# The time it takes to lay a sidewalk of a certain type varies directly as the length and inversely as the number of men working. If eight men take two days to lay 100 feet, how long will three men take to lay 150 feet?

Jul 2, 2016

$8$ days

#### Explanation:

As this question has both direct and inverse variation in it, let's do one part at a time:

Inverse variation means as one quantity increases the other decreases. If the number of men increases, the time taken to lay the sidewalk will decrease.

Find the constant: When 8 men lay 100 feet in 2 days:

$k = x \times y \Rightarrow 8 \times 2 , \text{ } k = 16$

The time taken for 3 men to lay 100 feet will be $\frac{16}{3} = 5 \frac{1}{3}$ days

We see that it will take more days, as we expected.

Now for the direct variation. As one quantity increases, the other also increases. It will take longer for the three men to lay 150 feet than 100 feet. The number of men stays the same.

For 3 men laying 150 feet, the time will be

$\frac{x}{150} = \frac{5 \frac{1}{3}}{100} \Rightarrow x = \frac{\frac{16}{3} \times 150}{100}$

= $\frac{16 \times 150}{3 \times 100} = \frac{16 \times {\cancel{150}}^{\cancel{3}}}{\cancel{3} \times {\cancel{100}}^{2}}$

= $\frac{16}{2} = 8$days