Problem
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.
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 a bigger dividend does too, so aim for the largest dividend and the smallest divisor.
Big on top, small underneath.
6.EE.A.2Guess And CheckA quotient grows when the dividend grows and when the divisor shrinks.
Why?
A quotient counts how many of the divisor fit into the dividend, so a bigger dividend holds more of them.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
Why?
A smaller divisor is a smaller piece, and smaller pieces fit into the same amount more often.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
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
The divisor is 1.2.
6.EE.A.2Make A Systematic List3STEP 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
The dividend is 6.54.
6.EE.A.2Make A Systematic List4STEP 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
The largest quotient is 5.45.
6.NS.B.3Work BackwardsAnswer
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