Question

one painter can paint a house in 7 days, yet a more experienced painter can paint the house in 5 days. how many days would it take the two painters, working together, to paint the house?
it will take them (square) days to paint the house together.
if needed, round answer to 3 decimal places.

Explanation:

Step1: Define work rates

Let the work (painting the house) be \( W = 1 \) (one house). The first painter's rate \( r_1=\frac{W}{t_1}=\frac{1}{7} \) (houses per day), and the second painter's rate \( r_2=\frac{W}{t_2}=\frac{1}{5} \) (assuming the more experienced painter takes 5 days, as there was a typo? Wait, original says "in 5 days"? Let's proceed with \( t_1 = 7 \), \( t_2 = 5 \)). When working together, their combined rate \( r = r_1 + r_2 \).

Step2: Calculate combined rate

\( r=\frac{1}{7}+\frac{1}{5}=\frac{5 + 7}{35}=\frac{12}{35} \) (houses per day).

Step3: Find time for combined work

Time \( t=\frac{W}{r}=\frac{1}{\frac{12}{35}}=\frac{35}{12}\approx2.917 \) days. Wait, wait, original problem: "more experienced painter can paint the house in 5 days"? Wait, the user's text: "in 5 days"? Let me check again. The problem says: "One painter can paint a house in 7 days, yet a more experienced painter can paint the house in 5 days. How many days would it take the two painters, working together, to paint the house?" So yes, \( t_1 = 7 \), \( t_2 = 5 \).

So combined rate: \( \frac{1}{7}+\frac{1}{5}=\frac{5 + 7}{35}=\frac{12}{35} \). Then time \( t = \frac{1}{\frac{12}{35}}=\frac{35}{12}\approx2.917 \) days. Wait, but maybe the second painter's time was a typo? Wait, the original user's text: "in 5 days" (I think that's what was intended, maybe a typo in the original problem statement, but assuming that). So the steps are:

1. Determine individual work rates (work per day).
2. Add the rates to get combined rate.
3. Take reciprocal of combined rate to get time.

Answer:

\( \frac{35}{12}\approx2.917 \) (or if the second painter's time was different, but based on the problem as presented, with 7 and 5 days, the answer is \( \frac{35}{12}\approx2.917 \))