Abhay working in Tele Bharti as a salesperson. His monthly salary is just Rs. 200. But he gets the bonus as per the given rule. If he sells the simcards of Rs. X then his bonus will be Rs. `[((x)/(100))^(2) + 10]`. In the first quarter of the year his average sale was Rs. 3000 per month. In the next 5 months his average sae was Rs. 5000 per month. In the last four months his average sale was Rs. 8000 per month. What is the average earning pre month for the whole year ?

Options

A. a. Rs. 3350
B. b. Rs. 3610
C. c. Rs. 3560
D. d. none of these

Answer

To find Abhay's average monthly earnings for the whole year, we need to calculate his total earnings from his salary and bonuses over the year and then divide that by 12 months. Let's break it down step by step.

### Step 1: Calculate the bonus for the first quarter
Abhay's average sales for the first quarter (3 months) is Rs. 3000 per month.

Using the bonus formula:
\[ \text{Bonus} = \left(\frac{X}{100}\right)^2 + 10 \]
where \( X = 3000 \).

Calculating the bonus:
\[ \text{Bonus} = \left(\frac{3000}{100}\right)^2 + 10 = (30)^2 + 10 = 900 + 10 = 910 \]

Total earnings for the first quarter (3 months):
\[ \text{Total for 3 months} = 3 \times (\text{Salary} + \text{Bonus}) = 3 \times (200 + 910) = 3 \times 1110 = 3330 \]

### Step 2: Calculate the bonus for the next 5 months
Abhay's average sales for the next 5 months is Rs. 5000 per month.

Calculating the bonus:
\[ \text{Bonus} = \left(\frac{5000}{100}\right)^2 + 10 = (50)^2 + 10 = 2500 + 10 = 2510 \]

Total earnings for the next 5 months:
\[ \text{Total for 5 months} = 5 \times (200 + 2510) = 5 \times 2710 = 13550 \]

### Step 3: Calculate the bonus for the last 4 months
Abhay's average sales for the last 4 months is Rs. 8000 per month.

Calculating the bonus:
\[ \text{Bonus} = \left(\frac{8000}{100}\right)^2 + 10 = (80)^2 + 10 = 6400 + 10 = 6410 \]

Total earnings for the last 4 months:
\[ \text{Total for 4 months} = 4 \times (200 + 6410) = 4 \times 6610 = 26440 \]

### Step 4: Calculate total earnings for the year
Now, we add up all the earnings from each period:
\[ \text{Total earnings for the year} = 3330 + 13550 + 26440 = 43320 \]

### Step 5: Calculate average monthly earnings
To find the average monthly earnings, we divide the total earnings by 12:
\[ \text{Average monthly earnings} = \frac{43320}{12} = 3610 \]

### Step 6: Add the monthly salary to the average bonus
Finally, we add Abhay's monthly salary to the average monthly bonus:
\[ \text{Total average monthly earnings} = 3610 + 200 = 3810 \]

Thus, Abhay's average earning per month for the whole year is Rs. 3810.