Work and Time Formula Tricks and How to Solve Questions 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 Formulas and Examples
1. What is the work and time concept in maths?
The work and time concept in maths deals with how much work a person or machine can complete in a given time and how long it takes to finish a task. It is based on the idea that:
- Work ∝ Time × Efficiency (Rate)
- If a person completes a job in T days, then one day’s work = 1/T
This concept is commonly used in competitive exams to solve problems involving individual work, combined work, pipes and cisterns, and efficiency comparison.
2. What is the basic formula for work and time?
The basic formula for work and time is Work = Time × Rate. In most problems, total work is taken as 1 unit.
- If a person completes work in T days, then rate = 1/T
- Time = Work ÷ Rate
These formulas help calculate how long a task takes or how much work is done in a given time.
3. How do you calculate time when two people work together?
When two people work together, their combined time is found by adding their individual work rates. If A takes T₁ days and B takes T₂ days, then:
- Rate of A = 1/T₁
- Rate of B = 1/T₂
- Combined rate = 1/T₁ + 1/T₂
Total time taken together = 1 ÷ (1/T₁ + 1/T₂). This method is used in combined work and efficiency problems.
4. What is one day’s work in work and time problems?
One day’s work is the fraction of total work completed in one day, calculated as 1 ÷ total days required. For example:
- If a person finishes a job in 5 days
- One day’s work = 1/5
This concept is fundamental in solving work and time questions efficiently.
5. How do you solve a work and time problem step by step?
To solve a work and time problem, convert time into rates, combine them if needed, and then calculate the required time or work.
- Step 1: Assume total work = 1 unit
- Step 2: Find individual rates = 1/T
- Step 3: Add or subtract rates (for combined or opposite work)
- Step 4: Use Time = Work ÷ Rate
This structured method helps avoid common calculation mistakes.
6. What is the difference between work rate and efficiency?
Work rate is the amount of work done per unit time, while efficiency compares how fast one worker is relative to another.
- Work rate = Work done in 1 day (e.g., 1/8)
- Efficiency is directly proportional to rate
If A is twice as efficient as B, then A’s rate is 2 times B’s rate, and A will take half the time taken by B.
7. How do you calculate work and time when one person leaves in between?
When one person leaves midway, calculate the work done together first, then add the remaining work done by the remaining person.
- Step 1: Find combined rate
- Step 2: Multiply by time worked together
- Step 3: Subtract from total work (1 unit)
- Step 4: Divide remaining work by individual rate
This approach ensures accurate calculation in mixed work scenarios.
8. What is the LCM method in work and time?
The LCM method assumes total work as the LCM of individual times to simplify calculations.
- If A takes 4 days and B takes 6 days
- LCM of 4 and 6 = 12 units
- A’s one day work = 12/4 = 3 units
- B’s one day work = 12/6 = 2 units
Combined one day work = 5 units, so total time = 12/5 days. This method avoids fractions.
9. How are pipes and cisterns related to work and time?
Pipes and cisterns problems are applications of work and time where filling a tank is treated as positive work and emptying as negative work.
- Filling pipe rate = 1/T
- Emptying pipe rate = −1/T
Net rate = sum of rates, and time = 1 ÷ net rate. This concept is widely used in aptitude exams.
10. Can you give a simple example of a work and time problem?
If A completes a job in 10 days and B completes it in 15 days, together they finish it in 6 days.
- A’s rate = 1/10
- B’s rate = 1/15
- Combined rate = 1/10 + 1/15 = 5/30 = 1/6
Time together = 1 ÷ (1/6) = 6 days. This demonstrates the standard combined work formula.
