2420.62 – Brick Wall

Tim has employed Kim and Nim as bricklayers for the summer. Jim wants a brick path leading from the street, through his garden, to the side door. Tim estimates that Nim could build the wall in 9 hours and Kim could build it in 10. However, he has learned that when they work together, their combined output decreases by 10 bricks per hour.

Nevertheless, being in a hurry, he puts them both to work on it and finds that it takes them exactly 5 hours, working together, to finish the path. How many bricks are in the path?

Solution

Let bb be the number of bricks in the wall. In one hour, N=b9N = \dfrac{b}{9} and K=b10K = \dfrac{b}{10}. Together, their hourly rate is b9+b10−10\dfrac{b}{9}+\dfrac{b}{10}-10. So,

5(b9+b10−10)=b5b9+5b10−50=b50b+45b90−b=5095b90−b=505b90=50b=50⋅905=900\begin{align*}5 \left(\dfrac{b}{9}+\dfrac{b}{10}-10\right) &= b \\[1em]\dfrac{5b}{9}+\dfrac{5b}{10}-50 &= b \\[1em]\dfrac{50b+45b}{90}-b &= 50 \\[1em]\dfrac{95b}{90}-b&=50 \\[1em]\dfrac{5b}{90}&=50 \\[1em]b&=\dfrac{50\cdot90}{5} = 900\\[1em]\end{align*}