AMC 10 · 2012 · #4
Grade 4 rate-ratioPick an answer.
The fractions are abstract and no real count is given, so Tool #9 (Solve an Easier Related Problem) replaces them with a concrete total of 5 marbles, turning fractions into objects you can count. Tool #16 (Count the Complement) gets the starting red share as everything that is not blue. Tool #3 (Eliminate Possibilities) guards the traps (A) 2/5 and (D) 3/5, which are just the original red and blue shares.
Find the starting red fraction
The other colour starts at 2/5.
Two colors that fill the whole bag must have shares that add up to one whole.
Two colours that fill the whole bag must have shares that add up to one whole.
▸ Why?
The bag is exactly its two piles put together, so nothing is left unaccounted for.
▸ Why?
Each share is a fraction of that one whole, so the two fractions have to complete each other.
Pick a friendly total of 5 marbles
A friendly total makes both counts whole.
Choosing a total that makes each fifth a whole marble turns fractions into things you can count on your fingers.
3.NF.A.1Solve An Easier Related ProblemDouble the red marbles
Doubling changes the total too.
Doubling only the red pile grows the total by exactly the reds you added, not the blues.
3.OA.C.7Solve An Easier Related ProblemWrite the new red fraction
The new share is 4/7, choice (C).
A fraction of a group is just the part you want written over the whole new group.
3.NF.A.1Eliminate PossibilitiesWhen a problem gives only fractions, pick a small total that makes them whole — here 5 marbles — count what happens, then read off the new fraction: 4/7.
- Find the starting red fraction
- Pick a friendly total of 5 marbles
- Double the red marbles
- Write the new red fraction