It takes two hours for a person X to mow the lawn. Y can mow the same lawn in four hours. How long (in minutes) will it take X and Y, if they work together to mow the lawn?
80
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.
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