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Arithmetic II

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Time & Work

Time And Work

MODULES

Basics & Worker-Day Method
Worker-Day Method
Unitary Method
Relative Efficiencies
Negative Work, Alternating Work & Past Questions

CONCEPTS & CHEATSHEET

Concept Revision Video

SPEED CONCEPTS

Time & Work 1
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Time & Work 2
-/10
Time & Work 3
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PRACTICE

Time & Work : Level 1
Time & Work : Level 2
Time & Work : Level 3
ALL MODULES

CAT 2025 Lesson : Time & Work - Unitary Method

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3. Unitary Method

These questions include two or more people with varying levels of efficiencies. Typical way in which this is provided is the total time taken by different individuals to complete the entire work. In these questions efficiency or rate of completion is to be taken as the portion of work a person can complete in 111 unit of time.

Portion of work completed in a day and time taken have a reciprocal relationship.

For example, if John can complete a piece of work in 202020 days, then his efficiency is the portion of work he can complete in 111 day, which is 120th\dfrac{1}{20}^{th}201​th of the work.

Similarly, if Juliet can complete 16th\dfrac{1}{6}^{th}61​th of the work in 111 day, then she will take 666 days to complete the work.

Example 8

Salman and Aabid take 101010 days and 202020 days respectively to build a car. How long would they if they worked together?

Solution

Standard Unitary Method (Recommended)

Salman can finish 110th\dfrac{1}{10}^{th}101​th of the work in 111 day and Aabid can finish 120th\dfrac{1}{20}^{th}201​th of the work in 111 day.

Portion of work completed by Salman and Aabid in 111 day =110+120=320th= \dfrac{1}{10} + \dfrac{1}{20} = \dfrac{3}{20}^{th}=101​+201​=203​th of the work

Days taken to complete the work is the reciprocal of the portion of work completed in 111 day.

Days taken by Salman and Aabid together =202=6.67= \dfrac{20}{2} = 6.67=220​=6.67 days

Alternatively (Percentage Method)

Total work is taken as 100%100 \%100% and the portions of work completed by each can be expressed in percentage.

Salman can finish 110th\dfrac{1}{10}^{th}101​th or 10% of the work in 111 day and Aabid can finish 120th\dfrac{1}{20}^{th}201​th or 5% of the work in 1 day.

Portion of work completed by Salman and Aabid in 111 day =10%+5%=15%= 10 \% + 5 \% = 15 \%=10%+5%=15%

Days taken =100%15%=6.67= \dfrac{100 \%}{15 \%} = 6.67=15%100%​=6.67 days

Alternatively (Parts Method)

The number of parts in the work is the LCM of the time taken.

In this question, LCM (10,20)=20(10, 20) = 20(10,20)=20. So, let the total work be 202020 parts.

In 111 day, Salman can finish 110th\dfrac{1}{10}^{th}101​th or 222 parts and Aabid can finish 120th\dfrac{1}{20}^{th}201​th or 111 part.

Parts completed by Salman and Aabid in 111 day =2+1=3= 2 + 1 = 3=2+1=3 parts

Days taken =203=6.67= \dfrac{20}{3} = 6.67=320​=6.67 days

Answer: 6.676.676.67 days

The percentage method is useful if the fractions can be easily converted to percentages.
The parts method is the recommended method when work is done alternately (covered in Section 5 of this lesson).
The standard unitary method will be used for the rest of this lesson as it is simple to understand and apply.

Example 9

Alex and Graham, working alone, take 666 days and 888 days respectively to complete a piece of work. How many days will they take to finish the work if they worked together?

Solution

Portion of work completed by Alex in 111 day =16= \dfrac{1}{6}=61​

Portion of work completed by Graham in 111 day =18= \dfrac{1}{8}=81​

Portion of work completed by both together in 111 day =16+18=724= \dfrac{1}{6} + \dfrac{1}{8} = \dfrac{7}{24}=61​+81​=247​

Therefore, time taken to complete the work together =247=3.43= \dfrac{24}{7} = 3.43=724​=3.43 days

Answer: 3.433.433.43 days

Note: In this question percentage method will be time consuming and difficult.


Example 10

A and B can finish a project working alone in 101010 days and 202020 days respectively. At the end of the fourth day, A leaves. At the beginning of the 6th6^{th}6th day, C joins and starts working with B on the project. If the project is completed at the end of the 7th7^{th}7th day, how long would C take to complete the project alone?

Solution

Portion of work completed by A and B in 111 day are 110\dfrac{1}{10}101​ and 120\dfrac{1}{20}201​ respectively.

Portion of work completed in 444 days =4×(110+120)=1220= 4 \times \left( \dfrac{1}{10} + \dfrac{1}{20} \right) = \dfrac{12}{20}=4×(101​+201​)=2012​

Portion of work completed in 555 days =1220+120=1320= \dfrac{12}{20} + \dfrac{1}{20} = \dfrac{13}{20}=2012​+201​=2013​

Portion of work completed on the 6th6^{th}6th and 7th7^{th}7th days =1−1320=720= 1 - \dfrac{13}{20} = \dfrac{7}{20}=1−2013​=207​

Let xxx be the number of days C takes to complete the work alone.

⇒2×(120+1x)=720 2 \times \left( \dfrac{1}{20} + \dfrac{1}{x} \right) = \dfrac{7}{20}2×(201​+x1​)=207​

⇒2x=520 \dfrac{2}{x} = \dfrac{5}{20}x2​=205​

⇒x=8 x = 8x=8 days

Answer: 888 days


Example 11

Shantanu can complete an assignment in 144144144 days, if he works for 999 hours everyday. Sayantan would take 727272 days to complete the same assignment if he worked for 121212 hours every day. If they are supposed to work together and complete the assignment in exactly 484848 days, how many hours should they work for every day? (Assume they are required to work for the same amount of time every day)

(1) 8258\dfrac{2}{5}852​ hours            (2) 999 hours            (3) 9359\dfrac{3}{5}953​ hours            (4) 104510\dfrac{4}{5}1054​ hours           

Solution

The prime factors of all the numbers given in the questions are 222 and 333 only. Therefore, we can prime factorise and represent these numbers for ease in calculations.

Time taken by Shantanu to complete the assignment =144×9=24×34= 144 \times 9 = 2^{4} \times 3^{4}=144×9=24×34

Time taken by Sayantan to complete the assignment =72×12=25×33= 72 \times 12 = 2^{5} \times 3^{3}=72×12=25×33

Portion of work completed by Shantanu and Sayantan in 111 hour =124×34+125×33=525×34= \dfrac{1}{2^{4} \times 3^{4}} + \dfrac{1}{2^{5} \times 3^{3}} = \dfrac{5}{2^{5} \times 3^{4}}=24×341​+25×331​=25×345​

Number of hours taken by Shantanu and Sayantan =25×345hours= \dfrac{2^{5} \times 3^{4}}{5} \text{hours}=525×34​hours

If it takes 484848 days to complete, then the hours/day they need to work for

=25×345hours48days=2×335=545= \dfrac{\dfrac{2^{5} \times 3^{4}}{5} \text{hours}}{48 \text{days}} = \dfrac{2 \times 3^{3}}{5} = \dfrac{54}{5}=48days525×34​hours​=52×33​=554​

=1045 = 10\dfrac{4}{5}=1054​ hours/day

Answer: 104510\dfrac{4}{5}1054​ hours

3.1 Groups with different efficiencies

Questions will include different groups of people or items. The efficiencies of individual members in a group will be constant. However, the efficiencies of the groups may be different and solved using linear equations.

Example 12

888 men and 333 women can assemble 202020 cars in 888 days. 111111 men and 666 women can assemble 202020 cars in 555 days. How long would it take 101010 men and 202020 women to assemble 404040 cars?

Solution

Let m and w be the portion of work completed by 111 man and 111 woman in 111 day respectively. And, let the work be defined as assembling 202020 cars.

8m+3w=188\text{m} + 3\text{w} = \dfrac{1}{8}8m+3w=81​ -----(1)\left(1 \right)(1)

11m+6w=1511\text{m} + 6\text{w} = \dfrac{1}{5}11m+6w=51​ -----(2)\left(2 \right)(2)

We proceed to find out the rate of completion of work of man and woman relative to each other.

8×(8m+3w)=5×(11m+6w) 8 \times (8\text{m} + 3\text{w}) = 5 \times ( 11\text{m} + 6\text{w})8×(8m+3w)=5×(11m+6w)
⇒ 9m=6w 9m = 6w9m=6w
⇒ w=1.5m w = 1.5mw=1.5m

Substituting in (2)(2)(2), 11m+9m=1511m + 9m = \dfrac{1}{5}11m+9m=51​
m=1100 m = \dfrac{1}{100}m=1001​

So, w=1.5100w = \dfrac{1.5}{100}w=1001.5​

Portion of work completed by 101010 men and 202020 women in 111 day

=10×1100+20×1.5100=25=2.5= \dfrac {10 \times 1}{100} + \dfrac{20 \times 1.5}{100} = \dfrac{2}{5} = 2.5=10010×1​+10020×1.5​=52​=2.5

As assembling 404040 cars is twice the defined work, they take 2×2.5=52 \times 2.5 = 52×2.5=5 days

Answer: 555 days


Example 13

In Bilaspur village, 121212 men and 181818 boys completed construction of a primary health center in 606060 days, by working for 7.57.57.5 hours a day. Subsequently the residents of the neighbouring Harigarh village also decided to construct a primary health center in their locality, which would be twice the size of the facility built in Bilaspur. If a man is able to perform the work equal to the same done by 222 boys, then how many boys will be required to help 212121 men to complete the work in Harigarh in 505050 days, working 999 hours a day? [IIFT 2011]

(1) 454545 boys            (2) 484848 boys            (3) 404040 boys            (4) 424242 boys           

Solution

As 111 man's work is equal to that of 222 boys, we can convert the number of men to their equivalent in boys.

In Bilaspur village, 121212 men (equivalent of 242424 boys) and 181818 boys worked on the center. Total boys equivalent is 24+18=4224 + 18 = 4224+18=42 boys.

Work to build a health center =40×60×7.5= 40 \times 60 \times 7.5=40×60×7.5 boy-hours

Work to build twice the size of this health center =2×42×60×7.5=42×60×15= 2 \times 42 \times 60 \times 7.5 = 42 \times 60 \times 15=2×42×60×7.5=42×60×15 boy-hours

Note that we needn't multiply the terms here as some of them will get cancelled later.

Let the number of boys working in Harigarh be bbb. Number of men is 212121, which is equivalent to 424242 boys. They are to complete the work in 505050 days working 999 hours a day.

(42+b)×50×9=42×60×15(42 + b) \times 50 \times 9 = 42 \times 60 \times 15(42+b)×50×9=42×60×15 boy-hours

⇒ b=42 b = 42b=42 boys

Answer: 424242 boys


3.2 Special Case: 2 people working together

If A and B together take x days to complete a piece of work, while individually they take a days longer and b days longer to complete the same amount of work, then x=abx = \sqrt{ab}x=ab​

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Example 14

Abhi and Ram individually take 161616 hours and 999 hours longer, respectively, than the time taken by them if they would have worked together. How long would Abhi, while working alone, take to complete work?

Solution

Let the time taken by Abhi and Ram while working together be xxx hours.

x=16×9=144=12x = \sqrt{16} \times \sqrt{9} = \sqrt{144} = 12x=16​×9​=144​=12 hours

∴ \therefore∴ Abhi, while working alone, takes 12+16=12 + 16 =12+16= 28 hours to complete the work.

Answer: 282828 hours

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