Problem 50
Question
Walt can mow a lawn in 1 hour, and his son, Malik, can mow the same lawn in 50 minutes. One day Malik started mowing the lawn by himself and worked for 30 minutes. Then Walt joined him and they finished the lawn. How long did it take them to finish mowing the lawn after Walt started to help?
Step-by-Step Solution
Verified Answer
It took approximately 10.91 minutes.
1Step 1: Determine the mow rate of Walt and Malik
Walt takes 1 hour to mow the lawn, so his rate is \( \frac{1}{60} \) lawns per minute. Malik takes 50 minutes, so his rate is \( \frac{1}{50} \) lawns per minute.
2Step 2: Calculate the amount of the lawn Malik mows alone
Malik works for 30 minutes alone. So, in 30 minutes, Malik mows \( 30 \times \frac{1}{50} = \frac{30}{50} = \frac{3}{5} \) of the lawn.
3Step 3: Determine the remaining portion of the lawn
The part of the lawn still needed to be mowed after Malik's 30 minutes is \( 1 - \frac{3}{5} = \frac{2}{5} \) of the lawn.
4Step 4: Calculate the combined mow rate of Walt and Malik
When Walt joins, their combined rate is \( \frac{1}{60} + \frac{1}{50} = \frac{5}{300} + \frac{6}{300} = \frac{11}{300} \) lawns per minute.
5Step 5: Determine the time taken to finish mowing
To mow \( \frac{2}{5} \) of the lawn at a rate of \( \frac{11}{300} \) lawns per minute, the time required is \( \frac{2/5}{11/300} = \frac{2 \times 300}{5 \times 11} = \frac{600}{55} = \frac{120}{11} \approx 10.91 \) minutes.
Key Concepts
Rate ProblemsWork and Time ProblemsAlgebraic Fractions
Rate Problems
In the context of collaborative work, understanding rate problems is crucial. Rate problems involve figuring out how much work can be completed in a specific amount of time.
In these problems, we often express the rate as a unit of work per time (e.g., lawns per minute or pages per hour). For instance, if Walt can mow a lawn in 1 hour, his rate is calculated as the fraction of the lawn he can mow per minute, which is \( \frac{1}{60} \) lawns per minute.
This problem-solving approach helps us determine how quickly different workers or machines complete tasks when working together or apart. It's essential to express individual rates first before analyzing combined efforts in collaborative scenarios.
In these problems, we often express the rate as a unit of work per time (e.g., lawns per minute or pages per hour). For instance, if Walt can mow a lawn in 1 hour, his rate is calculated as the fraction of the lawn he can mow per minute, which is \( \frac{1}{60} \) lawns per minute.
This problem-solving approach helps us determine how quickly different workers or machines complete tasks when working together or apart. It's essential to express individual rates first before analyzing combined efforts in collaborative scenarios.
Work and Time Problems
Work and time problems often require us to assess how long it takes different people or machines, working at different speeds, to complete a job together. In our example, Malik starts mowing alone, and after some time, Walt assists.
This type of problem is clearly illustrated in the exercise, where Malik can mow a lawn in 50 minutes, translating his work rate to \( \frac{1}{50} \) lawns per minute. Walt, on the other hand, has a rate of \( \frac{1}{60} \) lawns per minute. When working together, their task completion rate becomes the sum of their individual rates.
The key is understanding how much of the job was completed by Malik before Walt started and then determining how quickly they collectively finish the rest. This involves calculating the remaining work and dividing by the combined rate.
This type of problem is clearly illustrated in the exercise, where Malik can mow a lawn in 50 minutes, translating his work rate to \( \frac{1}{50} \) lawns per minute. Walt, on the other hand, has a rate of \( \frac{1}{60} \) lawns per minute. When working together, their task completion rate becomes the sum of their individual rates.
The key is understanding how much of the job was completed by Malik before Walt started and then determining how quickly they collectively finish the rest. This involves calculating the remaining work and dividing by the combined rate.
Algebraic Fractions
Algebraic fractions play a significant role in solving work and rate problems. They help us represent and analyze parts of the work done over time.
These fractions allow us to break down complex tasks into manageable pieces, particularly in collaborative settings where different workers contribute variably. For example, Malik mowing \( \frac{3}{5} \) of the lawn alone means we need to calculate the remainder, \( \frac{2}{5} \), that Walt and Malik must finish together.
The use of algebraic fractions simplifies calculating combined work rates and determining the time needed to complete a task. By using fractions, we can easily add, subtract, and compare different rates of work, ensuring we accurately calculate completion times.
These fractions allow us to break down complex tasks into manageable pieces, particularly in collaborative settings where different workers contribute variably. For example, Malik mowing \( \frac{3}{5} \) of the lawn alone means we need to calculate the remainder, \( \frac{2}{5} \), that Walt and Malik must finish together.
The use of algebraic fractions simplifies calculating combined work rates and determining the time needed to complete a task. By using fractions, we can easily add, subtract, and compare different rates of work, ensuring we accurately calculate completion times.
Other exercises in this chapter
Problem 49
For Problems 13-50, perform the indicated operations involving rational expressions. Express final answers in simplest form. \(\frac{a^{2}-4 a b+4 b^{2}}{6 a^{2
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For Problems 9-50, simplify each rational expression. \(\frac{-40 x^{3}+24 x^{2}+16 x}{20 x^{3}+28 x^{2}+8 x}\)
View solution Problem 50
Set up an algebraic equation and solve each problem. The sum of two numbers is 80 . If the larger is divided by the smaller, the quotient is 7 , and the remaind
View solution Problem 50
Perform the indicated divisions. $$ \left(x^{4}-1\right) \div(x-1) $$
View solution