A gardener can care for the Green’s property in 5 hours. If his helper assists him, they can complete the job in 4 hours. How long will it take the helper to do the job?
The correct answer and explanation is:
Correct Answer: 20 hours
Step-by-step Explanation:
This is a classic work problem where we calculate how long each person takes to do a job alone or together. Let’s define the variables and solve:
Let:
- The gardener alone takes 5 hours, so his work rate is 15 of the job per hour\frac{1}{5} \text{ of the job per hour}
- The helper’s time to complete the job alone is unknown, so let it be xx hours. Then the helper’s work rate is 1x of the job per hour\frac{1}{x} \text{ of the job per hour}
When they work together, they can finish the job in 4 hours, meaning their combined rate is 14 of the job per hour\frac{1}{4} \text{ of the job per hour}
Set up the equation:
Gardener’s rate+Helper’s rate=Combined rate\text{Gardener’s rate} + \text{Helper’s rate} = \text{Combined rate} 15+1x=14\frac{1}{5} + \frac{1}{x} = \frac{1}{4}
Solve for xx:
Subtract 15\frac{1}{5} from both sides: 1x=14−15\frac{1}{x} = \frac{1}{4} – \frac{1}{5}
Find a common denominator: 1x=5−420=120\frac{1}{x} = \frac{5 – 4}{20} = \frac{1}{20}
So, x=20x = 20
Final Answer:
It will take the helper 20 hours to complete the job alone.
Explanation:
This problem illustrates the inverse relationship between time and work rate. The faster someone works (less time needed), the higher their rate. When two people work together, their combined rate is the sum of their individual rates. By turning time into rate and solving algebraically, we can determine how long each person would need if working alone.
This type of problem is commonly seen in time-and-work scenarios, useful in real-life planning of jobs, especially when dealing with teams.