Math Detours · Grade 5 · Sharpen · 7

Half of a third

Multiplying fractions is taking a part of a part. One picture shows why you multiply the tops and multiply the bottoms.

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Tonight's idea

You are given a third of a cake. You eat half of your piece. How much of the whole cake did you eat?

a third of the cake half of the third: 1 piece of 6
Cut one way for thirds, the other way for halves. The cake ends up in 6 equal pieces.

Cut the cake into 3 columns and shade one. Then cut across into 2 rows. The cake is now in \(3 \times 2 = 6\) equal pieces, and you ate one. Half of a third is a sixth.

In math, "of" means multiply. So \(\tfrac12 \times \tfrac13 = \tfrac16\).

The bottoms multiply because columns times rows counts all the pieces. The tops multiply for the same reason. The pieces you keep also make a small rectangle, and you count it the same way.

To multiply two fractions, multiply the tops and multiply the bottoms.

When both fractions are less than 1, the answer is smaller than both. A part of a part is a smaller part. Multiplying does not always make things bigger.

Watch one: \(\tfrac23\) of \(\tfrac45\)

4 of the 5 columns 2 of the3 rows
8 pieces kept, out of 15.

Cut a rectangle into 5 columns and shade 4. That is \(\tfrac45\).

Cut across into 3 rows and keep 2 of them. That takes \(\tfrac23\) of the shaded part.

All the pieces: \(5 \times 3 = 15\). Pieces kept: \(4 \times 2 = 8\).

\[\tfrac23 \times \tfrac45 = \tfrac{2 \times 4}{3 \times 5} = \tfrac{8}{15}\]

Your turn

  1. (a) \(\tfrac12 \times \tfrac15\); (b) \(\tfrac13 \times \tfrac14\); (c) half of a half.
  2. Give each answer in its simplest form: (a) \(\tfrac23 \times \tfrac15\); (b) \(\tfrac34 \times \tfrac35\); (c) \(\tfrac12 \times \tfrac47\).
  3. Draw a rectangle that shows \(\tfrac34\) of \(\tfrac12\). What fraction of the whole rectangle is it?
  4. Three quarters of a class play an instrument. Two thirds of those children play the piano. (a) What fraction of the class play the piano? (b) There are 24 in the class. How many play the piano? (c) Is \(\tfrac23\) of \(\tfrac34\) the same as \(\tfrac34\) of \(\tfrac23\)? Say why.

Stretch: a long chain

Work out \(\tfrac12 \times \tfrac23 \times \tfrac34 \times \tfrac45\). Look for a shortcut before you multiply. What would the answer be if the chain ran on to \(\tfrac{99}{100}\)?

Answers

  1. (a) \(\tfrac{1}{10}\). (b) \(\tfrac{1}{12}\). (c) \(\tfrac12 \times \tfrac12 = \tfrac14\).
  2. (a) \(\tfrac{2}{15}\). (b) \(\tfrac{9}{20}\). (c) \(\tfrac{4}{14} = \tfrac27\).
  3. Cut a rectangle into 2 columns and shade one. Cut across into 4 rows and keep 3. That is 3 pieces out of 8: \(\tfrac38\).
  4. (a) \(\tfrac23 \times \tfrac34 = \tfrac{6}{12} = \tfrac12\). (b) Half of 24 is 12 children. (c) Yes. The tops give \(2 \times 3\) or \(3 \times 2\), and the bottoms give \(3 \times 4\) or \(4 \times 3\). The order does not change either product.

Stretch. \(\tfrac15\). The tops are \(1 \times 2 \times 3 \times 4\) and the bottoms are \(2 \times 3 \times 4 \times 5\). Divide top and bottom by 2, by 3 and by 4. Only the first top, 1, and the last bottom, 5, are left. Running on to \(\tfrac{99}{100}\) leaves \(\tfrac{1}{100}\).

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