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Arithmetic II

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Time & Work
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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 11 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 2020 days, then his efficiency is the portion of work he can complete in 11 day, which is 120th\dfrac{1}{20}^{th} of the work.

Similarly, if Juliet can complete 16th\dfrac{1}{6}^{th} of the work in 11 day, then she will take 66 days to complete the work.

Example 8

Salman and Aabid take 1010 days and 2020 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} of the work in 11 day and Aabid can finish 120th\dfrac{1}{20}^{th} of the work in 11 day.

Portion of work completed by Salman and Aabid in 11 day =110+120=320th= \dfrac{1}{10} + \dfrac{1}{20} = \dfrac{3}{20}^{th} of the work

Days taken to complete the work is the reciprocal of the portion of work completed in 11 day.

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

Alternatively (Percentage Method)

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

Salman can finish 110th\dfrac{1}{10}^{th} or 10% of the work in 11 day and Aabid can finish 120th\dfrac{1}{20}^{th} or 5% of the work in 1 day.

Portion of work completed by Salman and Aabid in 11 day =10%+5%=15%= 10 \% + 5 \% = 15 \%

Days taken =100%15%=6.67= \dfrac{100 \%}{15 \%} = 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. So, let the total work be 2020 parts.

In 11 day, Salman can finish 110th\dfrac{1}{10}^{th} or 22 parts and Aabid can finish 120th\dfrac{1}{20}^{th} or 11 part.

Parts completed by Salman and Aabid in 11 day =2+1=3= 2 + 1 = 3 parts

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

Answer: 6.676.67 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 66 days and 88 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 11 day =16= \dfrac{1}{6}

Portion of work completed by Graham in 11 day =18= \dfrac{1}{8}

Portion of work completed by both together in 11 day =16+18=724= \dfrac{1}{6} + \dfrac{1}{8} = \dfrac{7}{24}

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

Answer: 3.433.43 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 1010 days and 2020 days respectively. At the end of the fourth day, A leaves. At the beginning of the 6th6^{th} day, C joins and starts working with B on the project. If the project is completed at the end of the 7th7^{th} day, how long would C take to complete the project alone?

Solution

Portion of work completed by A and B in 11 day are 110\dfrac{1}{10} and 120\dfrac{1}{20} respectively.

Portion of work completed in 44 days =4×(110+120)=1220= 4 \times \left( \dfrac{1}{10} + \dfrac{1}{20} \right) = \dfrac{12}{20}

Portion of work completed in 55 days =1220+120=1320= \dfrac{12}{20} + \dfrac{1}{20} = \dfrac{13}{20}

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

Let xx 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}

⇒2x=520 \dfrac{2}{x} = \dfrac{5}{20}

⇒x=8 x = 8 days

Answer: 88 days


Example 11

Shantanu can complete an assignment in 144144 days, if he works for 99 hours everyday. Sayantan would take 7272 days to complete the same assignment if he worked for 1212 hours every day. If they are supposed to work together and complete the assignment in exactly 4848 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} hours            (2) 99 hours            (3) 9359\dfrac{3}{5} hours            (4) 104510\dfrac{4}{5} hours           

Solution

The prime factors of all the numbers given in the questions are 22 and 33 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}

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

Portion of work completed by Shantanu and Sayantan in 11 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}}

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

If it takes 4848 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}

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

Answer: 104510\dfrac{4}{5} 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

88 men and 33 women can assemble 2020 cars in 88 days. 1111 men and 66 women can assemble 2020 cars in 55 days. How long would it take 1010 men and 2020 women to assemble 4040 cars?

Solution

Let m and w be the portion of work completed by 11 man and 11 woman in 11 day respectively. And, let the work be defined as assembling 2020 cars.

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

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

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})
⇒ 9m=6w 9m = 6w
⇒ w=1.5m w = 1.5m

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

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

Portion of work completed by 1010 men and 2020 women in 11 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

As assembling 4040 cars is twice the defined work, they take 2×2.5=52 \times 2.5 = 5 days

Answer: 55 days


Example 13

In Bilaspur village, 1212 men and 1818 boys completed construction of a primary health center in 6060 days, by working for 7.57.5 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 22 boys, then how many boys will be required to help 2121 men to complete the work in Harigarh in 5050 days, working 99 hours a day? [IIFT 2011]

(1) 4545 boys            (2) 4848 boys            (3) 4040 boys            (4) 4242 boys           

Solution

As 11 man's work is equal to that of 22 boys, we can convert the number of men to their equivalent in boys.

In Bilaspur village, 1212 men (equivalent of 2424 boys) and 1818 boys worked on the center. Total boys equivalent is 24+18=4224 + 18 = 42 boys.

Work to build a health center =40×60×7.5= 40 \times 60 \times 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 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 bb. Number of men is 2121, which is equivalent to 4242 boys. They are to complete the work in 5050 days working 99 hours a day.

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

⇒ b=42 b = 42 boys

Answer: 4242 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}

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Example 14

Abhi and Ram individually take 1616 hours and 99 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 xx hours.

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

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

Answer: 2828 hours

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