Competition · AMC preparation · step 4 of 4
AMC 10 · 2020B · #3
Grade 6 rate-ratioPick an answer.
AMC 10 2020 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Tool #6 (Guess and Check) via WLOG: because ratios are scale-free, we are allowed to pick one variable to be a convenient number that satisfies the cleanest denominators. Letting x = 6 makes both w:x = 4:3 and z:x = 1:6 produce whole numbers. Tool #7 (Subproblems): split into three little jumps — x → w, x → z, z → y, then form w:y. Tool #15 (Reorganize): the ratios are given as w{:}x, y{:}z, z{:}x — re-order them along the chain w arrow x → z → y so the path from w to y is obvious.
Pick a convenient x
Pick a convenient value: let x = 6 so both z:x = 1:6 and w:x = 4:3 give whole numbers.
Ratios don't care about size — we choose the size that's nicest to compute with.
Ratios do not care about the actual size, so you may choose the size that is nicest to work with.
▸ Why?
One common factor scales every quantity together, so the ratios never change.
▸ Why?
Scaling both parts of a ratio leaves a different-looking pair naming the same relationship.
Find w from the first ratio
From w:x = 4:3 with x = 6, scale by 2 to get w = 8.
Multiply both sides of 4:3 by 2 to make the second part match x = 6.
6.RP.A.3Identify SubproblemsFind z from the second ratio
From z:x = 1:6 with x = 6, read off z = 1.
z is 1 for every 6 of x, and we set x = 6, so z = 1.
6.RP.A.3Identify SubproblemsFind y from the third ratio
From y:z = 3:2 with z = 1, halve to get y = .
For every 2 parts of z there are 3 parts of y — half-step gives y = 3/2.
6.RP.A.3Identify SubproblemsForm the ratio w to y
Form w:y = 8 : , then double both parts to clear the fraction: 16 : 3.
A ratio is unchanged when both parts are multiplied by the same number — clear fractions by scaling up.
6.RP.A.3Organize Information In More WaysThis AMC 10 problem only needs Grade 6 “ratios scale up and down the same way” you already know — set x = 6 to get w = 8, z = 1, y = , then w:y = 16:3.
- Pick a convenient x
- Find w from the first ratio
- Find z from the second ratio
- Find y from the third ratio
- Form the ratio w to y
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