## Introduction

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

1. What is the formula for work done?

Work done = Time taken by work × Rate of Doing work

2. How is energy related to work?

When work is done, a transaction of energy is required. Therefore, we can say that all the work that is done requires some energy, and it is its measurement or payment.

3. Is there another way to calculate work time relation without using unity?

No, unity is required to associate one quantity with another when more than one variable is concerned. However, we can solve the questions by cross-multiplying equal fractional relations when all the information is given.