A man gets an appointment with two options. Either he can accept Rs. 450 per day for 30 days or Rs. 300 on the first day with an increase of Rs. 15 per day for 30 days. Which of the options will be beneficial to him? How much will he gain by that choice?

Answer

To determine which option is more beneficial for the man, we will calculate the total earnings from both options and compare them.

### Step 1: Calculate total earnings for the first option
The first option is to earn Rs. 450 per day for 30 days.

\[ \text{Total earnings from option 1} = \text{Daily earnings} \times \text{Number of days} = 450 \times 30 \]

Calculating this gives:

\[ 450 \times 30 = 13500 \]

So, the total earnings from the first option is Rs. 13,500.

### Step 2: Calculate total earnings for the second option
The second option is to earn Rs. 300 on the first day, with an increase of Rs. 15 each subsequent day for 30 days. This forms an arithmetic progression (AP).

- First term (a) = Rs. 300
- Common difference (d) = Rs. 15
- Number of terms (n) = 30

The formula for the sum of the first n terms of an arithmetic progression is:

\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]

Substituting the values into the formula:

\[ S_{30} = \frac{30}{2} \times (2 \times 300 + (30 - 1) \times 15) \]

Calculating this step by step:

1. Calculate \(2 \times 300\):

\[ 2 \times 300 = 600 \]

2. Calculate \((30 - 1) \times 15\):

\[ (30 - 1) \times 15 = 29 \times 15 = 435 \]

3. Now substitute back into the sum formula:

\[ S_{30} = 15 \times (600 + 435) = 15 \times 1035 \]

4. Finally, calculate \(15 \times 1035\):

\[ 15 \times 1035 = 15525 \]

So, the total earnings from the second option is Rs. 15,525.

### Step 3: Compare the two options
Now we compare the total earnings from both options:

- Earnings from option 1: Rs. 13,500
- Earnings from option 2: Rs. 15,525

### Step 4: Determine the gain from the better option
To find out how much more beneficial the second option is, we subtract the earnings of the first option from the second option:

\[ \text{Gain} = \text{Earnings from option 2} - \text{Earnings from option 1} = 15525 - 13500 \]

Calculating this gives:

\[ 15525 - 13500 = 2025 \]

### Conclusion
The second option is more beneficial, and the man will gain Rs. 2,025 by choosing this option.

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