Two boys can paint a fence in 5 hours. How many hours would it take 3 boys to paint the same fence?

Options

A. `(3)/(2)`
B. `3`
C. `3(1)/(3)`
D. `4(2)/(3)`

Answer

To solve the problem of how long it would take 3 boys to paint the same fence that 2 boys can paint in 5 hours, we can follow these steps:

### Step 1: Determine the work done by 2 boys in 1 hour.
If 2 boys can paint the fence in 5 hours, then in 1 hour, they can complete:
\[ \text{Work done by 2 boys in 1 hour} = \frac{1}{5} \text{ of the fence} \]

### Step 2: Calculate the work done by 1 boy in 1 hour.
Since 2 boys complete \(\frac{1}{5}\) of the fence in 1 hour, 1 boy would complete half of that work in the same time:
\[ \text{Work done by 1 boy in 1 hour} = \frac{1}{5} \times \frac{1}{2} = \frac{1}{10} \text{ of the fence} \]

### Step 3: Calculate the work done by 3 boys in 1 hour.
If 1 boy can paint \(\frac{1}{10}\) of the fence in 1 hour, then 3 boys will do:
\[ \text{Work done by 3 boys in 1 hour} = 3 \times \frac{1}{10} = \frac{3}{10} \text{ of the fence} \]

### Step 4: Determine how many hours it will take for 3 boys to paint the entire fence.
Let \(n\) be the number of hours it takes for 3 boys to finish painting the fence. Since they complete \(\frac{3}{10}\) of the fence in 1 hour, in \(n\) hours they will complete:
\[ \text{Work done in } n \text{ hours} = n \times \frac{3}{10} \]
We need this to equal 1 whole fence:
\[ n \times \frac{3}{10} = 1 \]

### Step 5: Solve for \(n\).
To find \(n\), we can rearrange the equation:
\[ n = \frac{1}{\frac{3}{10}} = \frac{10}{3} \]

### Step 6: Convert \(\frac{10}{3}\) hours into hours and minutes.
\(\frac{10}{3}\) hours can be expressed as:
\[ \frac{10}{3} = 3 \text{ hours and } \frac{1}{3} \text{ hour} \]
To convert \(\frac{1}{3}\) hour into minutes, we multiply by 60:
\[ \frac{1}{3} \times 60 = 20 \text{ minutes} \]
Thus, \(n = 3\) hours and \(20\) minutes.

### Final Answer:
It would take 3 boys a total of **3 hours and 20 minutes** to paint the fence.

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