Dividing fractions and the four operations
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What is a reciprocal?
- The reciprocal of a number is what you multiply it by to get 1.
- For a fraction like \frac{a}{b}, its reciprocal is \frac{b}{a}.
- Example: The reciprocal of \frac{2}{3} is \frac{3}{2}, because \frac{2}{3} \times \frac{3}{2} = 1.
- The reciprocal of a whole number like 5 is \frac{1}{5}, because 5 \times \frac{1}{5} = 1.
- Zero has no reciprocal, because nothing times zero equals 1.
Dividing fractions: the rule
- To divide by a fraction, multiply by its reciprocal.
- So \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}.
- Example: \frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8}.
- This works because dividing by a number is the same as multiplying by its reciprocal.
Why does it work?
- Think of division as asking: how many of one fraction fit into another?
- For example, \frac{1}{2} \div \frac{1}{4} asks: how many quarters fit into a half? The answer is 2.
- Using the rule: \frac{1}{2} \times \frac{4}{1} = 2.
- Diagrams can help: draw a rectangle, split it into halves, then into quarters, and count.
Dividing whole numbers by fractions
- A whole number can be written as a fraction with denominator 1.
- Example: 3 \div \frac{1}{2} is the same as \frac{3}{1} \times \frac{2}{1} = 6.
- So 3 \div \frac{1}{2} = 6 because there are 6 halves in 3.
- Always convert the whole number to a fraction first.
Dividing fractions by whole numbers
- To divide a fraction by a whole number, write the whole number as a fraction with denominator 1.
- Example: \frac{2}{3} \div 4 = \frac{2}{3} \times \frac{1}{4} = \frac{2}{12} = \frac{1}{6}.
- So \frac{2}{3} \div 4 = \frac{1}{6}.
- You can also think: split \frac{2}{3} into 4 equal parts.
Mixed numbers and improper fractions
- A mixed number has a whole part and a fraction part, like 2\frac{1}{3}.
- An improper fraction has a numerator bigger than the denominator, like \frac{7}{3}.
- Before dividing, change mixed numbers to improper fractions.
- Example: 2\frac{1}{3} \div \frac{5}{6} becomes \frac{7}{3} \div \frac{5}{6} = \frac{7}{3} \times \frac{6}{5} = \frac{42}{15} = \frac{14}{5}.
- Always simplify your answer if you can.
The four operations with fractions
- Addition and subtraction: need the same denominator. Example: \frac{1}{4} + \frac{1}{6} = \frac{3}{12} + \frac{2}{12} = \frac{5}{12}.
- Multiplication: multiply the tops and the bottoms. Example: \frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}.
- Division: multiply by the reciprocal. Example: \frac{2}{3} \div \frac{3}{4} = \frac{2}{3} \times \frac{4}{3} = \frac{8}{9}.
- Always simplify your final answer.
Real-world problems
- Division of fractions can solve sharing problems.
- Example: If you have \frac{3}{4} of a pizza and each person gets \frac{1}{8}, how many people can you serve? \frac{3}{4} \div \frac{1}{8} = \frac{3}{4} \times \frac{8}{1} = 6.
- Rates: If a car uses \frac{1}{2} litre of petrol per kilometre, how far can it go on \frac{5}{2} litres? \frac{5}{2} \div \frac{1}{2} = 5 km.
- Check your answer: does it make sense in the story?
Common mistakes to avoid
- Do not forget to flip the second fraction when dividing.
- Do not flip the first fraction.
- Always convert mixed numbers to improper fractions before dividing.
- Always simplify your answer to lowest terms.
- Remember: zero has no reciprocal, so you cannot divide by zero.
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1.What is the reciprocal of 7?
Easy- A7
- B1/7
- C7/1
- D-7
2.What is the reciprocal of 3/4?
Easy- A3/4
- B4/3
- C1/4
- D3
3.The reciprocal of 0 is 0.
EasyTrue or false?
4.Which expression is equal to 2/3 ÷ 4/5?
Easy- A2/3 × 5/4
- B2/3 × 4/5
- C3/2 × 4/5
- D3/2 × 5/4
5.Match each number with its reciprocal.
Easy- 5
- 2/3
- 1/4
- 1/5
- 3/2
- 4
6.Put these steps in the correct order to divide 2/3 by 4/5.
Medium- Find the reciprocal of 4/5
- Multiply 2/3 by 5/4
- Simplify the product
7.Which of the following are equivalent to dividing by 3/5? (Select all that apply)
Medium- AMultiplying by 5/3
- BMultiplying by 3/5
- CDividing by 5/3
- DMultiplying by the reciprocal of 3/5
8.How many 1/2s are in 2?
Easy- A2
- B3
- C4
- D5
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