ATM Solution: Rs. 500 Preferred Denomination

Problem Setup

• Total: Rs. $5000$
• Constraint: Number of Rs. $500$ notes $>$ Number of other notes combined
• Let $z =$ Rs. $500$ notes, $x =$ Rs. $100$ notes, $y =$ Rs. $200$ notes
• Equation: $100x + 200y + 500z = 5000$
• Constraint: $z > x + y$

Analysis: Minimum Rs. 500 notes required

If $z = 7$ (Rs. $3500$), remaining Rs. $1500$ needs minimum $8$ notes:

• Best case: $7 \times 200 + 1 \times 100 = 1500$ uses $8$ notes
• Since $7 < 8$, this violates $z > x + y$

Therefore, minimum z = 8 (at least Rs. $4000$ in Rs. $500$ notes)

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Case 1: $z = 8$ (Rs. $4000$ in Rs. $500$ notes)

Remaining: Rs. $1000$ using Rs. $100$ and Rs. $200$ notes

Constraint: $8 > x + y$, so $x + y \leq 7$

Rs. $100$ $(x)$	Rs. $200$ $(y)$	Total notes $(x+y)$	Check: $8 > x+y$	Valid
$10$	$0$	$10$	$8 > 10$ ✗	No
$8$	$1$	$9$	$8 > 9$ ✗	No
$6$	$2$	$8$	$8 > 8$ ✗	No
$4$	$3$	$7$	$8 > 7$ ✓	Yes
$2$	$4$	$6$	$8 > 6$ ✓	Yes
$0$	$5$	$5$	$8 > 5$ ✓	Yes

Valid combinations for Case 1: 3

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Case 2: $z = 9$ (Rs. $4500$ in Rs. $500$ notes)

Remaining: Rs. $500$ using Rs. $100$ and Rs. $200$ notes

Constraint: $9 > x + y$

Solve: $200y + 100x = 500 \;\Rightarrow\; 2y + x = 5$

Rs. $100$ $(x)$	Rs. $200$ $(y)$	Total notes $(x+y)$	Check: $9 > x+y$	Valid
$5$	$0$	$5$	$9 > 5$ ✓	Yes
$3$	$1$	$4$	$9 > 4$ ✓	Yes
$1$	$2$	$3$	$9 > 3$ ✓	Yes

Valid combinations for Case 2: 3

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Case 3: $z = 10$ (Rs. $5000$ in Rs. $500$ notes)

No other notes needed: $x = 0, y = 0$

Constraint: $10 > 0 + 0$ ✓

Valid combinations for Case 3: 1

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Final Answer

Total valid ways $= 3 + 3 + 1 = 7$

All $7$ Valid Combinations:

Rs. $100$	Rs. $200$	Rs. $500$	Total Notes	Rs. $500 >$ Others?
$4$	$3$	$8$	$15$	$8 > 7$
$2$	$4$	$8$	$14$	$8 > 6$
$0$	$5$	$8$	$13$	$8 > 5$
$5$	$0$	$9$	$14$	$9 > 5$
$3$	$1$	$9$	$13$	$9 > 4$
$1$	$2$	$9$	$12$	$9 > 3$
$0$	$0$	$10$	$10$	$10 > 0$