AMC 10 · 2020 · #3
Grade 6 rate-ratioPick an answer.
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.
Anchor the shared quantity
Give the linking quantity a convenient value.
Ratios don't care about size — we choose the size that's nicest to compute with.
6.RP.A.3Guess And CheckFind the first quantity
The first ratio gives the first quantity.
Multiply both sides of 4:3 by 2 to make the second part match x = 6.
6.RP.A.3Identify SubproblemsFind the fourth quantity
The third ratio gives the fourth quantity.
z is 1 for every 6 of x, and we set x = 6, so z = 1.
6.RP.A.3Identify SubproblemsFind the third quantity
The second ratio gives the last one.
For every 2 parts of z there are 3 parts of y — half-step gives y = 3/2.
6.RP.A.3Identify SubproblemsTake the ratio
Tidying gives 16 to 3.
A ratio is unchanged when both parts are multiplied by the same number — clear fractions by scaling up.
A ratio is unchanged when both of its parts are multiplied by the same number.
▸ Why?
Scaling both parts together keeps their relative sizes exactly as they were.
▸ Why?
Multiplying top and bottom by the same amount renames a fraction without changing its value.
This AMC 12 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 = 3/2, then w:y = 16:3.