Problem
Use how multiplication and division undo each other
Two people's allowances are each described as a multiple of a third person's. We need one of them as a multiple of the other.
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
Brother 5/2, sibling 3/4.
6.RP.A.3Convert To AlgebraMy own allowance can be taken as one unit without losing anything.
Why?
Both other allowances are given only as multiples of mine, so each is a fixed number of copies of whatever mine happens to be.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
Why?
Asking how many times one is of the other compares two such counts, and scaling both by the same amount leaves that comparison unchanged.
🧱Multiplicative identity (one)One group of a number is just that number — multiplying by one changes nothing.
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
Sibling divided by brother.
6.RP.A.3Draw A Diagram3STEP 3
Do the division
Dividing by 5/2 is multiplying by 2/5.
3/4 × 2/5 = 6/20 = 3/10
The sibling gets 3/10 of what the brother gets.
6.NS.A.1Work BackwardsAnswer
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