Logical Reasoning

Math Puzzles

Medium

An ATM dispenses exactly Rs. $5000$ per withdrawal using $100$ , $200$ and $500$ rupee notes. The ATM requires every customer to give her preference for one of the three denominations of notes. It then dispenses notes such that the number of notes of the customer’s preferred denomination exceeds the total number of notes of other denominations dispensed to her.

In how many different ways can the ATM serve a customer who gives $500$ rupee notes as her preference?

Entered answer:

✅ Correct Answer: 7

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)

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

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

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

Final Answer

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

All $7$ Valid Combinations:

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

Question 6

If the ATM could serve only $10$ customers with a stock of fifty $500$ rupee notes and a sufficient number of notes of other denominations, what is the maximum number of customers among these $10$ who could have given $500$ rupee notes as their preferences?

Question 7

What is the maximum number of customers that the ATM can serve with a stock of fifty $500$ rupee notes and a sufficient number of notes of other denominations, if all the customers are to be served with at most $20$ notes per withdrawal?

Question 8

What is the number of $500$ rupee notes required to serve $50$ customers with $500$ rupee notes as their preferences and another $50$ customers with $100$ rupee notes as their preferences, if the total number of notes to be dispensed is the smallest possible?

In how many different ways can the ATM serve a customer who gives 500500500 rupee notes as her preference?

Solution

ATM Solution: Rs. 500 Preferred Denomination

Problem Setup

Analysis: Minimum Rs. 500 notes required

Final Answer

All 777 Valid Combinations:

More from this set:

Keyboard Shortcuts

CAT Past Year Questions (Topic-Wise)

In how many different ways can the ATM serve a customer who gives $500$ rupee notes as her preference?

All $7$ Valid Combinations: