CAT 2017

Slot 2

QA

Question & Solution

1. Concept Used

• Topic: Time and Work — Wage Distribution Proportional to Work Done
• Formula: $$\text{Individual's Share} = \frac{\text{Individual's Work Rate}}{\text{Combined Work Rate}} \times \text{Total Amount}$$

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2. Calculation

We are given that Amal completes the job in 10 days, so his work rate is (\frac{1}{10}) per day. Bimal completes it in 8 days, so his work rate is (\frac{1}{8}) per day. Together, all three complete it in 4 days, so the combined rate is (\frac{1}{4}) per day.

Let Kamal's time to complete the job alone be (x) days. Setting up the equation:

$$\frac{1}{10} + \frac{1}{8} + \frac{1}{x} = \frac{1}{4}$$

Solving for (\frac{1}{x}):

$$\frac{1}{x} = \frac{1}{4} - \frac{1}{10} - \frac{1}{8}$$

Finding the LCM of 4, 10, and 8, which is 40:

$$\frac{1}{x} = \frac{10}{40} - \frac{4}{40} - \frac{5}{40} = \frac{10 - 4 - 5}{40} = \frac{1}{40}$$

So (x = 40) days. Kamal alone would take 40 days to complete the job.

Now, the ratio of work done by Amal, Bimal, and Kamal in the same time period is proportional to their individual work rates:

$$\text{Amal} : \text{Bimal} : \text{Kamal} = \frac{1}{10} : \frac{1}{8} : \frac{1}{40}$$

Multiplying each term by 40 (LCM of 10, 8, 40) to simplify:

$$= 4 : 5 : 1$$

The total number of parts = (4 + 5 + 1 = 10).

Kamal's share of the total remuneration:

$$\text{Kamal's Share} = \frac{1}{10} \times 1000 = 100$$

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3. Solution

Answer = Option A ✅

The final calculated value is Rs. 100.