2012 AMC 10A Problem 1

The times are in seconds but the question is in minutes, so first put everything in seconds.

1. Five minutes is $5 \times 60 = 300$ seconds.
2. Cagney finishes one cupcake every $20$ seconds. In $300$ seconds she finishes $300 \div 20 = 15$ cupcakes.
3. Lacey finishes one cupcake every $30$ seconds. In $300$ seconds she finishes $300 \div 30 = 10$ cupcakes.
4. They each work on their own cupcakes at the same time, so the two counts simply add: $15 + 10 = 25$.

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

When two people work side by side, the amounts they finish add up, but the times per item do not. So turn each "one every so many seconds" into "how many in the given time", then add. Adding $20$ and $30$ seconds would describe one person doing both jobs one after the other, which is not what happens.

Look at one minute. In $60$ seconds Cagney frosts $3$ cupcakes and Lacey frosts $2$, so together they frost $5$ cupcakes per minute. In $5$ minutes that is $5 \times 5 = 25$.