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? [Official GMAT-2018]

Hint:

To calculate the percent increase in a quantity, find the difference between the new and old values, then divide by the old value and multiply by 100.

Answer 1

Step 1: Calculate the rate of toys assembled by the original machines. 
Each machine assembles 1 toy every 3 minutes. In 60 minutes, the number of toys assembled by one machine is: \[ \text{Toys assembled by one machine} = \frac{60}{3} = 20 \, \text{toys} \] Step 2: Calculate the rate of toys assembled by the new machines. 
Each new machine assembles 1 toy every 2 minutes. In 60 minutes, the number of toys assembled by one new machine is: \[ \text{Toys assembled by one new machine} = \frac{60}{2} = 30 \, \text{toys} \] Step 3: Determine the percentage of machines replaced. 
40 percent of the machines are being replaced by new machines, so the number of new machines is 40% of the total. Let the total number of machines be \( N \). The number of new machines is \( 0.4N \), and the number of old machines is \( 0.6N \). 
Step 4: Calculate the total number of toys assembled before the replacement. 
Before the replacement, all \( N \) machines are old, and each assembles 20 toys in an hour. Thus, the total number of toys assembled by all the machines before the replacement is: \[ \text{Total toys before replacement} = N \times 20 \] Step 5: Calculate the total number of toys assembled after the replacement. 
After the replacement, \( 0.6N \) old machines assemble 20 toys each, and \( 0.4N \) new machines assemble 30 toys each. Thus, the total number of toys assembled after the replacement is: \[ \text{Total toys after replacement} = (0.6N \times 20) + (0.4N \times 30) \] Simplifying: \[ \text{Total toys after replacement} = 12N + 12N = 24N \] Step 6: Calculate the percent increase. 
The percent increase in the number of toys assembled is: \[ \text{Percent increase} = \frac{\text{Total toys after replacement} - \text{Total toys before replacement}}{\text{Total toys before replacement}} \times 100 \] Substitute the values: \[ \text{Percent increase} = \frac{24N - 20N}{20N} \times 100 = \frac{4N}{20N} \times 100 = 20% \] Step 7: Conclusion. 
The percent increase in the number of toys assembled is 20%.