AMC 10 · 2006 · #3
Grade 6 rate-ratioThe ratio of Mary's age to Alice's age is 3:5. Alice is 30 years old. How old is Mary?
Pick an answer.
AMC 10 2006 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: Mary's age compared to Alice's age is in the ratio $3:5$. Alice is $30$ years old. Find Mary's age.
Givens: The ratio of Mary's age to Alice's age is $3:5$; Alice is $30$ years old; Answer choices: (A) $15$, (B) $18$, (C) $20$, (D) $24$, (E) $50$
Unknowns: Mary's age in years
Understand
Restated: Mary's age compared to Alice's age is in the ratio $3:5$. Alice is $30$ years old. Find Mary's age.
Givens: The ratio of Mary's age to Alice's age is $3:5$; Alice is $30$ years old; Answer choices: (A) $15$, (B) $18$, (C) $20$, (D) $24$, (E) $50$
Plan
Primary tool: #4 Introduce a Variable
Secondary: #13 Convert to Algebra, #3 Eliminate Possibilities
A ratio $3:5$ is really a set of equal-sized shares: Mary gets $3$ shares and Alice gets $5$. Tool #4 (Introduce a Variable) names the size of one share $x$, so Mary $=3x$ and Alice $=5x$. Tool #13 (Convert to Algebra) turns the fact 'Alice is $30$' into the equation $5x=30$, which pins down $x$. Tool #3 (Eliminate Possibilities) gives a fast reality check: Mary is the smaller side of the ratio, so she must be younger than $30$, which instantly kills (E) $50$.
Execute — Answer: B
6.RP.A.1 Step 1 Split the ages into equal shares
- Read the ratio $3:5$ as equal-sized shares of years.
- Mary's age is made of $3$ shares and Alice's age is made of $5$ shares, where every share is the same size.
- Call that share size $x$ years.
- Then Mary $=3x$ and Alice $=5x$.
💡 A ratio just chops both ages into identical blocks — Mary gets $3$ blocks, Alice gets $5$.
6.RP.A.3 Step 2 Use Alice's age to size one share
- Alice is $30$ years old and Alice is $5$ shares, so $5x = 30$.
- Dividing both sides by $5$ gives $x = 6$.
- One share is worth $6$ years.
💡 If $5$ equal blocks make $30$, each block must be $30 \div 5 = 6$.
6.RP.A.3 Step 3 Build Mary's age and confirm
- Mary is $3$ shares, and each share is $6$ years, so Mary $= 3 \times 6 = 18$ years.
- That is choice (B).
- Check: Mary is younger than Alice, as the ratio requires, and $18:30$ reduces to $3:5$.
- Since Mary must be smaller than $30$, the choice (E) $50$ was impossible from the start.
💡 Three blocks of $6$ years each stack up to $18$, and $18:30$ folds right back to $3:5$.
6.RP.A.1 Read the ratio $3:5$ as equal-sized shares of years. Mary's age is made of $3$ s 6.RP.A.3 Alice is $30$ years old and Alice is $5$ shares, so $5x = 30$. Dividing both sid 6.RP.A.3 Mary is $3$ shares, and each share is $6$ years, so Mary $= 3 \times 6 = 18$ yea Review
Reasonableness: Mary is the smaller side of the ratio, so her age must be below Alice's $30$ — this rules out (E) $50$ at a glance and tells us the answer sits in the low range. Testing the result: $18:30$ divides both by $6$ to give $3:5$, exactly the ratio in the problem, so $18$ is consistent. The other small choices fail this test — for example $15:30 = 1:2$ and $20:30 = 2:3$, neither of which is $3:5$.
Alternative: Skip the variable and scale the ratio directly. Alice's part of the ratio is $5$ and her real age is $30$, so the ratio was multiplied by $30 \div 5 = 6$ to reach real ages. Apply the same multiplier to Mary's part: $3 \times 6 = 18$. Or as a fraction: Mary is $\tfrac{3}{5}$ of Alice, and $\tfrac{3}{5} \times 30 = 18$.
CCSS standards used (min grade 6)
6.RP.A.1Understand the concept of a ratio and use ratio language (Reading $3:5$ as equal shares of years so that Mary $=3x$ and Alice $=5x$.)6.RP.A.3Use ratio and rate reasoning to solve real-world and mathematical problems (Sizing one share from $5x=30$ and scaling up to Mary's age $3 \times 6 = 18$.)
⭐ A ratio splits both amounts into equal shares — find the size of one share from the number you know, then multiply to get the one you want.
⭐ A ratio splits both amounts into equal shares — find the size of one share from the number you know, then multiply to get the one you want.
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