Operations & Word Problems

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

Use how multiplication and division undo each other

My older brother's allowance is 2122\frac{1}{2} times mine, and my younger sibling's is 34\frac{3}{4} of mine. My younger sibling's allowance is how many times my older brother's?

Number systemRatios
Your answer
How to solve
Strategy Convert to Algebra — Both allowances are measured against mine, so mine is the bridge between them. Since it never has to be known as a number, treat it as one unit and the comparison becomes a single division.
1STEP 1

Take my allowance as one unit

Both other allowances are given as multiples of mine, so setting mine to 1 costs nothing and makes both of them plain numbers.

2 1/2 = 5/2
2STEP 2

Write the comparison as a division

Asking how many times B is of A means dividing B by A.

3/4 ÷ 5/2
3STEP 3

Do the division

Dividing by 5/2 is multiplying by 2/5.

3/4 × 2/5 = 6/20 = 3/10
Answer
310\frac{3}{10} times
Try a number: if mine is 4000 won, the brother gets 10000 and the sibling 3000, and 3000 ÷ 10000 = 3/10.
Takeaway

When two things are both measured against a third, set that third to 1 and compare them directly.

  • Take my allowance as one unit
  • Write the comparison as a division
  • Do the division
Where next?
Level upsuggested
Gaps are one fewer than the objectsNumbers divisible by several are common multiples
Another one like thissuggested
12 practice problems
Step back and solidifysuggested
Dividing by a whole number is multiplying by its reciprocalDivision as repeated subtraction
1 reasoning problems grow out of this type — the bridge toward competition →

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