Joe and Jim together can mow a lawn in 6 hours. Joe alone can mow the same lawn in 10 hours. The question asks how long it would take Jim to mow the lawn alone.

1. Problem Description

Joe and Jim together can mow a lawn in 6 hours. Joe alone can mow the same lawn in 10 hours. The question asks how long it would take Jim to mow the lawn alone.

2. Solution Steps

Let $J$ be the rate at which Joe mows the lawn (fraction of lawn per hour), and $M$ be the rate at which Jim mows the lawn (fraction of lawn per hour).

We are given that Joe and Jim together can mow the lawn in 6 hours. This means their combined rate is $\frac{1}{6}$ lawns per hour. Therefore,

$J + M = \frac{1}{6}$

We are also given that Joe alone can mow the lawn in 10 hours. This means Joe's rate is $\frac{1}{10}$ lawns per hour. Therefore,

$J = \frac{1}{10}$

Substituting $J = \frac{1}{10}$ into the first equation, we get

$\frac{1}{10} + M = \frac{1}{6}$

$M = \frac{1}{6} - \frac{1}{10}$

To subtract the fractions, we need a common denominator, which is

3

0. $M = \frac{5}{30} - \frac{3}{30} = \frac{2}{30} = \frac{1}{15}$

So, Jim's rate is $\frac{1}{15}$ lawns per hour. Let $t$ be the time it takes Jim to mow the lawn alone. Then,

$M = \frac{1}{t}$

Therefore, $\frac{1}{15} = \frac{1}{t}$, which means $t = 15$ hours.

3. Final Answer

15 hours