TIME AND WORK

TIME AND WORK

 

Work is defined as something which has an effect or outcome; often the one desired or expected. The basic concept of Time and Work is similar to that across all Arithmetic topics, i.e. the concept of Proportionality.

Efficiency is inversely proportional to the Time taken when the amount of work done is constant.

This can be used to compare efficiencies and Time taken across different groups In Time-Speed-Distance, efficiency is replaced by Speed; i.e. Speed is inversely proportional to Time when the Distance is constant.

Pipes and Cisterns are just an application of Time and Work. Concept wise, it is one and the same. In the above proportionality, Efficiency is replaced by Rate of filling. The equation in this case becomes

Important Formulas:

1.     Work from Days:

If A can do a piece of work in n days, then A's 1 day's work =	1	.
	n

2.     Days from Work:

If A's 1 day's work =	1	,	then A can finish the work in n days.
	n

3.     Ratio:

If A is thrice as good a workman as B, then:

Ratio of work done by A and B = 3 : 1.

Ratio of times taken by A and B to finish a work = 1 : 3.

 

Approach 1: Using Fractions:

Ram can finish the work in 10 days i.e. in one day he will do 1/10th of the work.

Rahim can finish the work in 40 days i.e. in one day he will do 1/40th of the work.

So, in one day, both working together can finish= (1/10) + (1/40) = 5/40 = 1/8th of the work. So, to complete the work they will take 8 days.

 

Approach 2: Using Percentage (Shortcut- Recommended):

Rahim can finish 100 % of work in 10 days i.e. in one day he finishes 10% of the work.

Ram can finish 100% of the work in 40 days i.e. in one day he finishes 2.5 % of the work.

So, working together, in a single day they can finish 12.5% of the work. So, to complete 100% of the work, they will take 100/12.5 = 8 days.

 

 Negative Work:

Negative work increases the Time in which a work is to be completed. This application can be extended to cases involving Pipes and cisterns. Suppose there are two pipes in a Cistern. Pipe A is used to fill the Cistern and Pipe B is used to empty the Cistern. Here we say that Pipe B and Pipe A are working against each other. When a leak is developed in the Cistern, the leak forms the component of negative work, which slows down the completion of the task (in this case, the filling of the Cistern)

 

Example:

Pipe A can fill a tank in ‘an’ hours. On account of a leak at the bottom of the tank, it takes thrice as long to fill the tank. How long will the leak at the bottom of the tank take to empty a full tank, when pipe A is kept closed?

a) (3/2)hours

b) (2/3)hours

c) (4/3)hours

d) (3/4)hours

Solution:

Method 1: Using variables

The pipe can fill (1/a)^th of the tank in an hour. Because of the leak, it can only fill 1/3a of the tank per hour. Let X be the Time in which the leak can completely empty the tank, hence 1/x=1/a-1/3a

= x=3a/2hrs. Option (a)

Answer: Option (a)

Method 2: Using Numbers (Shortcut)

Assume a value for “a”- say 10 hours. Because of the leak, it will take 30 hours. Now, this means that in these 30 hours, the filling will occur at a rate of 3.33% and the leaking will slow down the process by 6.66% every hour Thus, the Time taken to empty a full tank= 100/6.66= 15 hours.

Answer: Only option (a) satisfies this value.

 

Inverse Proportionality of Efficiency and Time Taken:

If the product of the two variables is always constant, the two are said to be inversely proportional.

Efficiency and Time taken are inversely proportional implies that If A is twice as good as B then A will take half the Time that B will take.

If the efficiencies are in the ratio m: n then Time taken will be in the ratio, n: m. i.e. If A is thrice is as good as B then A will take (1/3)^rd of the Time.

 

 

Examples:

Illustration 1: 

Samir can do a job in 30 days. In how many days can he complete 70% of the job?
Sol: Now as per the question he finishes the work in 30 days, or he can do 100% of the work in 30 days. If he has to do only 70% of the work, he will require 70% of the time.
 Number of days required = 30 × 70/100 = 21 days.

Illustration 2:  

Reshma can do 75% job in 45 days. In how many days can she complete the job?
Sol: Every work is 100% in itself. Reshma does 75% of the work in 45 days. That means she does 1% of the work in 45/75 days and she will do 100% of the work in 100 × 45/75 = 60 days.

Illustration 3: 

John can do a piece of work in 60 days; he will do how much of the work in 40 days?
Sol: In 1 day, John does 1/60th of the work, so in 40 days he will do 40 × 1/60 = 2/3rd of the work.

Illustration 4:

 Anup can finish a piece of work in 30 days. He will finish what percent of the work in 15 days?
Sol: In 1 day, he does 1/30th of the work, and in 15 days, he will do 15/30th of the work which is 100 × 15/30 = 50%.

Illustration 5:

 Ria can do a piece of work in 40 days, she will take how many days to finish three-fourth of the work?
Sol:  Ria can complete the work in 40 days.  She will do ¾th of the work in ¾th of the total time. i.e. she will need 40 × 3/4 = 30 days.