Arithmetic

Time & Work

Medium

Sam can complete a job in 20 days when working alone. Mohit is twice as fast as Sam and thrice as fast as Ayna is the same job. The undertake a job with an arrangement where Sam and Mohit work together on the first day, Sam and Ayna on the second day, Mohit and Ayna on the third day, and this three-day pattern is repeated till the work gets completed. Then, the fraction of total work done by Sam is

$\frac{3}{20}$

$\frac{3}{10}$

$\frac{1}{5}$

$\frac{1}{20}$

✅ Correct Option: 2

Sam completes the job alone in 20 days, so his work rate = $\frac{1}{20}$ of the job per day.

If someone completes a job in 20 days, they complete $\frac{1}{20}$ of the job each day. This is a fundamental concept in work problems.

We are told:

Mohit is 2 times as fast as Sam

Mohit is 3 times as fast as Ayna

Let's convert this to ratios:

$M : S = 2 : 1$ (Mohit : Sam)

$M : A = 3 : 1$ (Mohit : Ayna)

To combine these ratios, we need the same value for Mohit in both ratios.

From $M : S = 2 : 1$ , we can write this as $M = 6, S = 3$

From $M : A = 3 : 1$ , we can write this as $M = 6, A = 2$

We need a number that's divisible by both 2 and 3. The smallest such number is 6.

Therefore: $S : M : A = 3 : 6 : 2$

Since Sam's efficiency is 3 units and he takes 20 days to complete the job:

Total work = Sam's daily work × Days taken

= 3 × 20

= 60 units

This means:

Sam works at 3 units per day

Mohit works at 6 units per day

Ayna works at 2 units per day

The work pattern repeats every 3 days:

Day 1: Sam + Mohit = $3 + 6 = 9$ units

Day 2: Sam + Ayna = $3 + 2 = 5$ units

Day 3: Mohit + Ayna = $6 + 2 = 8$ units

Work completed in each 3-day cycle = $9 + 5 + 8 = 22$ units

Let's see how the work progresses:

After 3 days: 22 units completed

After 6 days: $22 × 2 = 44$ units completed

Remaining work: $60 - 44 = 16$ units

Continuing the pattern:

Day 7 (Sam + Mohit): 9 units → Total: $44 + 9 = 53$ units

Day 8 (Sam + Ayna): 5 units → Total: $53 + 5 = 58$ units

Day 9 (Mohit + Ayna): Only need 2 more units to reach 60 → Work completed on Day 9

Sam works on the first 2 days in each cycle (total cycles = 3).

Sam's total working days = 2 $\times$ 3 = 6 days

Sam's total work = 6 days × 3 units per day = 18 units

Fraction of work done by Sam = $\frac{\text{Sam's work}}{\text{Total work}} = \frac{18}{60} = \frac{3}{10}$

In work problems with rotating patterns, we always track who works on which days and calculate their total contribution.

CAT 2018 Slot 1

When they work alone, $B$ needs $25 \%$ more time to finish a job than $A$ does. They two finish the job in $13$ days in the following manner: $A$ works alone till half the job is done, then $A$ and $B$ work together for four days, and finally $B$ works alone to complete the remaining $5\%$ of the job. In how many days can $B$ alone finish the entire job?

CAT 2018 Slot 2

A tank is emptied everyday at a fixed time point. Immediately thereafter, either pump $A$ or pump $B$ or both start working until the tank is full. On Monday, $A$ alone completed filling the tank at $8 \mathrm{pm}$ . On Tuesday, B alone completed filling the tank at $6 \mathrm{pm}$ . On Wednesday, $A$ alone worked till $5 \mathrm{pm}$ , and then B worked alone from $5 \mathrm{pm}$ to $7 \mathrm{pm}$ , to fill the tank. At what time was the tank filled on Thursday if both pumps were used simultaneously all along?

CAT 2019 Slot 1

Three men and eight machines can finish a job in half the time taken by three machines and eight men to finish the same job. If two machines can finish the job in $13$ days, then how many men can finish the job in $13$ days?

CAT 2018 Slot 2

Ramesh and Ganesh can together complete a work in $16$ days. After seven days of working together, Ramesh got sick and his efficiency fell by $30 \%$ . As a result, they completed the work in $17$ days instead of $16$ days. If Ganesh had worked alone after Ramesh got sick, in how many days would he have completed the remaining work?

CAT 2021 Slot 1

Anu, Vinu and Manu can complete a work alone in 15 days, 12 days and 20 days, respectively. Vinu works everyday. Anu works only on alternate days starting from the first day while Manu works only on alternate days starting from the second day. Then, the number of days needed to complete the work is

CAT 2020 Slot 2

John takes twice as much time as Jack to finish a job. Jack and Jim together take one-thirds of the time to finish the job than John takes working alone. Moreover, in order to finish the job, John takes three days more than that taken by three of them working together. In how many days will Jim finish the job working alone?

Solution

Related questions:

Keyboard Shortcuts

CAT Past Year Questions (Topic-Wise)