Each question requires finding the Lowest Common Denominator (LCD), converting the fractions, adding them, and simplifying when necessary.