TheCATExam
Sign in

CAT 2023 Slot 1 — QA Question 18

BasicsEasy

The amount of job that Amal, Sunil and Kamal can individually do in a day, are in harmonic progression. Kamal takes twice as much time as Amal to do the same amount of job. If Amal and Sunil work for 4 days and 9 days, respectively, Kamal needs to work for 16 days to finish the remaining job. Then the number of days Sunil will take to finish the job working alone, is

Answer & solution

Answer: 27

Solution

Easy

The key fact: if the daily outputs (efficiencies) are in Harmonic Progression, then the times taken are in Arithmetic Progression — they are reciprocals of each other. So Sunil's time is the average of Amal's and Kamal's. Express everything in Amal's time xx, then plug the work done by all three into the "one whole job" equation.

1

Name the times. Let Amal take xx days. Kamal takes twice as long, so 2x2x days. Efficiencies ∝1/time\propto 1/\text{time} are in HP ⇒\Rightarrow times are in AP:

Sunil’s time=x+2x2=1.5x\text{Sunil's time}=\frac{x+2x}{2}=1.5x
2

Sum the work. Each person's daily work is 1time\tfrac{1}{\text{time}}. The job is finished, so the fractions add to 11:

1x⋅4+11.5x⋅9+12x⋅16=1\frac{1}{x}\cdot 4+\frac{1}{1.5x}\cdot 9+\frac{1}{2x}\cdot 16=1
3

Solve for xx. Multiply through by xx:

4+91.5+162=x4+6+8=x ⇒ x=18\begin{aligned} &4+\frac{9}{1.5}+\frac{16}{2}=x\\ &4+6+8=x\ \Rightarrow\ x=18 \end{aligned}
4

Find Sunil's time from step 1:

1.5x=1.5×18=27 days1.5x=1.5\times 18=27\ \text{days}

Sunil alone needs 27\mathbf{27} days.

Remember the duality once and you skip all reciprocal juggling: efficiencies in HP ⇔\Leftrightarrow times in AP. That instantly gives Sunil's time as the mean 1.5x1.5x, and the work equation collapses to 4+6+8=x4+6+8=x.

CAT 2023 Slot 1 QA Q18: The amount of job that Amal, Sunil and Kamal can individually do in a day, are in harmonic progression. Kamal — Solution | TheCATExam