TIME AND WORK

Time and Work is a mathematical concept that deals with the relationship between the amount of work done, the time taken to complete the work, and the rate of work. It is commonly used in problems where people or machines work together to accomplish a task.

In the context of time and work problems:

Work refers to the total amount of job or task that needs to be completed.
Time is the duration or number of hours, days, or any other unit in which the work is completed.
Rate of Work is the speed at which the work is done. It is often expressed as the amount of work done per unit of time.

The fundamental formula relating time, work, and rate is:

Or, rearranging the formula to find the rate:

Time and Work problems involve determining the time required to complete a given amount of work, finding the rate of work, or calculating the total work done by a group of people or machines working together.

Typical scenarios in Time and Work problems include scenarios where multiple workers are collaborating to complete a task, each working at different rates, or situations where a task is started by one worker and continued by another. These problems often require setting up equations based on the work done by each individual or group and then solving for the unknowns, such as time, rate, or total work.

Understanding the basic principles of Time and Work is essential for solving quantitative aptitude questions in various competitive exams, including those conducted by the UPSC (Union Public Service Commission)

$M 1 D 1 T 1 = M 2 D 2 T 2$ is a key expression in time and work problems, reflecting the work equivalence principle. Let's break down the variables in this formula:

$M_{1}$ : The number of persons in the first scenario.
$D_{1}$ : The number of days taken in the first scenario.
$T_{1}$ : The number of hours worked per day in the first scenario.
$M_{2}$ : The number of persons in the second scenario.
$D_{2}$ : The number of days taken in the second scenario.
$T_{2}$ : The number of hours worked per day in the second scenario.

The formula is based on the principle that the work done is the same in both scenarios. It essentially says that the product of the number of persons ( $M$ ), the number of days ( $D$ ), and the number of hours per day ( $T$ ) is constant if the total work remains unchanged.

Solved Questions on Time and Work

Question 1: If 15 workers can complete a construction project in 12 days, how many days will it take for 10 workers to complete the same project, assuming the efficiency remains constant?

Solution: Let $M_{1}$ be the initial number of workers, $D_{1}$ be the initial number of days, and $W$ be the total work.

$M_{1} \times D_{1} = 15 \times 12 = 180$ (Work done by 15 workers in 12 days)

Let $M_{2}$ be the new number of workers and $D_{2}$ be the new number of days.

$M_{2} \times D_{2} = 10 \times D_{2}$ (Work done by 10 workers in $D_{2}$ days)

Since the work is constant:

$M_{1} \times D_{1} = M_{2} \times D_{2}$

$15 \times 12 = 10 \times D_{2}$

$D_{2} = 15 \times 12/10 = 18$ days

Therefore, it will take 10 workers 18 days to complete the project.

Question 2: A machine can produce 60 units in 4 hours. If the production rate remains constant, how many units can the machine produce in 6 hours?

Solution: Let $U$ be the number of units produced and $H$ be the number of hours.

The rate of production is $R = U/H$ .

$R = 60/4 = 15$ units per hour

To find the total units produced in 6 hours:

$U = R \times H = 15 \times 6 = 90$ units

Therefore, the machine can produce 90 units in 6 hours

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TIME AND WORK

TIME AND WORK

Solved Questions on Time and Work

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