Problem
Moving some across leaves the total unchanged
One person gives some of their candies to another, and afterwards the two amounts are in a known ratio. We need how many were given.
ExpressionsRatios
Your answer
How to solve
Strategy Change Focus / Count the Complement — Moving candies between two people changes each amount but not their total, and the total is the one number the new ratio can be applied to. Find what Jia is left with, then compare it with what she started with.
1STEP 1
See that the total does not change
Candies only moved from one to the other, so the two amounts still add to the same number.
20 + 20 = 40
There are still 40 candies in all.
6.RP.A.3Change Focus Count The ComplementThe total number of candies never changes.
Why?
Every candy that leaves Jia arrives with Minho, so each one is still there and is still counted exactly once.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
Why?
None is eaten and none is added, so the two counts always add back to the same forty.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
2STEP 2
Split the total in the new ratio
A ratio of 1 : 3 cuts the total into four equal parts, and Jia keeps one.
1 + 3 = 4, 40 ÷ 4 = 10
Jia is left with 10 candies.
6.RP.A.3Analyze The Units3STEP 3
Compare with what she started with
What she gave away is the difference between what she had and what she has left.
20 - 10 = 10
Jia gave away 10 candies.
6.EE.B.7Identify SubproblemsAnswer
10 candies
Check the ratio: Jia has 10 and Minho has 20 + 10 = 30, and 10 : 30 is 1 : 3.
Takeaway
Moving things between two people changes each amount but never the total -- so work from the total.
- See that the total does not change
- Split the total in the new ratio
- Compare with what she started with