Question

A certain machine drills 30 holes in 8 minutes. At that constant rate, how many holes will 4 such machines drill in \(1 \frac{1}{3}\) hours?

• A. 300
• B. 900
• C. 960
• D. 1,200
• E. 2,560

Hint:

In work-rate problems, always ensure your units are consistent before you multiply. It's usually easiest to convert all time units to the smallest unit mentioned (in this case, minutes).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This is a work-rate problem. We need to find the rate of a single machine and then scale it up for multiple machines and a different time period. It's crucial to keep units consistent (e.g., convert hours to minutes).
Step 2: Key Formula or Approach:
Total Work = Rate \(\times\) Time \(\times\) Number of Workers (or machines).
We'll first find the rate of one machine, then calculate the total work.
Step 3: Detailed Explanation:
1. Find the rate of one machine.
The rate is the number of holes drilled per unit of time.
\[ \text{Rate per machine} = \frac{30 \text{ holes}}{8 \text{ minutes}} \]
2. Convert the total time to minutes.
The time given is \(1 \frac{1}{3}\) hours. First, convert the mixed number to an improper fraction: \(1 \frac{1}{3} = \frac{4}{3}\).
Since there are 60 minutes in an hour:
\[ \text{Total time} = \frac{4}{3} \text{ hours} \times \frac{60 \text{ minutes}}{1 \text{ hour}} = 4 \times 20 = 80 \text{ minutes} \]
3. Calculate the total number of holes.
We have 4 machines, each working for 80 minutes at the rate calculated in Step 1.
\[ \text{Total Holes} = (\text{Rate per machine}) \times (\text{Total time}) \times (\text{Number of machines}) \]
\[ \text{Total Holes} = \left(\frac{30 \text{ holes}}{8 \text{ minutes}}\right) \times (80 \text{ minutes}) \times (4 \text{ machines}) \]
Let's simplify the calculation:
\[ \text{Total Holes} = \frac{30 \times 80 \times 4}{8} \]
We can cancel the 8 in the numerator and the denominator:
\[ \text{Total Holes} = 30 \times 10 \times 4 \]
\[ \text{Total Holes} = 300 \times 4 = 1,200 \]
Step 4: Final Answer:
The 4 machines will drill 1,200 holes in \(1 \frac{1}{3}\) hours.