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CAT 2025Slot 1QAQues & Sol

ArithmeticHard

Question

Arun, Varun and Tarun, if working alone, can complete a task in 24, 21, and 15 days, respectively. They charge Rs 2160, Rs 2400, and Rs 2160 per day, respectively, even if they are employed for a partial day. On any given day, any of the workers may or may not be employed to work. If the task needs to be completed in 10 days or less, then the minimum possible amount, in rupees, required to be paid for the entire task is

Options

Solution

1. Concept Used

  • Topic: Time and Work — Cost Optimization
  • Formula: $$\text{Total Cost} = \sum (\text{Days Employed} \times \text{Daily Charge})$$

2. Calculation

First, compute the total cost if each worker were to complete the entire task alone:

$$\text{Arun's Total Cost} = 24 \times 2160 = 51840$$

$$\text{Varun's Total Cost} = 21 \times 2400 = 50400$$

$$\text{Tarun's Total Cost} = 15 \times 2160 = 32400$$

Tarun is the cheapest overall, so the strategy is to maximize Tarun's usage first, then fill the remaining work with the next cheapest worker.

Step 1 — Maximize Tarun's contribution over 10 days:

Tarun completes $\frac{1}{15}$ of the work per day. In 10 days, he completes:

$$10 \times \frac{1}{15} = \frac{2}{3} \text{ of the total work}$$

Step 2 — Remaining work after Tarun:

$$1 - \frac{2}{3} = \frac{1}{3} \text{ of the total work remains}$$

Step 3 — Choose the next cheapest worker for remaining work:

Comparing Arun (Rs 2160/day) and Varun (Rs 2400/day), Arun charges less per day. However, Arun completes $\frac{1}{24}$ of work per day, and Varun completes $\frac{1}{21}$ of work per day.

Cost per unit of work: $$\text{Arun: } 2160 \times 24 = 51840 \text{ per full task}$$ $$\text{Varun: } 2400 \times 21 = 50400 \text{ per full task}$$

Since partial days are charged as full days, we must also consider how many days each would need for the remaining $\frac{1}{3}$ work:

Varun for remaining $(\frac{1}{3})$ work: $$\frac{1/3}{1/21} = 7 \text{ days (exact — no partial day penalty)}$$ $$\text{Cost} = 7 \times 2400 = 16800$$

Arun for remaining $(\frac{1}{3})$ work: $$\frac{1/3}{1/24} = 8 \text{ days (exact — no partial day penalty)}$$ $$\text{Cost} = 8 \times 2160 = 17280$$

Varun is cheaper for the remaining $\frac{1}{3}$ work. Note that Varun working 7 days is within the 10-day constraint (he can work alongside or after Tarun, or they work in parallel — the key constraint is completing within 10 days, and since both can work simultaneously or sequentially within 10 days, this is feasible).

Step 4 — Total Minimum Cost:

$$\text{Total Cost} = (10 \times 2160) + (7 \times 2400)$$

$$= 21600 + 16800 = 38400$$

3. Solution

Answer = Option A ✅

The minimum possible amount required to complete the task within 10 days is Rs 38,400.