GMAT Practice Question: Each machine at a toy factory assembles a certain kind of toy at a constant rate of one toy...
Question
Each machine at a toy factory assembles a certain kind of toy at a constant rate of one toy every 3 minutes. If 40 percent of the machines at the factory are to be replaced by new machines that assemble this kind of toy at a constant rate of one toy every 2 minutes, what will be the percent increase in the number of toys assembled in one hour by all the machines at the factory, working at their constant rates?
- 20 \%
- 25 \%
- 30 \%
- 40 \%
- 50 \%
Topics: word problems, percentages, rates
Solution
Step 1
We set up the variable for the machine count and turn \textbf{"one toy every 3 minutes"} and \textbf{"one toy every 2 minutes"} into hourly rates.
M = \text{total number of machines} \frac{60}{3} = 20 \text{ toys per hour per old machine} \frac{60}{2} = 30 \text{ toys per hour per new machine}
Step 2
We write the original hourly output when all machines are old.
Q_{\text{old}} = 20M
Step 3
We use \textbf{"40 percent of the machines"} to split the machines into old and new groups and then add their outputs.
Q_{\text{new}} = (0.6M) \times 20 + (0.4M) \times 30 = 12M + 12M = 24M
Step 4
We compare the change to the original to get the percent increase, then turn the resulting fraction into a percent.
\text{Percent increase} = \frac{Q_{\text{new}} - Q_{\text{old}}}{Q_{\text{old}}} \times 100\% = \frac{24M - 20M}{20M} \times 100\% = \frac{4\cancel{M}}{20\cancel{M}} \times 100\% = \frac{1}{5} \times 100\% = 20\%
Answer
20%
