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2012 AMC 10A Problem 1

2012 AMC 10A Problem 1AMC 10 Step by Step

Name: Date: September 13, 2026

Linear Equations & Word Problems · difficulty 1 of 5 · about 1 min · position P1-10

Cagney can frost a cupcake every $20$ seconds and Lacey can frost a cupcake every $30$ seconds. Working together, how many cupcakes can they frost in $5$ minutes?

A)
$10$
B)
$15$
C)
$20$
D)
$25$
E)
$30$

Answer

Key insight

Convert each frosting time to a rate per minute (3 and 2 cupcakes per minute), add the rates, and multiply by 5 minutes.

Solution

Five minutes is $300$ seconds.

Cagney frosts one cupcake every $20$ seconds, so in $300$ seconds she frosts $300/20 = 15$ cupcakes. Lacey frosts one every $30$ seconds, so she frosts $300/30 = 10$ cupcakes.

They work independently, so their outputs simply add: $15 + 10 = 25$ .

The answer is $\boxed{\textbf{(D)}\ 25}$ .

Why this works

Work problems become easy once every worker is described by a rate (cupcakes per second) rather than a time per cupcake. Rates of people working side by side add; times per item do not. Convert to a common time unit first, then count.

Alternative approach

Combined rate: $\frac{1}{20} + \frac{1}{30} = \frac{3 + 2}{60} = \frac{1}{12}$ cupcakes per second, i.e. one cupcake every $12$ seconds. In $300$ seconds that is $300/12 = 25$ .

The trap

Adding the times (20 + 30 = 50 seconds per cupcake) instead of adding the rates, which gives 6 cupcakes.

Common mistakes

Adding the times (20 + 30 = 50 seconds per cupcake) instead of adding the rates, which gives 6 cupcakes.
Forgetting to convert $5$ minutes into $300$ seconds and computing with $5$ as the time.

Techniques

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