A man has 1680 pens and he categorized all of the total selling the pens into three types such that he earns a profit of 22%, 26% and 30% respectively on each type. He earns 23% profit on first two types and earns overall profit of 25% on all types. Find the number of pens in first types.

Options

A. a)900
B. b)400
C. c)450
D. d)200

Answer

To solve the problem step by step, we need to find the number of pens in the first type (Type A) given the overall profit percentages and the total number of pens.

### Step 1: Define Variables
Let:
- A = Number of pens in Type A
- B = Number of pens in Type B
- C = Number of pens in Type C

Given that the total number of pens is:
\[ A + B + C = 1680 \]

### Step 2: Profit Percentages
The profit percentages for each type are:
- Type A: 22%
- Type B: 26%
- Type C: 30%

### Step 3: Profit on Types A and B
The combined profit on Types A and B is given as 23%. This means we can set up the following equation based on the profit percentages:
- Profit from A and B combined:
\[ \frac{(22A + 26B)}{(A + B)} = 23 \]
Cross-multiplying gives:
\[ 22A + 26B = 23(A + B) \]
This simplifies to:
\[ 22A + 26B = 23A + 23B \]
Rearranging gives:
\[ A = 3B \quad \text{(1)} \]

### Step 4: Overall Profit Calculation
The overall profit from all three types is 25%. Therefore, we can write:
\[ \frac{(22A + 26B + 30C)}{(A + B + C)} = 25 \]
Substituting \(A + B + C = 1680\):
\[ 22A + 26B + 30C = 25 \times 1680 \]
Calculating \(25 \times 1680\):
\[ 25 \times 1680 = 42000 \]
Thus, we have:
\[ 22A + 26B + 30C = 42000 \quad \text{(2)} \]

### Step 5: Express C in terms of A and B
From equation (1), substitute \(A = 3B\) into the total pens equation:
\[ 3B + B + C = 1680 \]
This simplifies to:
\[ 4B + C = 1680 \]
Rearranging gives:
\[ C = 1680 - 4B \quad \text{(3)} \]

### Step 6: Substitute C into Equation (2)
Now substitute equation (3) into equation (2):
\[ 22(3B) + 26B + 30(1680 - 4B) = 42000 \]
This simplifies to:
\[ 66B + 26B + 50400 - 120B = 42000 \]
Combining like terms:
\[ -28B + 50400 = 42000 \]
Rearranging gives:
\[ -28B = 42000 - 50400 \]
\[ -28B = -8400 \]
Dividing by -28:
\[ B = 300 \]

### Step 7: Find A and C
Using \(B = 300\) in equation (1):
\[ A = 3B = 3 \times 300 = 900 \]

Using \(B = 300\) in equation (3):
\[ C = 1680 - 4B = 1680 - 4 \times 300 = 1680 - 1200 = 480 \]

### Step 8: Final Answer
Thus, the number of pens in the first type (Type A) is:
\[ \boxed{900} \]