A student scored `70%,80%,50% and 48%` in four papers. It was decided to give double weight age to the 1st and the 3rd paper and half to the 2nd and 4th paper. What would have been the average had the weightage system been inter changed?

Answer

To solve the problem step by step, we need to find the average score of the student when the weightage system is interchanged.

### Step 1: Understand the given scores and weightages
The student scored:
- Paper 1: 70%
- Paper 2: 80%
- Paper 3: 50%
- Paper 4: 48%

Initially, the weightages are:
- Paper 1: Double weightage (2)
- Paper 2: Half weightage (0.5)
- Paper 3: Double weightage (2)
- Paper 4: Half weightage (0.5)

### Step 2: Calculate the total score with the initial weightage system
Using the formula for average:
\[ \text{Average} = \frac{(w_1 \cdot x_1) + (w_2 \cdot x_2) + (w_3 \cdot x_3) + (w_4 \cdot x_4)}{w_1 + w_2 + w_3 + w_4} \]
Substituting the values:
\[ \text{Average} = \frac{(2 \cdot 70) + (0.5 \cdot 80) + (2 \cdot 50) + (0.5 \cdot 48)}{2 + 0.5 + 2 + 0.5} \]

### Step 3: Calculate the numerator
Calculating each term:
- For Paper 1: \(2 \cdot 70 = 140\)
- For Paper 2: \(0.5 \cdot 80 = 40\)
- For Paper 3: \(2 \cdot 50 = 100\)
- For Paper 4: \(0.5 \cdot 48 = 24\)

Adding these together:
\[ 140 + 40 + 100 + 24 = 304 \]

### Step 4: Calculate the denominator
Calculating the total weightage:
\[ 2 + 0.5 + 2 + 0.5 = 5 \]

### Step 5: Calculate the average
Now, substituting back into the average formula:
\[ \text{Average} = \frac{304}{5} = 60.8 \]

### Step 6: Interchange the weightage system
Now, we interchange the weightages:
- Paper 1: Half weightage (0.5)
- Paper 2: Double weightage (2)
- Paper 3: Half weightage (0.5)
- Paper 4: Double weightage (2)

### Step 7: Calculate the new average with the interchanged weightage
Using the same average formula:
\[ \text{Average} = \frac{(0.5 \cdot 70) + (2 \cdot 80) + (0.5 \cdot 50) + (2 \cdot 48)}{0.5 + 2 + 0.5 + 2} \]

### Step 8: Calculate the numerator for the new average
Calculating each term:
- For Paper 1: \(0.5 \cdot 70 = 35\)
- For Paper 2: \(2 \cdot 80 = 160\)
- For Paper 3: \(0.5 \cdot 50 = 25\)
- For Paper 4: \(2 \cdot 48 = 96\)

Adding these together:
\[ 35 + 160 + 25 + 96 = 316 \]

### Step 9: Calculate the denominator for the new average
Calculating the total weightage:
\[ 0.5 + 2 + 0.5 + 2 = 5 \]

### Step 10: Calculate the new average
Now, substituting back into the average formula:
\[ \text{Average} = \frac{316}{5} = 63.2 \]

### Final Answer
The average score with the interchanged weightage system is **63.2%**.