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Dividing fractions

Numbers · level 4, lesson 3 of 7

A question to start

You have 3 pizzas, and each person at a party eats a quarter of a pizza. How many people can you feed?

Each pizza feeds 4 people, so 3 pizzas feed \(3 \times 4 = 12\). That question was \(3 \div \tfrac14\), and the answer was 12: dividing by a quarter gave the same answer as multiplying by 4. Coincidence? No. This lesson shows why it always happens, and turns it into the famous rule "to divide by a fraction, multiply by its reciprocal", with two different reasons why the rule is true. A rule you can explain is a rule you will never get backwards.

The idea

What dividing asks

\(12 \div 3\) asks two questions that have the same answer:

  • Sharing: 12 shared into 3 equal groups; how much in each? (4)
  • Measuring: how many 3s fit into 12? (4)

For dividing by a fraction, the measuring question is the one that makes sense. \(3 \div \tfrac14\) asks: how many quarters fit into 3?

0123 12 hops of 1/4 fit into 3

Each whole holds 4 quarters, so 3 wholes hold 12. Dividing by \(\tfrac14\) is multiplying by 4.

Notice something new: dividing by a number smaller than 1 makes the answer bigger. That is the twin of the last lesson's surprise. Small things fit into a big thing many times.

Same denominators: just divide the counts

How many eighths fit into \(\tfrac34\)? Change \(\tfrac34\) into eighths: \(\tfrac68\). Now the question is "how many 1-eighths fit into 6-eighths?", and the answer is plainly 6.

\[\frac34 \div \frac18 = \frac68 \div \frac18 = 6 \div 1 = 6\]

This works for any pair. Once two fractions have the same denominator, they are counts of the same unit, and dividing them is just dividing the counts. It is like asking "how many 3-metre poles fit in 12 metres?": the "metres" do not matter, only \(12 \div 3\) does.

What if the counts do not divide evenly? \(\tfrac72 \div \tfrac34\). In quarters: \(\tfrac{14}{4} \div \tfrac34\). How many 3-quarters fit into 14-quarters? \(14 \div 3 = \tfrac{14}{3} = 4\tfrac23\). Four whole times, and two thirds of another time.

The reciprocal

Two numbers whose product is 1 are reciprocals of each other. You find the reciprocal of a fraction by turning it upside down:

  • the reciprocal of \(\tfrac34\) is \(\tfrac43\), because \(\tfrac34 \times \tfrac43 = \tfrac{12}{12} = 1\);
  • the reciprocal of \(\tfrac14\) is \(\tfrac41 = 4\);
  • the reciprocal of 5 (which is \(\tfrac51\)) is \(\tfrac15\).

Zero has no reciprocal: nothing times 0 makes 1.

The rule, and why it works

To divide by a fraction, multiply by its reciprocal.

\[\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}\]

So \(\tfrac72 \div \tfrac34 = \tfrac72 \times \tfrac43 = \tfrac{28}{6} = \tfrac{14}{3}\), the same answer as before. Here are two reasons this is always true.

Reason 1: common denominators. Give both fractions the denominator \(b \times d\):

\[\frac{a}{b} \div \frac{c}{d} = \frac{a \times d}{b \times d} \div \frac{c \times b}{d \times b}\]

Now they are counts of the same unit, so divide the counts: the answer is \(\dfrac{a \times d}{c \times b}\). And that is exactly \(\dfrac{a}{b} \times \dfrac{d}{c}\). The rule is just the common-denominator method, done in one step.

Reason 2: division undoes multiplication. \(12 \div 3 = 4\) because \(4 \times 3 = 12\). In the same way, \(\tfrac{a}{b} \div \tfrac{c}{d}\) is the number that, multiplied by \(\tfrac{c}{d}\), gives back \(\tfrac{a}{b}\). Let us test the number \(\tfrac{a}{b} \times \tfrac{d}{c}\):

\[\left(\frac{a}{b} \times \frac{d}{c}\right) \times \frac{c}{d} = \frac{a}{b} \times \left(\frac{d}{c} \times \frac{c}{d}\right) = \frac{a}{b} \times 1 = \frac{a}{b}\]

It gives back \(\tfrac{a}{b}\), so it is the answer. The reciprocal is exactly the thing that cancels out the \(\tfrac{c}{d}\).

A third way to see it, without letters: dividing by \(\tfrac14\) asks how many quarters fit, and every whole holds 4 of them, so multiply by 4. Dividing by \(\tfrac34\): each whole holds 4 quarters, and every 3 quarters make one \(\tfrac34\), so each whole holds \(\tfrac43\) of a "three-quarters". Multiply by \(\tfrac43\).

Mixed numbers and whole numbers

Change mixed numbers to improper fractions, and write whole numbers over 1:

\[3\tfrac34 \div \frac58 = \frac{15}{4} \times \frac85 = \frac{120}{20} = 6\]

Dividing a fraction by a whole number: \(\tfrac43 \div 2 = \tfrac43 \times \tfrac12 = \tfrac46 = \tfrac23\). That is just "half of \(\tfrac43\)", as it should be.

What does a fraction left over mean?

A recipe uses \(\tfrac34\) cup of rice per serving. You have \(2\tfrac12\) cups. How many servings?

\[2\tfrac12 \div \frac34 = \frac52 \times \frac43 = \frac{10}{3} = 3\tfrac13\]
0122 1/23 1231/3 of a hop

Three full servings use \(3 \times \tfrac34 = 2\tfrac14\) cups. The \(\tfrac14\) cup left over is a third of a serving, not "a third of a cup". The fraction in the answer counts servings, the thing we divided by.

Worked examples

1. How many fit? \(5 \div \tfrac13\). Each whole holds 3 thirds; 5 wholes hold 15. (\(5 \times 3 = 15\).)

2. Common denominators. \(\tfrac56 \div \tfrac23\). In sixths: \(\tfrac56 \div \tfrac46 = 5 \div 4 = \tfrac54 = 1\tfrac14\). With the rule: \(\tfrac56 \times \tfrac32 = \tfrac{15}{12} = \tfrac54\). ✓

3. The rule with cancelling. \(\tfrac{15}{4} \div \tfrac58 = \tfrac{15}{4} \times \tfrac85\). Cancel 5 (15 and 5) and 4 (4 and 8): \(\tfrac31 \times \tfrac21 = 6\).

4. Check by multiplying. Is \(\tfrac{9}{10} \div \tfrac35 = \tfrac32\)? Check: \(\tfrac32 \times \tfrac35 = \tfrac{9}{10}\). ✓

5. A story. A ribbon is 6 m long. How many pieces of \(\tfrac34\) m can you cut? \(6 \div \tfrac34 = 6 \times \tfrac43 = 8\) pieces, none left over.

6. Bigger or smaller? Without calculating, is \(\tfrac25 \div \tfrac37\) more or less than \(\tfrac25\)? Dividing by a number less than 1 makes it bigger, so more. Exactly: \(\tfrac25 \times \tfrac73 = \tfrac{14}{15}\), which is indeed more than \(\tfrac25 = \tfrac{6}{15}\).

Try it

  1. Write the reciprocal of: (a) \(\tfrac27\) (b) 9 (c) \(1\tfrac12\).
  2. How many sixths are in 4? Write it as a division and answer it.
  3. \(\tfrac78 \div \tfrac18\) and \(\tfrac{9}{10} \div \tfrac{3}{10}\) (no rule needed).
  4. \(\tfrac23 \div \tfrac49\) by the rule. Check your answer by multiplying.
  5. \(4\tfrac12 \div 1\tfrac12\).
  6. You have \(3\tfrac13\) litres of juice and glasses that hold \(\tfrac14\) litre. How many full glasses can you pour, and what fraction of a glass is left?
  7. Which is bigger, \(10 \div \tfrac12\) or \(10 \times \tfrac12\)? Explain without calculating, then calculate.
  8. Harder: explain with Reason 1 why \(\tfrac{a}{b} \div \tfrac{a}{b} = 1\) for any fraction that is not zero, and then show it with the rule.
Answers
  1. (a) \(\tfrac72\) (b) \(\tfrac19\) (c) \(1\tfrac12 = \tfrac32\), so its reciprocal is \(\tfrac23\).
  2. \(4 \div \tfrac16 = 4 \times 6 = 24\) sixths.
  3. 7 eighths ÷ 1 eighth = 7. 9 tenths ÷ 3 tenths = 3.
  4. \(\tfrac23 \times \tfrac94 = \tfrac{18}{12} = \tfrac32\). Check: \(\tfrac32 \times \tfrac49 = \tfrac{12}{18} = \tfrac23\). ✓
  5. \(\tfrac92 \div \tfrac32\): both are halves, so \(9 \div 3 = 3\). (Rule: \(\tfrac92 \times \tfrac23 = \tfrac{18}{6} = 3\).)
  6. \(3\tfrac13 \div \tfrac14 = \tfrac{10}{3} \times 4 = \tfrac{40}{3} = 13\tfrac13\). 13 full glasses, and a third of a glass left over (which is \(\tfrac13 \times \tfrac14 = \tfrac{1}{12}\) litre).
  7. \(10 \div \tfrac12\) asks how many halves fit in 10: 20. \(10 \times \tfrac12\) is half of 10: 5. So \(10 \div \tfrac12\) is bigger (dividing by a number less than 1 makes bigger; multiplying by it makes smaller).
  8. Reason 1: both fractions already have the same denominator, so divide the counts: \(a \div a = 1\) (as long as \(a\) is not 0). With the rule: \(\tfrac{a}{b} \times \tfrac{b}{a} = \tfrac{ab}{ba} = 1\).

Common mistake

Flipping the wrong fraction. In \(\tfrac23 \div \tfrac49\) you flip the second fraction (the one you divide by), not the first. Flipping the first gives \(\tfrac32 \times \tfrac49 = \tfrac23\), which is wrong. Check by multiplying back: the answer times \(\tfrac49\) must give \(\tfrac23\).

Misreading the leftover. In \(2\tfrac12 \div \tfrac34 = 3\tfrac13\), the \(\tfrac13\) means a third of a serving (\(\tfrac34\) cup), not a third of a cup. The fraction in a division answer is always measured in the thing you divided by.

Where it leads