   SEARCH HOME Math Central Quandaries & Queries  Question from Felicia, a student: Sarah takes 3 hours longer to paint a floor than it takes Kate. When they work together it takes 2 hours. How long would each take to do the job alone? Hi Felicia,

Suppose it takes Sarah s hours to paint a floor and it takes Kate k hours to paint a floor. The first sentence in your problem tells you that s = k + 3.

For the "working together" part of the problem you need to think in terms of rates. Sarah works at the rate of s hours per floor or equivalently 1/s floors per hour. Kate's rate is 1/k floors per hour. Thus together they work at the rate of 1/s + 1/k floors per hour. Use the fact that s = k + 3 to express this rate in terms of k alone.

Use a common denominator and write this sum of two fractions as one fraction. Invert this fraction to find their combines rate in hours per floor. But you know that their combined rate is 2 hours per floor so this gives you an equation you can solve for k.

I hope this helps,
Penny     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.