Numbers & Place Value

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

Dividing a bigger number by a smaller one grows the quotient

Five cards fill the digits of a two-decimal-place number divided by a one-decimal-place number. We need the largest quotient possible.
1 2 4 5 6
Number systemExpressions
Your answer
How to solve
Strategy Make a Systematic List — A quotient grows when the dividend grows and when the divisor shrinks, and the two choices do not fight each other -- the cards spent on a small divisor are exactly the ones a large dividend does not want. So both can be pushed to their extreme at once.
1STEP 1

Decide what makes a quotient large

Dividing by a smaller number gives a bigger answer, and so does a bigger dividend, so aim for the largest dividend and smallest divisor.

2STEP 2

Build the smallest divisor

Two cards make the divisor, and the smallest number comes from putting the two smallest cards there with the smaller one in the ones place.

1.2
3STEP 3

Build the largest dividend from what is left

The remaining cards are 4, 5 and 6, and the largest number puts the biggest card in the highest place.

6.54
4STEP 4

Do the division

Multiply both by 10 to clear the divisor's decimal point, then divide.

6.54 ÷ 1.2 = 65.4 ÷ 12 = 5.45
Answer
5.45
Try the nearest rival: keeping the divisor at 1.2 but using 6.45 gives 5.375, and moving to a divisor of 1.4 with dividend 6.52 gives about 4.66 -- both smaller.
Takeaway

To make a quotient large, push the dividend up and the divisor down -- and check whether the two aims compete.

  • Decide what makes a quotient large
  • Build the smallest divisor
  • Build the largest dividend from what is left
  • Do the division
Where next?
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8 reasoning problems grow out of this type — the bridge toward competition →

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▶ Practice — 12 problems

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