X's Rate of Work

Person X takes 2 hours to mow the lawn. This means X completes $\frac{1}{2}$ of the lawn per hour. This is X's work rate.

Y's Rate of Work

Person Y takes 4 hours to mow the same lawn. Therefore, Y completes $\frac{1}{4}$ of the lawn per hour. This is Y's work rate.

Calculating Combined Rate

When X and Y work together, their individual rates add up to find their combined rate.

Combined Rate = X's Rate + Y's Rate

Combined Rate = $\frac{1}{2} + \frac{1}{4}$

To add these fractions, we find a common denominator, which is 4.

Combined Rate = $\frac{2}{4} + \frac{1}{4} = \frac{3}{4}$

So, together they complete $\frac{3}{4}$ of the lawn per hour.

Determining Total Time Together

The time it takes to complete a job when working together is the inverse of their combined rate.

Time = $\frac{\text{Total Work}}{\text{Combined Rate}}$

Assuming the total work is mowing 1 lawn:

Time = $\frac{1 \text{ lawn}}{\frac{3}{4} \text{ lawn/hour}} = \frac{4}{3}$ hours

Converting Time to Minutes

The question asks for the time in minutes. Since 1 hour is equal to 60 minutes, we convert the time:

Time in minutes = $\frac{4}{3} \text{ hours} \times 60 \frac{\text{minutes}}{\text{hour}}$

Time in minutes = $\frac{4 \times 60}{3} = \frac{240}{3}$

Time in minutes = 80 minutes

Therefore, it will take X and Y 80 minutes to mow the lawn if they work together.