Question

If a line worker can install 40 insulators in 18 fraction numerator 3 over denominator 4 end fraction hours, how many insulators should they be able to install in 28 fraction numerator 1 over denominator 8 end fraction hours?

Answer 1

To determine how many insulators a line worker can install in \(28 \frac{1}{8}\) hours, we first need to convert the mixed numbers \(18 \frac{3}{4}\) hours and \(28 \frac{1}{8}\) hours into improper fractions.

1. Convert \(18 \frac{3}{4}\) hours to an improper fraction:
\[
18 \frac{3}{4} = 18 + \frac{3}{4} = \frac{18 \times 4 + 3}{4} = \frac{72 + 3}{4} = \frac{75}{4} \text{ hours}
\]

2. Convert \(28 \frac{1}{8}\) hours to an improper fraction:
\[
28 \frac{1}{8} = 28 + \frac{1}{8} = \frac{28 \times 8 + 1}{8} = \frac{224 + 1}{8} = \frac{225}{8} \text{ hours}
\]

Next, we find the rate at which the insulator is installed. The worker installs 40 insulators in \( \frac{75}{4} \) hours.

3. Calculate the installation rate in insulators per hour:
\[
\text{Rate} = \frac{40 \text{ insulators}}{\frac{75}{4} \text{ hours}} = 40 \times \frac{4}{75} = \frac{160}{75} = \frac{32}{15} \text{ insulators/hour}
\]

Now, we can use this rate to find out how many insulators can be installed in \(28 \frac{1}{8}\) hours (which is \(\frac{225}{8}\) hours).

4. Calculate the total number of insulators installed in \( \frac{225}{8} \) hours:
\[
\text{Total insulators} = \text{Rate} \times \text{Time} = \frac{32}{15} \text{ insulators/hour} \times \frac{225}{8} \text{ hours}
\]
To compute this product:
\[
\text{Total insulators} = \frac{32 \times 225}{15 \times 8}
\]

5. Simplifying the calculation:
- First, calculate \( \frac{225}{15} = 15 \), so:
\[
\text{Total insulators} = \frac{32 \times 15}{8} = \frac{480}{8} = 60
\]

Therefore, the line worker should be able to install **60 insulators** in \(28 \frac{1}{8}\) hours.