Dividing fractions

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1. Dividing fractions

Warm-up

Three quick questions from earlier lessons.

Discover

How many 14\tfrac{1}{4}-cup scoops are in 3 cups?

  1. 12
  2. 34\tfrac{3}{4}

Key idea

Divide by a fraction: multiply by its reciprocal.

ab÷cd=ab⋅dc\dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \cdot \dfrac{d}{c}. Division answers “how many groups?” or “how much in one group?”. Dividing by a number less than 1 gives a result larger than the starting number.

Worked example

23÷45\tfrac{2}{3} \div \tfrac{4}{5}.

23⋅54\tfrac{2}{3} \cdot \tfrac{5}{4}
Reciprocal of 4/5.
1012=56\tfrac{10}{12} = \tfrac{5}{6}
Multiply.

Your turn, guided

Your turn: 34÷12\tfrac{3}{4} \div \tfrac{1}{2}.

32\tfrac{3}{2}
3/4 · 2.

Practice ★

  1. How many 12\tfrac{ 1 }{ 2 }-cup servings are in 7 cups of rice?
  2. How many 14\tfrac{ 1 }{ 4 }-cup servings are in 2 cups of rice?
  1. Compute 78÷59\tfrac{ 7 }{ 8 } \div \tfrac{ 5 }{ 9 }.

Watch out!

Practice ★★

  1. 59\tfrac{ 5 }{ 9 } of a bag of flour weighs 30 kg. How much does the whole bag weigh (kg)?
  2. 25\tfrac{ 2 }{ 5 } of a bag of flour weighs 6 kg. How much does the whole bag weigh (kg)?
  1. Boss problem. A ribbon is 4 yards long. It is cut into pieces 24\tfrac{ 2 }{ 4 } yard long.

Exit ticket

Three questions to check you have got it.

  1. Compute 78÷59\tfrac{ 7 }{ 8 } \div \tfrac{ 5 }{ 9 }.
  2. Compute 16÷46\tfrac{ 1 }{ 6 } \div \tfrac{ 4 }{ 6 }.
  3. Compute 23÷45\tfrac{ 2 }{ 3 } \div \tfrac{ 4 }{ 5 }.

Summary card

Dividing fractions

Multiply by the reciprocal of the divisor.

Watch out: flip the second fraction, not the first.

2. Area and volume with fractional lengths

Warm-up

Three quick questions from earlier lessons.

Discover

A rectangle is 12\tfrac{1}{2} m by 13\tfrac{1}{3} m. Its area is what part of a square meter?

  1. 16\tfrac{1}{6}
  2. 56\tfrac{5}{6}

Key idea

Area and volume formulas work with fractions too.

Area of a rectangle =ℓ⋅w= \ell \cdot w and volume of a box =ℓ⋅w⋅h= \ell \cdot w \cdot h also for fractional lengths: multiply numerators and denominators. A unit cube holds k3k^3 cubes of edge 1k\tfrac{1}{k}.

Worked example

A box 12\tfrac{1}{2} by 3 by 23\tfrac{2}{3}.

12⋅3⋅23=1\tfrac{1}{2} \cdot 3 \cdot \tfrac{2}{3} = 1 cubic unit
Multiply.

Your turn, guided

Your turn: a square with side 34\tfrac{3}{4}. Its area.

916\tfrac{9}{16}
3/4 · 3/4.

Practice ★

  1. A rectangle is 15\tfrac{ 1 }{ 5 } m by 25\tfrac{ 2 }{ 5 } m. What is its area (m²)?
  2. A rectangle is 45\tfrac{ 4 }{ 5 } m by 56\tfrac{ 5 }{ 6 } m. What is its area (m²)?
  1. A box is 12\tfrac{ 1 }{ 2 } ft by 7 ft by 13\tfrac{ 1 }{ 3 } ft. What is its volume (ft³)?

Watch out!

Practice ★★

  1. How many cubes with edge 13\tfrac{ 1 }{ 3 } unit fill a box 2 by 1 by 3 units?
  2. How many cubes with edge 12\tfrac{ 1 }{ 2 } unit fill a box 2 by 1 by 1 units?

Exit ticket

Three questions to check you have got it.

  1. A rectangle is 15\tfrac{ 1 }{ 5 } m by 25\tfrac{ 2 }{ 5 } m. What is its area (m²)?
  2. A rectangle is 45\tfrac{ 4 }{ 5 } m by 56\tfrac{ 5 }{ 6 } m. What is its area (m²)?
  3. A rectangle is 16\tfrac{ 1 }{ 6 } m by 23\tfrac{ 2 }{ 3 } m. What is its area (m²)?

Summary card

Fractional lengths

Multiply lengths for areas and volumes, fractions included.

Watch out: a 1/k cube is 1/k³ of a unit cube.

Answers: Dividing fractions

1. Dividing fractions

A

  1. 1414
  2. 88

B

  1. 6340\frac{63}{40}

C

  1. 5454
  2. 1515

D

  1. 14\frac{1}{4}

E

  1. 6340\frac{63}{40}
  2. 14\frac{1}{4}
  3. 56\frac{5}{6}

2. Area and volume with fractional lengths

A

  1. 225\frac{2}{25}
  2. 23\frac{2}{3}

B

  1. 76\frac{7}{6}

C

  1. 162162
  2. 1616

D

  1. 225\frac{2}{25}
  2. 23\frac{2}{3}
  3. 19\frac{1}{9}