Unit 5

Dividing fractions

Find out how many of one fraction fit inside another, and see why "flip and multiply" is not a magic trick.

1. Read

   How many fit?

   A tailor in Surat has a roll of ribbon 3 metres long. Each party favour needs a piece $\tfrac14$ of a metre long. How many favours can she make?

   You can probably see it without any rule. Each metre gives 4 pieces, so 3 metres give $3 \times 4 = 12$ pieces. In symbols, $3 \div \tfrac14 = 12$.

   Notice something odd. We divided, and the answer (12) is much bigger than what we started with (3). By the end of this reading you will know why that is not odd at all, and why a rule that sounds like a magic trick ("flip the second fraction and multiply") always works.

   Division means "how many fit?"

   There are two ways to think about $a \div b$:

   • Sharing: split $a$ into $b$ equal groups. How big is each group?
   • Measuring: how many copies of $b$ fit inside $a$?

   Sharing a quantity into "$\tfrac14$ of a group" does not make much sense. Measuring does. So for dividing by a fraction we use the measuring meaning:

   $\tfrac34 \div \tfrac18 \quad\text{asks:}\quad \text{how many } \tfrac18\text{s fit inside } \tfrac34\,?$

   Method 1: draw it with same-sized pieces

   

   Cut one whole into eighths. Three quarters is the same as six eighths, so six pieces are shaded. Each shaded piece is one $\tfrac18$. Count them: 6. So

   $\tfrac34 \div \tfrac18 = 6.$

   Example. $\tfrac56 \div \tfrac13$. Write $\tfrac13$ as $\tfrac26$. Now $\tfrac56$ is 5 sixths and the divisor is 2 sixths. How many groups of 2 fit in 5? Two whole groups and half of another, so $2\tfrac12$.

   Once both fractions are written with the same denominator, the denominators are like units that can be ignored, and you are just dividing numerators: "5 sixths ÷ 2 sixths = 5 ÷ 2".

   When the divisor does not fit exactly

   

   Now $\tfrac23 \div \tfrac45$. Use fifteenths, because $3 \times 5 = 15$:

   $\tfrac23 = \tfrac{10}{15}, \qquad \tfrac45 = \tfrac{12}{15}.$

   So the question is "how many groups of 12 fit in 10?" Not even one whole group. Only $\tfrac{10}{12}$ of a group fits:

   $\tfrac23 \div \tfrac45 = \tfrac{10}{12} = \tfrac56.$

   Misconception. "Dividing always makes the answer smaller." That is true when you divide by a number greater than 1 (sharing 6 sweets among 3 children). It is false when you divide by a number less than 1: the pieces you are counting are smaller than 1, so more of them fit. $3 \div \tfrac14 = 12$ is bigger than 3. And $\tfrac23 \div \tfrac45 = \tfrac56$ is smaller than 1 because the group (four fifths) is bigger than what we have (two thirds).

   A quick check you can always make: if the divisor is less than 1, the answer is bigger than the dividend; if the divisor is more than 1, it is smaller.

   Method 2: where "multiply by the reciprocal" comes from

   The reciprocal of a fraction is what you get by turning it upside down. The reciprocal of $\tfrac38$ is $\tfrac83$; the reciprocal of 5 is $\tfrac15$. A number times its reciprocal is always 1.

   Look at the ribbon again. How many $\tfrac14$s fit in 1 whole? Four. So how many fit in 3 wholes? $3 \times 4$. Dividing by $\tfrac14$ is the same as multiplying by 4, which is the reciprocal.

   The same works for any fraction. How many $\tfrac{c}{d}$ fit in 1? Cut 1 into $d$ parts: each part is $\tfrac1d$, and $\tfrac cd$ is $c$ of them. So one group takes $c$ parts out of $d$, and $\tfrac dc$ groups fit in 1. (Check: $\tfrac38$ in 1: $\tfrac83 = 2\tfrac23$ groups. Two full groups use $\tfrac68$, and $\tfrac28$ is left, which is $\tfrac23$ of a group. Yes.)

   So the number of $\tfrac cd$s that fit in $\tfrac ab$ is $\tfrac ab$ times as many as fit in 1:

   $\frac ab \div \frac cd = \frac ab \times \frac dc .$

   Only the second fraction (the divisor) is turned upside down. Flip the wrong one and you answer a different question.

   Example. $\tfrac34 \div \tfrac18 = \tfrac34 \times \tfrac81 = \tfrac{24}{4} = 6$. It matches the picture.

   Example. $\tfrac23 \div \tfrac45 = \tfrac23 \times \tfrac54 = \tfrac{10}{12} = \tfrac56$. It matches the fifteenths.

   Example, with a mixed number. $1\tfrac12 \div \tfrac34$. Change $1\tfrac12$ to $\tfrac32$ first: $\tfrac32 \times \tfrac43 = \tfrac{12}{6} = 2$. Check: two groups of $\tfrac34$ make $\tfrac64 = 1\tfrac12$. ✓

   Words, and the other direction

   Division and multiplication undo each other. If a rectangle of area $\tfrac34\ \text{m}^2$ is $\tfrac12$ m wide, its length is

   $\tfrac34 \div \tfrac12 = \tfrac34 \times 2 = \tfrac32 \text{ m} = 1\tfrac12 \text{ m},$

   because $\tfrac12 \times \tfrac32 = \tfrac34$. In general $a \div b$ is the number that, multiplied by $b$, gives $a$.

   Example (a rate). A cyclist covers $\tfrac38$ km in $\tfrac14$ hour. How far does she go in a whole hour? A whole hour holds $1 \div \tfrac14 = 4$ quarter-hours, so she covers $4$ times as far:

   $\tfrac38 \div \tfrac14 = \tfrac38 \times 4 = \tfrac{12}{8} = 1\tfrac12 \text{ km in an hour}.$

   Her speed is $1\tfrac12$ km/h.

   Which method when?

   Situation	Handy method
   Small, friendly fractions	Draw it, or use a common denominator
   Big or awkward fractions	Multiply by the reciprocal
   You are not sure the answer is sensible	Estimate: is the divisor above or below 1?

   Whatever the method, finish by asking: does the size of my answer make sense?

   Summary

   • $a \div b$ with fractions asks "how many $b$s fit in $a$?"
   • With a common denominator, $\tfrac{m}{n} \div \tfrac{k}{n} = m \div k$.
   • Dividing by a fraction is multiplying by its reciprocal, because $\tfrac dc$ copies of $\tfrac cd$ fit in 1.
   • Dividing by a number less than 1 gives a bigger result.
   • Only the divisor is flipped.

2. Lab

   How many fit?

   Each problem shows a bar cut into equal pieces with the first fraction shaded. Take groups the size of the second fraction until the shaded part is used up, then type how many groups you took, including any part of a group. Afterwards the lab shows the same answer from the reciprocal shortcut.

   Your goals

   • Solve all six fraction divisions by taking groups on the bar.
   • Notice that the picture and the "multiply by the reciprocal" shortcut always agree.

   🔒 The interactive lab opens once you register. Register to do this.

3. Quick check

   Dividing fractions check

   Eight questions on dividing by fractions. Draw a bar if it helps.

   1. 1.A 3 metre ribbon is cut into pieces that are 1/4 metre long. How many pieces?

      Answer with a number

   2. 2.3/4 ÷ 1/8 = ?

      Answer with a number

   3. 3.What is 2/3 ÷ 4/5?

      • 5/6
      • 8/15
      • 6/5
      • 15/8

      Choose one

   4. 4.5/6 ÷ 1/3 = ? Give your answer as a decimal.

      Answer with a number

   5. 5.Which of these equal 6 ÷ 2/3? Choose all that apply.

      • 6 × 3/2
      • 9
      • 6 × 2/3
      • 4
      • 18/2

      Choose all that apply

   6. 6.Without working it out fully, which division has an answer greater than 5?

      • 5 ÷ 1/2
      • 5 ÷ 2
      • 5 ÷ 1
      • 5 ÷ 3/2

      Choose one

   7. 7.One batch of parathas uses 3/4 cup of flour. Rani has 4 1/2 cups. How many batches can she make?

      Answer with a number

   8. 8.Match each division to its answer.

      • 1/2 ÷ 1/4
      • 3/4 ÷ 1/2
      • 1/4 ÷ 3/4
      • 2 ÷ 1/2

      Match each to: 1/3 · 4 · 2 · 1 1/2

   🔒 Answering and checking are for registered learners. Register to do this.

4. Quick check

   New puzzles with fraction division

   Use fraction division in situations that are new. Decide first what is being divided by what.

   1. 1.A rope 3 1/2 m long is cut into pieces of 7/8 m each. How many pieces are there?

      Answer with a number

   2. 2.A rectangular mat has area 3/4 square metres and is 1/2 m wide. How long is it, in metres? Give a decimal.

      Answer with a number

   3. 3.Which story is described by 3/4 ÷ 1/2?

      • How many half-litre bottles can be filled from 3/4 of a litre of juice?
      • What is half of 3/4 of a litre of juice?
      • What is 3/4 of a half-litre bottle?
      • How much juice does each of 3/4 of a child get from half a litre?

      Choose one

   4. 4.A snail crawls 3/8 m in 1/4 hour at a steady pace. How many metres does it crawl in one hour? Give a decimal.

      Answer with a number

   🔒 Answering and checking are for registered learners. Register to do this.

5. Your turn

   Explain dividing fractions

   Write in your own words. Describe or sketch bars where it helps.

   1. A friend says: "Division always makes things smaller, so $3 \div \tfrac14$ must be less than 3." Work out $3 \div \tfrac14$ and explain, with a real situation, why the friend's rule fails.
   2. Work out $\tfrac23 \div \tfrac45$ in two ways: (a) with a picture or common denominators, (b) by multiplying by a reciprocal. Explain why the two answers must agree.
   3. Explain why dividing by $\tfrac38$ is the same as multiplying by $\tfrac83$. Start from "how many $\tfrac38$ fit in 1?".

   How it's judged

   • Dividing by a number less than 1
   • Two methods agree
   • Why the reciprocal works

   🔒 Writing and submitting your answer is for registered learners. Register to do this.