My father-in-law used to have a garden in our backyard. We worried about him weeding the whole garden by himself so we usually sent one of our children out to help. Now, if my son worked on the garden by himself it would take him 3 1/2 hours longer to weed the garden than his grandfather took. However, if he worked with my father-in-law, my son worked twice as fast. If it took an hour and twenty minutes for them to weed the garden together, how long did it take my son working by himself?

The rate at which each person works is proportional to 1 over the time it takes for the person to work alone. The combined rate of both working together is the sum of the two rates. Therefore the time it takes when both work together is 1 over the sum of the rates, or:


T = \frac 1 {\frac 1 {T_1} + \frac 1 {T_2}} = \frac {T_1 T_2}{T_1 + T_2}


For your problem you have values for T1 and T2, but note that when they work together the son works faster than by himself, so use half the value of time that you were given for the son working alone. Combine with the fact that T2=T1+3.5, and solve for T2.