A donation box can receive only cheques of ₹100, ₹250, and ₹500. On one good day, the donation box was found to contain exactly 100 cheques amounting to a total sum of ₹15250. Then, the maximum possible number of cheques of ₹500 that the donation box may have contained, is

Question

Accepted Answer

#### Step 1: Define Variables and Equations  
Let:  
- The number of ₹100 cheques = $$x$$,  
- The number of ₹250 cheques = $$y$$,  
- The number of ₹500 cheques = $$z$$.

The total number of cheques is:  
$$
x + y + z = 100 	ag{1}.
$$

The total value of the cheques is:  
$$
100x + 250y + 500z = 15,250 	ag{2}.
$$

#### Step 2: Simplify the Value Equation  
Divide Equation (2) by 50 to simplify:  
$$
2x + 5y + 10z = 305 	ag{3}.
$$

#### Step 3: Eliminate $$x$$ to Solve for $$y$$ and $$z$$  
Multiply Equation (1) by 2 and subtract it from Equation (3):  
$$
(2x + 5y + 10z) - (2x + 2y + 2z) = 305 - 200.
$$  
Simplify:  
$$
3y + 8z = 105 	ag{4}.
$$

#### Step 4: Maximize $$z$$ (Number of ₹500 Cheques)  
From Equation (4):  
$$
3y = 105 - 8z \implies y = \frac{105 - 8z}{3}.
$$  
For $$y$$ to be an integer, $$105 - 8z$$ must be divisible by 3.

#### Step 5: Check Divisibility Condition for $$z$$  
Since $$z \geq 0$$ and $$x, y, z \geq 0$$:  
- Substitute values of $$z$$ such that $$105 - 8z \equiv 0 \pmod{3}.$$

Simplify modulo condition:  
$$
105 \equiv 0 \pmod{3}, \quad -8z \equiv 0 \pmod{3}.
$$  
Thus, $$z$$ must be a multiple of 3.

#### Step 6: Test Maximum Value of $$z$$ Under Constraints  
From Equation (1): $$x + y + z = 100.$$  
From Equation (4): $$3y + 8z = 105.$$

Substitute values for $$z$$ starting from the maximum possible under $$x, y, z \geq 0$$:

- **Case $$z = 12$$:**  
    $$
    y = \frac{105 - (8)(12)}{3} = \frac{105 - 96}{3} = \frac{9}{3} = 3.
    $$
    From Equation (1):  
    $$
    x + y + z = x + 3 + 12 = 100 \implies x = 85.
    $$
    All values are valid integers.

- **Case $$z > 12$$:** Results in negative or non-integer values for $$y$$ or $$x$$.

Thus, the maximum value of $$z$$ is **12**.

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### Final Answer:  
The maximum possible number of ₹500 cheques is **12**.