A person purchases 1 kg of tomatoes from each of the 4 places at the rate of 1 kg, 2 kg, 3 kg, 4 kg per rupee respectively. On an average, he has purchased `x` kg of tomatoes per rupee. Then the value of `x` is (a) 1.92 (b) 2 (c) 2.5 (d) None of these

Options

A. 2
B. 25
C. 1.92
D. none of these

Explanation

To solve the problem, we will use the concept of the harmonic mean since we are dealing with rates (kg per rupee). Here’s the step-by-step solution:

### Step 1: Identify the rates
The rates at which tomatoes are purchased from the four places are:
- Place 1: 1 kg per rupee
- Place 2: 2 kg per rupee
- Place 3: 3 kg per rupee
- Place 4: 4 kg per rupee

### Step 2: Set up the formula for harmonic mean
The formula for the harmonic mean (HM) of n observations is given by:
\[ HM = \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \frac{1}{x_3} + \frac{1}{x_4}} \]
where \(x_1, x_2, x_3, x_4\) are the rates of tomatoes per rupee.

### Step 3: Calculate the harmonic mean
Here, \(n = 4\) (the number of places). The rates are:
- \(x_1 = 1\)
- \(x_2 = 2\)
- \(x_3 = 3\)
- \(x_4 = 4\)

Now substituting these values into the formula:
\[ HM = \frac{4}{\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4}} \]

### Step 4: Calculate the sum of the reciprocals
Calculating the sum of the reciprocals:
\[ \frac{1}{1} = 1, \quad \frac{1}{2} = 0.5, \quad \frac{1}{3} \approx 0.333, \quad \frac{1}{4} = 0.25 \]
Adding these together:
\[ 1 + 0.5 + 0.333 + 0.25 = 2.083 \]

### Step 5: Substitute back into the harmonic mean formula
Now substituting back into the harmonic mean formula:
\[ HM = \frac{4}{2.083} \approx 1.92 \]

### Step 6: Conclusion
Thus, the average amount of tomatoes purchased per rupee, \(x\), is approximately 1.92 kg per rupee.

### Final Answer
The value of \(x\) is (a) 1.92.

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