Painting Houses

Will can paint a house in 3 hours. Sam can paint a house in 5 hours. How long will it take for them to paint it together?

\(\begin{array}{lll} \text{Will painting } & \text{per hour} & \frac13 \text{ house }\\ \text{Sam painting } & \text{per hour} & \frac15 \text{ house }\\\\ \end{array}\\ \boxed{~ \begin{array}{lll} \frac13 \frac{\text{house}}{\text{hour}} \cdot t + \frac15 \frac{\text{house}}{\text{hour}} \cdot t & =& 1\ \text{ house } \end{array} ~}\\ \begin{array}{rcl} \\ t \cdot \left( \frac13 \frac{\text{house}}{\text{hour}} + \frac15 \frac{\text{house}}{\text{hour}} \right) & =& 1\ \text{ house }\\ t \cdot \left( \frac13 + \frac15 \right) \frac{\text{house}}{\text{hour}} & =& 1\ \text{ house }\\ t \cdot \left( \frac13\cdot \frac55 + \frac15\cdot \frac33 \right) \frac{\text{house}}{\text{hour}} & =& 1\ \text{ house }\\ t \cdot \left( \frac{5}{15} + \frac{5}{15} \right) \frac{\text{house}}{\text{hour}} & =& 1\ \text{ house }\\ t \cdot \left( \frac{8}{15} \right) \frac{\text{house}}{\text{hour}} & =& 1\ \text{ house }\\ t & =& \frac{15}{8}\ \text{ house }\frac{\text{hour}}{\text{house}}\\ t & =& \frac{15}{8}\ \text{ hour }\\ t & =& 1\ \text{ hour } 52.5\ \text{ minutes }\\ \end{array}\)

Here's an easy way to solve this kind of problem:

Add the fractions together

1/3  + 1/5  =  

5/15 + 3/15  =

8/15

Take the reciprocal of this  =

15 / 8  = 

1 7/8   hours