Question

1. kelly can paint a kitchen in 6 hours and tom can paint in 8 hours. how long will it take them to paint a kitchen if they work together?

Explanation:

Step1: Define work rates

Let the total work (painting a kitchen) be \( W = 1 \). Kelly's rate \( r_{k}=\frac{W}{t_{k}}=\frac{1}{6} \) per hour (since she takes 6 hours). Tom's rate \( r_{t}=\frac{W}{t_{t}}=\frac{1}{8} \) per hour (since he takes 8 hours).

Step2: Combined work rate

When working together, their combined rate \( r = r_{k}+r_{t}=\frac{1}{6}+\frac{1}{8} \). Find a common denominator (24): \( \frac{1}{6}+\frac{1}{8}=\frac{4}{24}+\frac{3}{24}=\frac{7}{24} \) per hour.

Step3: Time to complete together

Let \( t \) be the time together. Then \( r\times t = W \), so \( t=\frac{W}{r}=\frac{1}{\frac{7}{24}}=\frac{24}{7}\approx 3.43 \) hours.

Answer:

\(\frac{24}{7}\) hours (or approximately \( 3.43 \) hours)