Question

1. to build a simple dog house, it would take dan four hours working alone. if fran helps dan, it will take them two and a half hours to build the same dog house. how long would it take fran to do the job alone? express the answer in hours and minutes.

Explanation:

Step1: Define work rates

Let \( D \) be Dan's work rate (jobs per hour) and \( F \) be Fran's work rate (jobs per hour). The total work to build a dog house is 1 job. Dan takes 4 hours alone, so his rate \( D=\frac{1}{4} \) job per hour. When they work together, they take \( 2.5=\frac{5}{2} \) hours, so their combined rate is \( \frac{1}{\frac{5}{2}}=\frac{2}{5} \) job per hour. The combined rate is also \( D + F \).

Step2: Solve for Fran's rate

We know \( D + F=\frac{2}{5} \) and \( D=\frac{1}{4} \). Substitute \( D \) into the equation:
\( \frac{1}{4}+F=\frac{2}{5} \)
Subtract \( \frac{1}{4} \) from both sides:
\( F=\frac{2}{5}-\frac{1}{4} \)
Find a common denominator (20):
\( F=\frac{8}{20}-\frac{5}{20}=\frac{3}{20} \) job per hour.

Step3: Find Fran's time alone

Time is reciprocal of rate, so Fran's time \( t=\frac{1}{F}=\frac{1}{\frac{3}{20}}=\frac{20}{3} \) hours. Convert \( \frac{20}{3} \) hours to hours and minutes. \( \frac{20}{3}=6+\frac{2}{3} \) hours. \( \frac{2}{3} \) of an hour is \( \frac{2}{3}\times60 = 40 \) minutes.

Answer:

6 hours and 40 minutes