AMC 10 · 2020 · #3
Grade 6 rate-ratioProblem
The ratio of w to x is 4:3, the ratio of y to z is 3:2, and the ratio of z to x is 1:6. What is the ratio of w to y?
Pick an answer.
AMC 10 2020 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: Three ratios link four quantities: $w:x = 4:3$, $y:z = 3:2$, and $z:x = 1:6$. Find $w:y$.
Givens: $w:x = 4:3$; $y:z = 3:2$; $z:x = 1:6$; Answer choices: (A) $4:3$, (B) $3:2$, (C) $8:3$, (D) $4:1$, (E) $16:3$
Unknowns: The ratio $w:y$
Understand
Restated: Three ratios link four quantities: $w:x = 4:3$, $y:z = 3:2$, and $z:x = 1:6$. Find $w:y$.
Givens: $w:x = 4:3$; $y:z = 3:2$; $z:x = 1:6$; Answer choices: (A) $4:3$, (B) $3:2$, (C) $8:3$, (D) $4:1$, (E) $16:3$
Plan
Primary tool: #6 Guess and Check
Secondary: #7 Identify Subproblems, #15 Reorganize Information
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 — $x \to w$, $x \to z$, $z \to y$, then form $w:y$. Tool #15 (Reorganize): the ratios are given as $w{:}x$, $y{:}z$, $z{:}x$ — re-order them along the chain $w \leftarrow x \to z \to y$ so the path from $w$ to $y$ is obvious.
Execute — Answer: E
6.RP.A.3 Step 1 - Pick a convenient value.
- Since $x$ appears in two of the ratios, let $x = 6$ — that makes $z:x = 1:6$ give $z = 1$ and $w:x = 4:3$ give $w$ a whole number.
- (Allowed because ratios stay the same under scaling.)
💡 Ratios don't care about size — we choose the size that's nicest to compute with.
6.RP.A.3 Step 2 From $w:x = 4:3$ with $x = 6$: scale by $2$, so $w = 4 \times 2 = 8$.
💡 Multiply both sides of $4:3$ by $2$ to make the second part match $x = 6$.
6.RP.A.3 Step 3 From $z:x = 1:6$ with $x = 6$: $z = 1$ directly.
💡 $z$ is $1$ for every $6$ of $x$, and we set $x = 6$, so $z = 1$.
6.RP.A.3 Step 4 From $y:z = 3:2$ with $z = 1$: $y = \frac{3}{2} \cdot z = \frac{3}{2}$.
💡 For every $2$ parts of $z$ there are $3$ parts of $y$ — half-step gives $y = \frac{3}{2}$.
6.RP.A.3 Step 5 - Form $w:y = 8 : \tfrac{3}{2}$.
- Multiply both sides by $2$ 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.3 Pick a convenient value. Since $x$ appears in two of the ratios, let $x = 6$ — t 6.RP.A.3 From $w:x = 4:3$ with $x = 6$: scale by $2$, so $w = 4 \times 2 = 8$. 6.RP.A.3 From $z:x = 1:6$ with $x = 6$: $z = 1$ directly. 6.RP.A.3 From $y:z = 3:2$ with $z = 1$: $y = \frac{3}{2} \cdot z = \frac{3}{2}$. 6.RP.A.3 Form $w:y = 8 : \tfrac{3}{2}$. Multiply both sides by $2$ to clear the fraction: Review
Reasonableness: Try a different starting value. Let $x = 12$: then $w = 16$, $z = 2$, $y = 3$. Check $w:y = 16:3$ — same ratio, confirming (E). Sanity: $w$ is much bigger than the others while $y$ is small, so $w:y$ ought to be large; $16:3$ matches that feel.
Alternative: Tool #5 (Look for a Pattern) — chain the ratios algebraically: $\frac{w}{y} = \frac{w}{x} \cdot \frac{x}{z} \cdot \frac{z}{y} = \frac{4}{3} \cdot 6 \cdot \frac{2}{3} = \frac{48}{9} = \frac{16}{3}$.
CCSS standards used (min grade 6)
6.RP.A.3Use ratio and rate reasoning to solve real-world and mathematical problems (Scaling each given ratio to a common variable $x = 6$ and combining to get $w:y = 16:3$.)
⭐ This AMC 10 problem only needs Grade 6 “ratios scale up and down the same way” you already know — set $x = 6$ to get $w = 8$, $z = 1$, $y = \tfrac{3}{2}$, then $w:y = 16:3$.
⭐ This AMC 10 problem only needs Grade 6 “ratios scale up and down the same way” you already know — set $x = 6$ to get $w = 8$, $z = 1$, $y = \tfrac{3}{2}$, then $w:y = 16:3$.
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