2006 AMC 10A Problem 3

P1-10~1 minPrint

Ratios, Percents & Averages Set up the equation/formula and compute; no special trick needed

The ratio of Mary's age to Alice's age is $3:5$ . Alice is $30$ years old. How old is Mary?

Show solution without answering

Key insight

Think of the ages as 3 equal parts and 5 equal parts. Alice's 5 parts are 30 years, so one part is 6 years.

Solution

A ratio of $3:5$ means Mary's age is $3$ equal parts and Alice's age is $5$ of the same parts.

Alice's $5$ parts make $30$ years, so one part is $30 \div 5 = 6$ years.
Mary has $3$ parts: $3 \times 6 = 18$ years.
Check: $18 : 30$ simplifies (divide both by $6$ ) to $3 : 5$ . Correct.

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

Why this works

A ratio tells you how many equal "parts" each quantity has. Once you know one quantity, you know the size of one part, and then everything else follows. A quick sense check helps: Mary's ratio number $3$ is smaller than Alice's $5$ , so Mary must be younger than $30$ .

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

Think of the ages as $3$ parts and $5$ parts of the same size. Alice's $5$ parts equal $30$ years, so one part is $30 \div 5 = 6$ years.

Mary has $3$ parts: $3 \cdot 6 = 18$ years.

Equivalently, $\dfrac{\text{Mary}}{30} = \dfrac{3}{5}$ gives $\text{Mary} = \dfrac{3 \cdot 30}{5} = 18$ .

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

Review in the AoPS books

Introduction to Algebra, Chapter 6: Ratios and Percents
Introduction to Algebra, Chapter 7: Proportion (§7.1, §7.2, §7.3)
Prealgebra, Chapter 13: Data and Statistics (§13.1)