Patterns & Reasoning

Depth · one concept from first principles · step 2 of 4

Subtract the remainder and the division comes out exact

Dividing $36$ by $17$ does not come out exactly. When $0.\bigcirc\triangle$ is subtracted from $36$ first, the division by $17$ ends exactly at the second decimal place. Find the smallest such number $0.\bigcirc\triangle$ .

Number system

Your answer

How to solve

Strategy Work Backwards — A division ends at the second decimal place exactly when what is being divided is 17 times some two-place decimal. So look for the largest such multiple that does not pass 36 -- the closer it gets, the less has to be taken off.

1STEP 1

See why it does not come out

Dividing 36 by 17 gives 2.11 and still leaves something over, so 36 is not 17 times a two-place decimal.

17 × 2.11 = 35.87

2STEP 2

Look for the closest multiple below 36

The next two-place quotient up is 2.12, and 17 × 2.12 = 36.04, which is already past 36. So 35.87 is as close as we can get from below.

17 × 2.12 = 36.04

3STEP 3

Take the difference

Subtract that multiple from 36 to see how much has to come off.

36 - 35.87 = 0.13

Answer

0.13

Check it: 35.87 ÷ 17 = 2.11, which stops exactly at the second decimal place.

Takeaway

A division ends neatly exactly when the dividend is a whole number of the divisor's steps -- so aim for the nearest step below.

See why it does not come out
Look for the closest multiple below 36
Take the difference

Where next?

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