Multiplying and Dividing in Your Head

Multiplying and Dividing by 10, 100 and 1,000 #

The digits move; the zeros just fill the gaps.

Multiplying by ten moves every digit one place up, and a zero fills the gap

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45 is 4 tens and 5 ones.

When we multiply by 10, tens become hundreds and ones become tens. 45 × 10 = 450.

Similarly with 100, every part moves up 2 places and two zeros fill the gaps.

A thousand is three moves. And dividing runs the moves backwards: 4500 ÷ 100 = 45.

A multiple splits into its digit and its zeros: × 300 is × 3, then × 100.

Multiplying by Splitting #

Ten of them, then the rest, then together.

Split one number at the ten, multiply each piece, and add the pieces back

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The length is 13 = 10 + 3. Let's multiply 10 × 7 first, which gives 70.

Then multiply 3 × 7 = 21. Add each part 70 + 21 = 91.

Doubling to Multiply #

By four is double double; by eight, once more.

Four is two twos, so multiplying by four is doubling twice

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15 × 4 = 15 × (2 × 2) = (15 × 2) × 2.

Halving to Divide #

By four is halve and halve again.

Dividing by four is halving twice, because four is two twos

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To calculate 84 ÷ 4, we can halve it twice. 84 ÷ 4 = (84 ÷ 2) ÷ 2 = 42 ÷ 2 = 21.

Dividing by Splitting #

Peel off ten of the divisor, share the rest.

Split the number being shared into two friendly pieces, and share each piece

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91 ÷ 7 looks hard. But 91 = 70 + 21, and 7 divides both of those easily.

70 ÷ 7 is 10. 21 ÷ 7 is 3.

10 + 3 = 13, so 91 ÷ 7 = 13.

Rounding to Estimate #

Know the size of the answer before the answer.

Round each number to something friendly and you learn the size of the answer

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Before working out 49 × 21 exactly, learn its size: it is nearly 50 × 20.

50 × 20 is 1000, so the true answer lives close by. It is 1029.

A friend claims 49 × 21 is 249. You can estimate: $49\times 21\approx 50\times 20\approx 1000$49 × 21 ≈ 50 × 20 ≈ 1000.