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If Johnny can mow the lawn in 30 minutes...

If Johnny can mow the lawn in 30 minutes and with the help of his brother, Bobby, they can mow the lawn 20 minutes, how long would it take Bobby working along to mow the lawn?

A

1/2 hour

B

3/4 hour

C

1 hour

D

`3//2` hour

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem step by step, we can follow these instructions: ### Step 1: Understand the problem We know that Johnny can mow the lawn in 30 minutes, and together with Bobby, they can mow the lawn in 20 minutes. We need to find out how long it would take Bobby to mow the lawn alone. ### Step 2: Determine the work rates - **Johnny's work rate**: Since Johnny can mow the lawn in 30 minutes, his work rate is: \[ \text{Work rate of Johnny} = \frac{1 \text{ lawn}}{30 \text{ minutes}} = \frac{1}{30} \text{ lawns per minute} \] - **Combined work rate of Johnny and Bobby**: Since they can mow the lawn together in 20 minutes, their combined work rate is: \[ \text{Work rate of Johnny and Bobby} = \frac{1 \text{ lawn}}{20 \text{ minutes}} = \frac{1}{20} \text{ lawns per minute} \] ### Step 3: Set up the equation for Bobby's work rate Let \( x \) be the time in minutes it takes Bobby to mow the lawn alone. Therefore, Bobby's work rate is: \[ \text{Work rate of Bobby} = \frac{1 \text{ lawn}}{x \text{ minutes}} = \frac{1}{x} \text{ lawns per minute} \] ### Step 4: Write the equation based on the combined work rates According to the problem, the sum of Johnny's work rate and Bobby's work rate equals their combined work rate: \[ \frac{1}{30} + \frac{1}{x} = \frac{1}{20} \] ### Step 5: Solve for \( \frac{1}{x} \) Rearranging the equation gives: \[ \frac{1}{x} = \frac{1}{20} - \frac{1}{30} \] ### Step 6: Find a common denominator The least common multiple of 20 and 30 is 60. We can rewrite the fractions: \[ \frac{1}{20} = \frac{3}{60} \quad \text{and} \quad \frac{1}{30} = \frac{2}{60} \] Thus, we have: \[ \frac{1}{x} = \frac{3}{60} - \frac{2}{60} = \frac{1}{60} \] ### Step 7: Solve for \( x \) Taking the reciprocal gives: \[ x = 60 \text{ minutes} \] ### Step 8: Convert minutes to hours Since 60 minutes is equal to 1 hour, we conclude that: \[ \text{Bobby takes 1 hour to mow the lawn.} \] ### Final Answer Bobby takes **1 hour** to mow the lawn. ---
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