Question

3
a painter takes 3 hours to paint \\(\frac{1}{8}\\) of a mural.
which statement correctly explains how much of the mural can be painted in 10 hours at the same rate?
a. the painter paints \\(\frac{1}{24}\\) of the mural per hour and will paint \\(\frac{5}{12}\\) of the mural in 10 hours.
b. the painter paints \\(\frac{1}{24}\\) of the mural per hour and will paint \\(\frac{1}{240}\\) of the mural in 10 hours.
c. the painter paints \\(\frac{3}{8}\\) of the mural per hour and will paint \\(\frac{15}{4}\\) of the mural in 10 hours.
d. the painter paints \\(\frac{3}{8}\\) of the mural per hour and will paint \\(\frac{3}{80}\\) of the mural in 10 hours.
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Explanation:

Step1: Find the rate per hour

The painter takes 3 hours to paint $\frac{1}{8}$ of a mural. To find the rate per hour, we divide the amount painted by the time taken. So the rate $r$ is $\frac{\frac{1}{8}}{3}$. Using the rule of dividing by a number is the same as multiplying by its reciprocal, we get $r=\frac{1}{8}\times\frac{1}{3}=\frac{1}{24}$ of the mural per hour. Wait, no, wait. Wait, $\frac{1}{8}$ mural in 3 hours, so per hour is $\frac{1}{8}\div3=\frac{1}{8}\times\frac{1}{3}=\frac{1}{24}$? Wait, no, that can't be. Wait, no, let's recalculate. Wait, 3 hours for $\frac{1}{8}$ mural, so 1 hour is $\frac{1}{8}\div3=\frac{1}{24}$? Wait, but let's check option A. Option A says the painter paints $\frac{1}{24}$ per hour? Wait, no, wait, $\frac{1}{8}$ divided by 3 is $\frac{1}{24}$? Wait, no, $\frac{1}{8}\div3=\frac{1}{24}$? Wait, 3 times $\frac{1}{24}$ is $\frac{3}{24}=\frac{1}{8}$, which matches. Then in 10 hours, the amount painted is rate times time, so $\frac{1}{24}\times10=\frac{10}{24}=\frac{5}{12}$. Let's check the options. Option A: "The painter paints $\frac{1}{24}$ of the mural per hour and will paint $\frac{5}{12}$ of the mural in 10 hours." Wait, wait, earlier I miscalculated the rate. Wait, 3 hours for $\frac{1}{8}$ mural, so rate is $\frac{1}{8}$ mural / 3 hours = $\frac{1}{24}$ mural per hour. Then 10 hours: $\frac{1}{24}\times10=\frac{10}{24}=\frac{5}{12}$. So option A is correct. Wait, let's check the other options. Option B: rate $\frac{1}{240}$ is wrong. Option C: rate $\frac{3}{8}$ is wrong because 3 hours for $\frac{1}{8}$, so per hour is less than $\frac{1}{8}$. Option D: rate $\frac{3}{8}$ is wrong. So option A is correct.

Wait, let's re-express the steps:

Step1: Calculate the hourly rate

The amount painted in 3 hours is $\frac{1}{8}$ of the mural. To find the rate (amount per hour), we divide the amount by the time:
Rate $= \frac{\frac{1}{8}}{3} = \frac{1}{8} \times \frac{1}{3} = \frac{1}{24}$ of the mural per hour.

Step2: Calculate the amount painted in 10 hours

Using the rate, the amount painted in 10 hours is:
Amount $= \text{Rate} \times \text{Time} = \frac{1}{24} \times 10 = \frac{10}{24} = \frac{5}{12}$.

Answer:

A. The painter paints $\frac{1}{24}$ of the mural per hour and will paint $\frac{5}{12}$ of the mural in 10 hours.