Math Detours · Grade 5 · Around the corner · 21

Half, and half again

Walk half way to the wall, then half of what is left, for ever. The pieces add up to exactly one.

Name ____________________Date ____________

Tonight's idea

Stand one metre from a wall. Walk half way. Then walk half of what is left. Then half of that, and keep going.

Your steps are \(\tfrac12\), \(\tfrac14\), \(\tfrac18\), \(\tfrac1{16}\) of a metre, and so on. Each step is half the one before. Where do you end up?

wall 1 2 1 4 1 8 0 1 2 3 4 7 8 1 start 1 2 1 4 1 8 1 16 a square of area 1
Left: the walk to the wall. Right: the same pieces cut from a square of area 1.

Look at the square. Shade half. Then shade half of what is left, again and again. Each time, the part still white is the same size as the piece you just shaded.

Total so far = 1 − the last piece.

So the total never goes past 1. But the white part keeps halving. It gets smaller than any size you can name. The only number the total can be heading for is 1. That is what this line means:

\[\tfrac12+\tfrac14+\tfrac18+\tfrac1{16}+\cdots = 1\]

The three dots say "and so on, for ever".

Watch one: four steps

Add the first four pieces. Write each one in sixteenths, so the pieces are the same size.

\[\tfrac12+\tfrac14+\tfrac18+\tfrac1{16} = \tfrac8{16}+\tfrac4{16}+\tfrac2{16}+\tfrac1{16} = \tfrac{15}{16}\]

Now the shortcut. The last piece was \(\tfrac1{16}\), and \(1-\tfrac1{16}=\tfrac{15}{16}\). It agrees.

After four steps you are \(\tfrac1{16}\) of a metre from the wall. That is about 6 cm.

Your turn

  1. (a) Write the next two pieces after \(\tfrac1{16}\). (b) Add \(\tfrac12+\tfrac14+\tfrac18\) by writing each one in eighths.
  2. Use the shortcut to find the total of the first six pieces. How far are you from the wall then?
  3. Now the wall is 8 metres away. Your steps are 4 m, 2 m, 1 m, half a metre, and so on. (a) How far have you walked after five steps? (b) How far away is the wall then?
  4. Back to the 1 metre walk. One metre is 1000 mm, so after one step you are 500 mm from the wall. How many steps does it take to be less than 1 mm from the wall?

Stretch: sharing a cake for ever

Two friends cut a cake into three equal pieces. They take one each. They cut the last piece into three and do the same again, for ever. Each friend gets \(\tfrac13+\tfrac19+\tfrac1{27}+\cdots\) of the cake. What does that add up to?

Answers

  1. (a) \(\tfrac1{32}\) and \(\tfrac1{64}\). Each piece is half the one before, so the bottom number doubles. (b) \(\tfrac48+\tfrac28+\tfrac18=\tfrac78\).
  2. The sixth piece is \(\tfrac1{64}\). The total is \(1-\tfrac1{64}=\tfrac{63}{64}\). You are \(\tfrac1{64}\) of a metre from the wall, the same as the last step.
  3. (a) \(4+2+1+\tfrac12+\tfrac14=7\tfrac34\) m. (b) \(8-7\tfrac34=\tfrac14\) m. Again the gap is the same as the last step.
  4. The gap halves each step: 500, 250, 125, then about 63, 31, 16, 8, 4, 2 and 1 mm. The tenth gap is just under 1 mm (it is 0.977 mm). So it takes 10 steps.

Stretch. One half. The leftover piece shrinks towards nothing, so in the end the whole cake is shared out. The two friends always take the same amount, so each gets half. This shows \(\tfrac13+\tfrac19+\tfrac1{27}+\cdots=\tfrac12\).

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