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2006 AMC 10A Problem 3

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2006 AMC 10A Problem 3AMC 10 Step by Step
Name:                             Date: September 13, 2026
Ratios, Percents & Averages · difficulty 1 of 5 · about 1 min · position P1-10

The ratio of Mary's age to Alice's age is 3:53:5 . Alice is 3030 years old. How old is Mary?

  1. A)

    1515

  2. B)

    1818

  3. C)

    2020

  4. D)

    2424

  5. E)

    5050

Answer

B

Key insight

A 3:5 ratio with the 5-part equal to 30 makes each part 6, so the 3-part is 18.

Solution

Think of the ages as 33 parts and 55 parts of the same size. Alice's 55 parts equal 3030 years, so one part is 30÷5=630 \div 5 = 6 years.

Mary has 33 parts: 3⋅6=183 \cdot 6 = 18 years.

Equivalently, Mary30=35\dfrac{\text{Mary}}{30} = \dfrac{3}{5} gives Mary=3⋅305=18\text{Mary} = \dfrac{3 \cdot 30}{5} = 18.

The answer is (B) 18\boxed{\textbf{(B)}\ 18}.

Why this works

A ratio a:ba : b says the two quantities are aa and bb copies of a common unit. Once one quantity is known, the unit is fixed and the other follows. Sanity check: Mary's ratio number is the smaller one, so her age must be less than 3030.

The trap

Inverting the ratio and computing 30 times 5/3 = 50 (choice (E)).

Common mistakes

  • Inverting the ratio and computing 30⋅5/3=5030 \cdot 5/3 = 50 (choice (E)).
  • Treating 3:53:5 as a difference of 22 parts and subtracting something from 3030.

Techniques

Set up the equation/formula and compute; no special trick needed