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Dividing fractions and the four operations

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Notes de leçon

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.

Diapos

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Questions d'entraînement

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  1. 1.What is the reciprocal of 7?

    Easy
    • A7
    • B1/7
    • C7/1
    • D-7

  2. 2.What is the reciprocal of 3/4?

    Easy
    • A3/4
    • B4/3
    • C1/4
    • D3

  3. 3.The reciprocal of 0 is 0.

    Easy

    True or false?

  4. 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. 5.Match each number with its reciprocal.

    Easy
    • 5
    • 2/3
    • 1/4
    • 1/5
    • 3/2
    • 4

  6. 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. 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. 8.How many 1/2s are in 2?

    Easy
    • A2
    • B3
    • C4
    • D5

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