A company offers total 150 pens to its customers. As per the scheme one pen will be offered on the purchase of a "Quantitative Aptitude" book. Out of 150 pens the cost of some pens is Rs. 3 and the cost of the rest pens is Rs 5. Maximum how many customers can avail a pen of Rs. 5 as an ofer from the company if the total cost of the pens cannot exceed Rs. 745.

Options

A. 45
B. 120
C. can't be determined
D. none of these

Answer

To solve the problem step by step, we need to find out how many pens of Rs. 5 can be offered to customers while ensuring that the total cost of all pens does not exceed Rs. 745.

### Step 1: Define Variables
Let:
- \( x \) = number of pens costing Rs. 3
- \( y \) = number of pens costing Rs. 5

From the problem, we know:
1. The total number of pens is 150.
\[ x + y = 150 \quad (1) \]

2. The total cost of the pens cannot exceed Rs. 745.
\[ 3x + 5y \leq 745 \quad (2) \]

### Step 2: Express \( x \) in terms of \( y \)
From equation (1), we can express \( x \) in terms of \( y \):
\[ x = 150 - y \quad (3) \]

### Step 3: Substitute \( x \) in the cost equation
Now, substitute equation (3) into equation (2):
\[ 3(150 - y) + 5y \leq 745 \]
Expanding this gives:
\[ 450 - 3y + 5y \leq 745 \]
Combining like terms:
\[ 450 + 2y \leq 745 \]

### Step 4: Solve for \( y \)
Now, isolate \( y \):
\[ 2y \leq 745 - 450 \]
\[ 2y \leq 295 \]
\[ y \leq \frac{295}{2} \]
\[ y \leq 147.5 \]

Since \( y \) must be a whole number (you can't have half a pen), the maximum integer value for \( y \) is 147.

### Step 5: Check if \( y = 147 \) is valid
If \( y = 147 \):
\[ x = 150 - 147 = 3 \]
Now, check the total cost:
\[ 3x + 5y = 3(3) + 5(147) = 9 + 735 = 744 \]
This is less than Rs. 745, so this combination is valid.

### Step 6: Conclusion
Thus, the maximum number of customers that can avail a pen costing Rs. 5 is 147.

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