Using a riding lawnmower, Je can cut his lawn in 45 minutes. Using a push power, Joe's son can cut the lawn in 3 hours. If they work together, how long will it take them to cut the lawn?

Options

A. 24 minutes
B. 30 minutes
C. 36 minutes
D. 40 minutes

Explanation

To solve the problem of how long it will take Joe and his son to cut the lawn together, we can follow these steps:

### Step 1: Determine the rates of work for Joe and his son.
- Joe can cut the lawn in 45 minutes. Therefore, his rate of work (R_J) is:
\[ R_J = \frac{W}{45} \]
- Joe's son can cut the lawn in 3 hours (which is 180 minutes). Therefore, his rate of work (R_S) is:
\[ R_S = \frac{W}{180} \]

### Step 2: Set up the equation for their combined work rate.
When they work together, their combined rate of work is the sum of their individual rates:
\[ R_J + R_S = \frac{W}{T} \]
where T is the time taken for them to cut the lawn together.

Substituting the rates we found:
\[ \frac{W}{45} + \frac{W}{180} = \frac{W}{T} \]

### Step 3: Simplify the equation.
Since W is common in all terms, we can cancel it out:
\[ \frac{1}{45} + \frac{1}{180} = \frac{1}{T} \]

### Step 4: Find a common denominator.
The least common multiple of 45 and 180 is 180. We can rewrite the fractions:
\[ \frac{1 \times 4}{45 \times 4} + \frac{1}{180} = \frac{4}{180} + \frac{1}{180} = \frac{5}{180} \]

### Step 5: Set the equation to solve for T.
Now we have:
\[ \frac{5}{180} = \frac{1}{T} \]
Taking the reciprocal gives:
\[ T = \frac{180}{5} \]

### Step 6: Calculate T.
Now, divide 180 by 5:
\[ T = 36 \]

### Conclusion:
Therefore, it will take Joe and his son 36 minutes to cut the lawn together.

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