Time Problem

Question

It takes 3 hours less for the painter to paint a house than his brother. Working together, they paint the same house in 6 hours and 40 minutes. How long will each of them take to paint the whole house on their own.

Answer 1

Let's say the painter takes p hours to paint the house on his own, then his brother takes p+3 hours.

Let's suppose the house needs P litres of paint to paint it. The paint rate for the painter is P/p litres per hour; his brother's rate is P/(p+3) l/hr. If the house takes 6 2/3=20/3 hrs to paint when they work together then the paint rate is P÷20/3=3P/20. When we add the paint rates together we have P/p+P/(p+3)=3P/20 so we can cancel P out leaving 1/p+1/(p+3)=3/20. Multiply through by 20p(p+3):

20p+60+20p=3p(p+3)=3p^2+9p; 3p^2-31p-60=0=(3p+5)(p-12). We only need the positive root p=12 hours.

The painter takes 12 hours to paint the house on his own and his brother takes 15 hours.