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Near Enough

Friendly numbers for grown-ups: round and fix, make a ten, a tip as ten per cent and half again, and the habit of asking whether an answer is the right size.

Spark & Whetstone · sparkwhetstone.org/learn/near-enough · CC BY-NC-SA 4.0. Answers are at the end, so you can try each question first.

Lesson 1. Round and fix

A number close to a round one can borrow the round one's ease. 49 is one less than 50, and 50 is easy to multiply.

So 49 × 6 is 50 × 6, which is 300, with one 6 taken away again: 294. Round, do the easy sum, then fix what rounding changed.

Children learn this as a strategy now. It is also how most quick adult arithmetic works, whether or not anyone named it.

Question. What is the quickest exact way to work out 49 × 6?

  1. 50 × 6 = 300, then take away one 6: 294
  2. 50 × 6 = 300, then take away 1: 299
  3. 40 × 6 = 240, then add 9: 249

Say it back. In your own words: after you round, what do you have to fix?

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Lesson 2. Make a ten

Adding is easier when you land on a ten. For 8 + 7, move 2 from the 7 to the 8: that makes 10 + 5, which is 15.

The same move works on bigger numbers. For 38 + 27, give 2 of the 27 to the 38 to make 40. That leaves 25 to add: 65.

Question. Use make-a-ten: what is 38 + 27?

  1. 65
  2. 55
  3. 75

Say it back. Say it back: what does "make a ten" mean, and why does it help?

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Lesson 3. Ten per cent and half again

Ten per cent is the easiest percentage there is: move the decimal point one place left. 10% of $46 is $4.60.

Fifteen per cent is ten per cent and half of it again. Half of $4.60 is $2.30, so 15% of $46 is $4.60 + $2.30.

Twenty per cent is ten per cent doubled. Most of the percentages you meet at a counter are built from these two moves.

Question. What is 15% of a $46 bill?

  1. $6.90
  2. $4.60
  3. $9.20

Say it back. In your own words: how do you find fifteen per cent without a calculator?

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Lesson 4. Is it the right size?

Before you trust a total, ask what size it should be. Round everything, add the round numbers, and see whether the real answer is anywhere near.

Three things at $19.99 are about three things at $20, which is about $60. A total of $89.97 is not near $60, so something on the bill is wrong, perhaps an item rung up twice.

This takes seconds and catches the mistakes that matter: a missed decimal point, a doubled line, a number typed in the wrong box.

Question. A till says three items at $19.99 come to $89.97. Is that the right size?

  1. No, it should be close to $60.
  2. Yes, it is close enough.
  3. No, it should be close to $30.

Say it back. Say it back: what is the quick check you do before trusting a total?

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At the table: The receipt guess

Players: 2 to 6. You'll need: a shopping list or a till receipt with the total covered, paper, a pencil each.

  1. One person reads the prices aloud, one at a time.
  2. Everyone keeps a rounded running total in their head and writes a guess at the end.
  3. Uncover the total. Whoever is closest explains how they rounded, and everyone tries their way on the next receipt.

Seat sheets and a run-through: sparkwhetstone.org/todays-guess

Answers

  1. 50 × 6 = 300, then take away one 6: 294 — Rounding 49 to 50 adds one more 6 to the product, so you take one 6 back off.
  2. 65 — 38 needs 2 to reach 40. That leaves 25 of the 27, and 40 + 25 is the answer.
  3. $6.90 — 10% is $4.60 and half of that is $2.30. Together they make 15%.
  4. No, it should be close to $60. — Each item is about $20, so three are about $60. $89.97 is about one extra item's worth too much.