ICSE Class 8 Maths Commercial Arithmetic Chapter 14 Time and Work

Time And Work

Relation Between Time And Work

Let us find out the relation between time and work in different cases:

1. number of workers is fixed;

2. work to be done is fixed; and

3. time is fixed

Case I: When Number Of Workers Is Fixed

The amount of work done varies directly with the time spent on doing the work, provided the number of workers remains constant.

or Work done ∝ Time taken or more work will get done in more time

If a₁ number of articles are made in time t₁ by a certain number of workers and a₂ number of articles are made in time t₂, then

a₁ : a₂ :: t₁ : t₂

Example 1: A Weaver Takes 2 Days To Weave 3 Shawls. How Many Days Will He Take To Weave 15 Shawls?

Method I: Using Direct Proportion

Let the weaver take x number of days to weave 15 shawls.

Then 3 : 15 :: 2 : x

⇒ 3x = 2 × 15

⇒ x = 30/3 = 10 days

Thus, the weaver will take 10 days to weave 15 shawls.

Method II: Using Unitary Method

If 3 shawls can be woven in 2 days

1 shawl can be woven in 2/3 days

and 15 shawls can be woven in 2/3 × 15 = 10 days

Thus, the weaver will take 10 days to weave 15 shawls.

Case II: When Work To Be Done Is Fixed

The time taken to do a job varies inversely with the number of workers doing the job, provided the amount of work involved in doing the job remains constant.

or Time taken ∝ 1/(Number of workers)

or more workers will do the same job in less time.

If w₁ number of workers take time t₁ to complete a work and w₂ number of workers take time t₂ to complete the same amount of work, then

w₁ : w₂ = inverse of t₁ : t₂

or w₁ : w₂ :: t₂ : t₁

Example 2: 3 Gardeners Take 90 Minutes To Weed A Garden. If The Job Has To Be Done In Only 15 Minutes, How Many More Gardeners Need To Be Put On The Job?

Method I: Using Inverse Proportion

Let x number of gardeners be needed to weed the garden in 15 minutes.

Then 3 : x = inverse of 90 : 15

⇒ 3 : x :: 15 : 90

⇒ 15x = 90 × 3

⇒ x = 270/15 = 18 gardeners

Thus, 18 - 3 = 15 more gardeners need to be put on the job.

Method II: Using Unitary Method

Weeding the garden in 90 minutes takes: 3 gardeners

or Weeding the garden in 1 minute will take: 3 × 90 = 270 gardeners

or Weeding the garden in 15 minutes will take: 270/15 = 18 gardeners

Thus, 18 - 3 = 15 more gardeners need to be put on the job.

Case III: When Time Is Fixed

The number of workers required varies directly with the amount of work to be done when the time in which to do the work is constant.

Number of workers ∝ Work to be done

or more workers will do more work, and less work will need a less number of workers.

If w₁ number of workers can make a₁ number of articles in a certain time and w₂ number of workers can make a₂ number of articles in the same time, then

w₁ : w₂ :: a₁ : a₂

Example 3: 8 Mechanics Can Assemble 4 TV Sets In 2 Days. How Many Mechanics Will Assemble 3 TV Sets In 2 Days?

Method I: Using Direct Proportion

Let x number of mechanics take 2 days to assemble 3 TV sets.

Then 8 : x :: 4 : 3

⇒ 4x = 3 × 8

⇒ x = 24/4 = 6 mechanics

Thus, 6 mechanics will assemble 3 TV sets in 2 days.

Method II: Using Unitary Method

If 4 TV sets can be assembled by 8 mechanics in 2 days,

then 1 TV set can be assembled by 8/4 or 2 mechanics in 2 days

and 3 TV sets can be assembled by 2 × 3 = 6 mechanics in 2 days

Thus in 2 days, 6 mechanics will assemble 3 TV sets.

Example 4: A Takes 9 Minutes While B Takes 12 Minutes To Make A Candle. Working Together How Long Will A And B Take To Make 35 Candles?

In 9 minutes, A can make 1 candle

In 1 minute, A can make 1/9 candle

In 12 minutes, B can make 1 candle

In 1 minute, B can make 1/12 candle

In 1 minute A and B can make

1/9 + 1/12 = 7/36 candle

If 7/36 candle can be made by A and B in 1 minute,

then 1 candle can be made by A and B in

1 × 36/7 minutes

and 35 candles can be made by A and B in

36/7 × 35 = 180 minutes

Thus, working together, A and B will take 180 minutes or 3 hours to make 35 candles.

Example 5: A Can Paint A Shed In 5 Hours While B Takes 6 Hours To Do The Same Job. A And B Begin Painting The Shed Together, But B Is Called Away On Some Other Work After 2 Hours. How Long Will A Take To Finish Painting The Rest Of The Shed?

In 5 hours A can paint 1 shed

In 1 hour A can paint 1/5 of the shed

In 6 hours B can paint 1 shed

In 1 hour B can paint 1/6 of the shed

In 1 hour A and B can paint 1/5 + 1/6 = 11/30 of the shed

In 2 hours A and B would have painted

11/30 × 2 = 11/15 of the shed

Part of the shed remaining to be painted is 1 - 11/15 = 4/15

To paint 1 shed, A takes 5 hours.

To paint 4/15 of the shed, A will take 5 × 4/15

= 1 1/3 hours or 1 h 20 minutes

Thus, A will take another 1 h and 20 minutes to finish the job.

Example 6: Carpenters A And B Can Make An Almirah, Working Together, In 10 Hours. B And C Take 15 Hours While A And C Take 7.5 Hours To Do The Same Job. How Long Will A, B, And C, Working Together, Take To Make An Almirah? If After Selling The Almirah, A Profit Of Rs 1260 Is Made, How Should It Be Shared Among A, B, And C?

A and B make an almirah in 10 hours.

Thus in 1 hour, A + B make 1/10 of the almirah

B and C make an almirah in 15 hours.

Thus in 1 hour, B + C make 1/15 of the almirah.

A and C make an almirah in 15/2 hours.

Thus in 1 hour, A + C make 2/15 of the almirah.

Thus in 1 hour, A + B + B + C + A + C make

(1/10 + 1/15 + 2/15) of the almirah.

⇒ 2(A + B + C) make (3 + 2 + 4)/30 of the almirah,

or A + B + C make 9/(30 × 2) = 9/60 of the almirah

Now working together, A + B + C make 9/60 of the almirah in 1 hour.

So working together, A + B + C make 1 almirah in

60/9 = 6 2/3 hours.

But the efficiency of A, B, and C is not the same. In 1 hour A's share of work

= Joint work - (B + C)'s share of work

= 9/60 of the almirah - 1/15 of the almirah

= 5/60 or 1/12 of the almirah

In 1 hour B's share of work

= Joint work - (A + C)'s share of work

= 9/60 of the almirah - 2/15 of the almirah

= 1/60 of the almirah

In 1 hour C's share of work

= Joint work - (A + B)'s share of work

= 9/60 of the almirah - 1/10 of the almirah

= 3/60 or 1/20 of the almirah

Thus, the contribution of A, B, and C in the completion of the work is in the ratio 1/12 : 1/60 : 1/20 or

5 : 1 : 3. The profit made should be shared in this ratio as well.

Thus A's share of profit = 5/9 × 1260 = Rs 700

B's share of profit = 1/9 × 1260 = Rs 140

C's share of profit = 3/9 × 1260 = Rs 420

Thus, after working for 6 h and 40 min the almirah will be made, and after selling it A will earn Rs 700, B will earn Rs 140, and C will earn Rs 420 as profit.

Teacher's Note

Understanding how time, work, and efficiency relate helps in planning daily tasks like cooking multiple recipes or organizing group projects at school.

Problems On Pipes And Cisterns

The cistern shown below has two inlet pipes X and Y and one outlet pipe Z, all of different diameters through which water flows at different rates.

If inlet X takes x minutes to fill the cistern, then in 1 minute 1/x part of the cistern will be filled by X alone. If inlet Y takes y minutes to fill the cistern, then in 1 minute 1/y part of the cistern will be filled by Y alone.

In 1 minute 1/x + 1/y part of the cistern will be filled by X and Y together. If outlet Z takes z minutes to empty the cistern, then in 1 minute 1/z part of the cistern will be emptied by Z.

In 1 minute 1/x + 1/y - 1/z part of the cistern will be filled if both X and Z are open and the diametre of X is greater than the diametre of Z.

Example 7: Two Inlet Pipes Can Fill Up A Cistern In 3 Hours And 5 Hours, Respectively. If Both Pipes Start Filling The Cistern Together, How Long Will It Take For The Cistern To Get Filled?

In 1 hour the first pipe will fill up 1/3 part of the cistern.

In 1 hour the second pipe will fill up 1/5 part of the cistern.

In 1 hour both pipes will fill up 1/3 + 1/5 = 8/15 part of the cistern.

If 8/15 part of the cistern is filled by both pipes in 1 hour, then 1 whole of the cistern is filled by both pipes in 1 × 15/8 hours = 17/8 hours = 1 h 52.5 min.

Example 8: An Overhead Tank Springs A Leak That Can Empty The Full Tank In 16 Hours. If Its Inlet Pipe Could Fill It In 4 Hours Before The Leak Developed, How Long Would It Now Take To Get Filled Up, When Empty?

In 1 hour, the inlet pipe fills up 1/4 part of the tank.

In 1 hour, the leak empties 1/16 part of the tank.

In 1 hour 1/4 - 1/16 = 3/16 part of the tank gets filled up.

If 3/16 part of the tank is filled up in 1 hour, the whole tank is filled up in 1 × 16/3 = 5 1/3 hours or 5 h 20 min.

Teacher's Note

Pipes and cisterns problems help us understand how long it takes to fill water tanks or pools, which is important for managing water resources in homes and communities.

This is a preview of the first 3 pages. To get the complete book, click below.