Question

If Sally can paint a house in 4 hours, and John can paint the same house in 6 hour, how long will it take for both of them to paint the house together?

• A. 2 hours and 24 minutes
• B. 3 hours and 12 minutes
• C. 3 hours and 44 minutes
• D. 4 hours and 10 minutes
• E. 4 hours and 33 minutes

Hint:

When two people work together, their combined time will always be less than the faster person's individual time. In this case, it must be less than 4 hours, which can help you eliminate some options.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This is a combined work problem. To find the time it takes for them to work together, we need to add their individual work rates.
Step 2: Key Formula or Approach:
The formula for combined work is: \[ \frac{1}{T_{\text{total}}} = \frac{1}{T_1} + \frac{1}{T_2} \] where \( T_1 \) and \( T_2 \) are the times taken by each individual, and \( T_{\text{total}} \) is the time taken when working together.
Step 3: Detailed Explanation:
First, calculate the individual rates:
Sally's rate = \( \frac{1 \text{ house}}{4 \text{ hours}} = \frac{1}{4} \) of the house per hour.
John's rate = \( \frac{1 \text{ house}}{6 \text{ hours}} = \frac{1}{6} \) of the house per hour.
Next, add their rates to get the combined rate:
\[ \text{Combined Rate} = \frac{1}{4} + \frac{1}{6} \] To add these fractions, find a common denominator, which is 12.
\[ \text{Combined Rate} = \frac{3}{12} + \frac{2}{12} = \frac{5}{12} \text{ of the house per hour} \] The time it takes to complete the house together is the reciprocal of the combined rate.
\[ \text{Time together} = \frac{1}{\text{Combined Rate}} = \frac{1}{5/12} = \frac{12}{5} \text{ hours} \] Now, convert the time into hours and minutes.
\( \frac{12}{5} \) hours = 2.4 hours.
The whole number part is 2 hours. The decimal part is 0.4 hours.
To convert 0.4 hours to minutes, multiply by 60:
\[ 0.4 \times 60 = 24 \text{ minutes} \] Step 4: Final Answer:
Working together, it will take them 2 hours and 24 minutes to paint the house. The correct option is (A).