Max can mow a lawn in 45 minutes. Jan takes twice as long to mow the same lawn. If they work together, the situation can be modeled by the following equation, where $t$ is the number of minutes it would take to mow the lawn together.
[tex]\frac{1}{45}+\frac{1}{90}=\frac{1}{t}[/tex]
How long will it take Maxine and Jan to mow the lawn together?
A. 30.0 minutes
B. 45.0 minutes
C. 67.5 minutes
D. 135.0 minutes
Answer (2)
Max and Jan can mow the lawn together in 30 minutes. This is calculated by combining their work rates and solving the equation for t . Therefore, the correct answer is 30.0 minutes.
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Find a common denominator and rewrite the fractions: 45 1 + 90 1 = 90 2 + 90 1 .
Combine the fractions: 90 2 + 90 1 = 90 3 = 30 1 .
Set up the equation: 30 1 = t 1 .
Solve for t by taking the reciprocal: t = 30 . The time it takes for them to mow the lawn together is 30.0 minutes.
Explanation
Problem Analysis Let's analyze the problem. We are given an equation that models the combined work rate of Max and Jan mowing a lawn together. Max can mow the lawn in 45 minutes, and Jan takes twice as long, which is 90 minutes. The equation is:
Equation Setup 45 1 + 90 1 = t 1 where t is the time it takes for them to mow the lawn together. We need to solve for t .
Finding Common Denominator To solve the equation, we first find a common denominator for the fractions on the left side. The least common denominator for 45 and 90 is 90. So we rewrite the equation as:
Rewriting Fractions 90 2 + 90 1 = t 1
Combining Fractions Now, we combine the fractions on the left side:
Simplified Equation 90 3 = t 1
Simplifying Fraction Simplify the fraction on the left side:
Further Simplification 30 1 = t 1
Solving for t Now, we take the reciprocal of both sides of the equation to solve for t :
Final Calculation t = 30
Conclusion Therefore, it will take Max and Jan 30 minutes to mow the lawn together.
Examples
Understanding how to combine work rates is useful in many real-life situations. For example, if you and a friend are painting a room, and you know how long it takes each of you to paint the room individually, you can use this type of equation to estimate how long it will take to paint the room together. This concept applies to various tasks, from construction projects to manufacturing processes, where multiple people or machines are working together to complete a job.
