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.

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?

Entered answer:

✅ Correct Answer: 6

Maximum Rs. 500 Preference Customers

Constraints

Total customers: $10$
Available Rs. $500$ notes: $50$
Each Rs. $500$ preference customer needs minimum $8$ Rs. $500$ notes
Non-Rs. $500$ preference customers can use $0$ to $7$ Rs. $500$ notes

Rs. 500 Usage by Preference Type

Customers preferring Rs. 500 notes:

Minimum $8$ : If using only $7$ Rs. $500$ notes (Rs. $3500$ ), remaining Rs. $1500$ needs at least $8$ other notes (best case: $7 \times 200 + 1 \times 100$ ). But $7 < 8$ violates the preference constraint.
Maximum $10$ : $10 \times 500 = 5000$ (uses all money)
Valid ranges:
- $8$ Rs. $500$ notes: Rs. $1000$ remaining, can use $\leq 7$ other notes
- $9$ Rs. $500$ notes: Rs. $500$ remaining, can use $\leq 8$ other notes
- $10$ Rs. $500$ notes: Rs. $0$ remaining, $0$ other notes

Customers preferring Rs. 100 or Rs. 200 notes:

The maximum Rs. $500$ notes they can use while still having their preferred denomination dominate.

For Rs. $100$ preference ( $x > y+z$ ):

Maximum $z$ occurs when $x$ is minimized relative to constraint.
If $z=7$ , need $x \geq 8$ , remaining amount $= 5000 - 500 \times 7 = 1500$
$100x + 200y = 1500 \;\Rightarrow\; 100 \times 8 + 200y = 1500 \;\Rightarrow\; y=4$
Total other notes $= 12$ , check: $8 > 4+7=11$ ✗

Testing smaller $z$ values:

$z=6$ : $x \geq 7$ , remaining $=2000$ , gives $x=10$ , $y=5$ , check: $10 > 5+6=11$ ✗
$z=5$ : $x \geq 6$ , remaining $=2500$ , gives valid solutions
Maximum $z$ for Rs. $100$ preference: $4$ notes

For Rs. $200$ preference ( $y > x+z$ ):

Similar analysis shows maximum $z=4$ notes

Customer Type	Rs. $500$ Notes Range
Rs. $500$ preference	$8,9,10$
Rs. $100$ preference	$0$ to $4$
Rs. $200$ preference	$0$ to $4$

Optimisation Strategy

To maximise Rs. $500$ preference customers, minimise Rs. $500$ notes used by others.

Strategy: Use $8$ Rs. $500$ notes per Rs. $500$ preference customer (minimum required)

Scenario	Rs. $500$ Pref	Others	Rs. $500$ Used	Rs. $500$ Remaining	Valid?
$6$ customers prefer Rs. $500$	$6$	$4$	$6 \times 8 = 48$	$2$	✓
$7$ customers prefer Rs. $500$	$7$	$3$	$7 \times 8 = 56$	$-6$	✗

Check $6$ customers scenario:

$6$ customers: $8$ Rs. $500$ notes each $= 48$ notes
$4$ remaining customers: Can use $0-2$ Rs. $500$ notes each
This works within the $50$ note limit

Check if $7$ is possible with optimal distribution:

Minimum for $7$ customers: $7 \times 8 = 56$ notes
Available: $50$ notes
Deficit: $6$ notes → Not possible

Final Answer

Maximum customers who can prefer Rs. $500$ notes: $6$

Question 5

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

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?

If the ATM could serve only 101010 customers with a stock of fifty 500500500 rupee notes and a sufficient number of notes of other denominations, what is the maximum number of customers among these 101010 who could have given 500500500 rupee notes as their preferences?

Solution