Worked Problems
Simple Working Rate Problems
Problem #1
It takes 3 days for 7 men to paint a house. How long will it take 3 men painting at the same rate?
A. 7 days
B. 6 days
C. 10 days
D. 14 days
How I Solve This
I solve this in three steps.
Step 1: Unit Time.
First, I determine how long it will take 1 man to paint the house. If it takes 3 days for 7 men, then it will take only 1 man about seven times as long. This means it will take a single guy 3 • 7 = 21 days to paint the house.
Step 2: New Crew's Time.
Now that I know it takes 21 days for a single man to paint the house, I can determine how long it will take 3 men working at the same rate. If there are 3 men instead of one, it will take a third of the time. This means that it will take the men ⅓ • 21, or 21/3 = 7 days to paint the house.
Step 3: Answer.
It will take 7 days for 3 men to paint the house, provided that they work at the same rate of the original 7 men. The correct answer is thus A.
Extra Step (For A+ Students): Check.
I check my answer by reasoning whether the answer makes sense.
If it takes 7 men 3 days, then it should take longer for 3 men working at the same rate.
Problem #2
If it takes 7 days for 5 men to plow a large commercial field, how many men should the managing company hire to get the job done in 3 days?
How I Solve This
I solve this in three steps.
Step 1: Unit Time.
How long will it take a single worker to plow the field?
Since it takes a crew of 5 men 7 days to plow the field, it would take a single man 5 times as long, or 7 • 5 = 35 days.
Step 2: New Crew's Size.
Now that I know it takes 35 days for a single worker to plow the large field, I can calculate how many workers are needed for a 3-day completion time by dividing 35 by 3.
35/3 ≈ 12 (rounded to the next whole number because you can’t have a fraction of a person)
Step 3: Answer.
The managing company will need to hire 12 men to plow the field in 3 days.
Extra Step (For A+ Students): Check.
I check my answer by reasoning whether the answer makes sense.
If it takes 5 men 7 days, then it makes sense that it would take 12 men 3 days. If there were 10 workers, they would get the job done in half the time (3.5 days). This is a close approximation to the answer.
Summary of My Method
To solve these kinds of problems, I follow 3 steps:
- Unit Time: I calculate the time it would take a single worker to complete the job working at the same rate.
- New Crew’s Size/Time: Depending on what I need to find, I either calculate how long it would take the crew of given size to complete the same job, or how many people are needed to complete a job in a given period of time.
- Answer: I write the answer I found in Step 2. Note that rounding may sometimes be required in these types of problems.
Also, I make sure to check my answers to ensure I calculated correctly.
If you have any questions or would like some help with problems like these, feel free to send me a message. I hope these calculations will help you as you continue on your journey of math mastery 🙂
~ Your Humble Study Buddy,
Ace