How to Solve Work and Time Problems Step by Step
Have you ever heard someone tell you that you could work faster if you did something alone? Surely there is a distraction factor, but can one person complete a task faster alone when it requires more people? There is a way to determine this through a simple understanding of the time and basic work concepts. One must relate work done with the amount of time taken. This can be calculated in ideal scenarios through ratios and simplifying to unity.
What Is The Relation Between Work And Time?
The amount of work done and the time taken to do the same is the main concept required to find the relation between the two. The speed at which a person does work, even the energy spent on the task, can be determined when these two variables are provided.
How To Solve Work Time Problems?
One must keep the following two key points in mind to solve work time problems.
When a person has done a certain amount of work in ‘x’ days, it means he has done $\frac{1}{x}$amount of work in one day.
If a person has done $\frac{1}{y}$ amount of work in one day, it means he will take ‘y’ days to complete it.
Therefore, we can see the relationship between work and time works on the principle of reciprocals.
Word Problems With Time And Work
Q1. If Betty takes 3 hours to cook a meal for 10 people, how many hours will she take to cook a meal for 32 people?
Solution: Time is taken Betty to cook for 10 people = 3 hours
Time taken by her to cook for 1 person = $\frac{3}{10}$hours
To cook for 32 people Betty will take = $32\times \frac{3}{10}$
∴ Time taken by Betty is $9.6$hours = $9$hours and $36$minutes ($\because 1hr=60m$)
Q2. If Raj is 4 times faster than Shyam in shovelling snow, how much time will Raj take to shovel all the snow when they can complete it together in 5 days?
Solution:
Raj’s one day work: Shyam’s one day work = $4:1$
Total work done by both in one day = $\frac{1}{5}$
Total work done by Raj = $\frac{4}{5}$
Work done by Raj in one day = $\frac{5}{4}$
∴ Time required by Raj to do the work alone = $\frac{5}{4}\times 5=\frac{25}{4}=6.25$days
Q3. If Sakshi, Disha, Hari and Ramu can do a task in 12, 18, 15 and 24 days, respectively. How many days will it take to do the task if they all do it together?
Solution:
Time taken by Sakshi = 12 days
Time taken by Disha = 18 days
Time taken by Hari = 15 days
Time taken by Ramu = 24 days
Ratio of work done by each in one day $=\frac{1}{12}:\frac{1}{18}:\frac{1}{15}:\frac{1}{24}$
$=12:18:15:24=4:6:5:8$
Therefore, total number of days required = sum of the ratio terms = $4+6+5+8$
Ans. 23 days
Q4. If Rita does some work in 2 days and Mina does two times that work in 5 days, compare the work done by each in one day.
Solution:
Work done by Rita in one day = $\frac{1}{2}$
Work done by Mina in one day = $\frac{2}{5}$
$\therefore $Ratio of work done by both in one day = $\frac{1}{2}:\frac{2}{5}$
FAQs on Work and Time Concept Explained with Examples
1. What is the fundamental concept of 'Work and Time' in Maths?
The fundamental concept of 'Work and Time' is based on the inverse relationship between the number of people doing a job and the time it takes to complete it. Essentially, if you increase the number of workers (assuming they work at the same rate), the time required to finish the work decreases, and vice versa. All calculations in this topic are based on determining the rate of work.
2. What is the basic formula that connects work, rate, and time?
The core relationship in work and time problems is defined by the formula: Work = Rate × Time. From this, we can derive two other essential formulas:
- Rate = Work / Time (This tells you how much work is done per unit of time).
- Time = Work / Rate (This tells you how long it takes to complete a certain amount of work at a given rate).
3. Can you explain the work and time relationship with a real-world example?
Certainly. Imagine a painter takes 10 hours to paint one wall. Here, the 'work' is painting 1 wall and the 'time' is 10 hours. The painter's rate of work is 1/10 of the wall per hour. If another painter who works at the same rate joins him, their combined rate becomes (1/10 + 1/10) = 2/10 or 1/5 of the wall per hour. Together, they can paint the entire wall in just 5 hours, demonstrating how more workers reduce the time taken.
4. How is the 'Unitary Method' used to solve work and time problems?
The Unitary Method is the most common technique for solving these problems. It involves two main steps:
- First, you calculate the amount of work an individual can do in one unit of time (like one day or one hour). For example, if a person finishes a task in 'x' days, their one-day's work is 1/x. This value (1/x) represents their work rate.
- Second, you use this per-unit-time rate to calculate the total time required or the work done in a specific period.
5. How do you calculate the combined work rate when two or more people work together?
When multiple people work together, their individual work rates are added to find the combined work rate. If Person A can complete a job in 'A' days (rate = 1/A) and Person B can complete it in 'B' days (rate = 1/B), their combined work rate is (1/A + 1/B). The time they would take to complete the work together is the reciprocal of this combined rate, i.e., 1 / (1/A + 1/B).
6. What is the difference between positive and negative work in these problems?
Positive and negative work refer to whether work is being done to complete a task or to undo it.
- Positive Work: This is constructive work, like filling a tank, building a wall, or completing an assignment. The work rates are positive values.
- Negative Work: This is destructive or emptying work, like a leak draining a tank or a person demolishing a wall. The work rate is treated as a negative value in calculations.
7. What is the MDH formula, and in what type of questions is it useful?
The MDH formula is a shortcut for problems involving groups of workers. It stands for Men, Days, and Hours. The formula is expressed as: M₁D₁H₁ / W₁ = M₂D₂H₂ / W₂, where 'M' is the number of workers, 'D' is the number of days, 'H' is the hours per day, and 'W' is the amount of work. This formula is extremely useful for comparing two scenarios where the number of workers, days, or hours changes, allowing you to find the missing variable quickly.
8. What should be the first step when a problem states one person is 'n times as efficient' as another?
When a problem mentions efficiency, the first step is to establish a ratio of their work rates. If person A is 'n times as efficient' as person B, it means A can do 'n' times the work B can do in the same amount of time. Therefore, the ratio of their work rates is A:B = n:1. This also implies that the time taken will be in the inverse ratio, i.e., Time(A):Time(B) = 1:n.
9. Why is the rate of work assumed to be constant in these problems?
The assumption of a constant work rate is a key simplification used in 'Work and Time' problems to make them solvable with basic arithmetic. In reality, a person's efficiency might decrease due to fatigue or increase with practice. However, for mathematical purposes, we assume each person or machine works at a steady, unchanging pace throughout the task. This allows us to create predictable and solvable equations.
10. If two people complete a job in 8 hours together, does it mean each person works for 4 hours?
No, this is a common misconception. When two people complete a job together in 8 hours, it means they are both working simultaneously for the full 8 hours. The total duration is 8 hours for both of them. Their individual contribution to the 'work' might differ based on their respective work rates, but the 'time' they spend working together is the same.
