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?
Correct Answer: 7
Solution and Explanation
We are to find the number of different ways an ATM can distribute a total of ₹5000 using ₹100, ₹200, and ₹500 notes, giving preference to ₹500 notes.
Let:
$x$ = number of ₹500 notes
$y$ = number of ₹200 notes
$z$ = number of ₹100 notes
The total value must satisfy:
$500x + 200y + 100z = 5000$
Also, the number of ₹500 notes must be greater than the number of ₹200 and ₹100 notes combined:
$x > y + z$
Approach:
Since ₹500 notes are preferred, we try to maximize $x$ and work our way down.
Case 1: $x = 10$
$500 \times 10 = 5000$
No other notes needed ⇒ $y = 0$, $z = 0$
Condition: $10 > 0 + 0$ ✅
Valid way = 1
Case 2: $x = 9$
$500 \times 9 = 4500$
Remaining ₹500 to be distributed with $y$ and $z$
Try combinations:
- $y = 1, z = 0$ ⇒ $200 \times 1 = 200$, total = ₹4700 ❌
- $y = 0, z = 2$ ⇒ $100 \times 2 = 200$, total = ₹4700 ❌
- $y = 1, z = 0$ ⇒ $500 \times 9 + 200 \times 1 = 5000$ ✅
Check: $9 > 1$ ✅ - $y = 0, z = 2$ ⇒ $500 \times 9 + 100 \times 2 = 5000$ ✅
Check: $9 > 2$ ✅
Valid ways = 2
Case 3: $x = 8$
$500 \times 8 = 4000$
Remaining ₹1000 to be filled
- $y = 5, z = 0$ ⇒ $200 \times 5 = 1000$ ⇒ total = ₹5000
Check: $8 > 5$ ✅ - $y = 4, z = 2$ ⇒ $800 + 200 = 1000$ ⇒ total = ₹5000
Check: $8 > 6$ ✅ - $y = 3, z = 4$ ⇒ $600 + 400 = 1000$ ⇒ total = ₹5000
Check: $8 > 7$ ✅
Valid ways = 3
Case 4: $x = 7$
$500 \times 7 = 3500$
Remaining ₹1500 to be filled
- $y = 7, z = 1$ ⇒ $1400 + 100 = 1500$ ⇒ total = ₹5000
Check: $7 > 8$ ❌ - $y = 6, z = 3$ ⇒ $1200 + 300 = 1500$ ⇒ total = ₹5000
Check: $7 > 9$ ❌ - $y = 5, z = 5$ ⇒ $1000 + 500 = 1500$ ⇒ total = ₹5000
Check: $7 > 10$ ❌
Only 1 valid way when $y = 5, z = 5$ and $x = 7$ ✅
Total Valid Ways:
Case 1: 1 way
Case 2: 2 ways
Case 3: 3 ways
Case 4: 1 way
Total = 1 + 2 + 3 + 1 = 7 ways
Final Answer: 7
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?
Correct Answer: 6
Solution and Explanation
A customer prefers to receive ₹500 notes. It is required that the number of ₹500 notes dispensed must be greater than the number of notes of any other denomination.
We are to determine the maximum number of customers that can be served with a stock of 50 notes of ₹500, considering this condition.
Case Analysis:
Initial Observation:
If ₹3500 is dispensed using ₹500 notes (7 notes), then ₹1500 remains to be dispensed using ₹100 and ₹200 notes.
Minimum number of other denomination notes needed:
- Use one ₹100 note and seven ₹200 notes → total 8 notes
So, other denomination notes = 8, which is more than ₹500 notes = 7 ⇒ Invalid.
Hence, at least ₹4000 must be dispensed using ₹500 notes.
Case 1: ₹4000 dispensed as ₹500 notes
₹4000 using ₹500 notes ⇒ 8 notes.
Remaining ₹1000 to be dispensed using ₹100 and ₹200 notes.
Try combinations:
- ₹1000 as 10 × ₹100 ⇒ 10 notes → not valid (same number as ₹500 notes)
- ₹1000 as 8 × ₹100 + 1 × ₹200 ⇒ 9 notes → not valid
- ₹1000 as 6 × ₹100 + 2 × ₹200 ⇒ 8 notes → not valid
- ₹1000 as 4 × ₹100 + 3 × ₹200 ⇒ 7 notes → valid ✅
- ₹1000 as 2 × ₹100 + 4 × ₹200 ⇒ 6 notes → valid ✅
- ₹1000 as 5 × ₹200 ⇒ 5 notes → valid ✅
Therefore, for ₹4000 in ₹500 notes (8 notes), we have 3 valid ways to dispense the remaining ₹1000.
Case 2: ₹4500 dispensed as ₹500 notes
₹4500 using ₹500 notes ⇒ 9 notes.
Remaining ₹500 to be dispensed using:
- 5 × ₹100 = 5 notes
- or 1 × ₹100 + 2 × ₹200 = 3 notes
- or 2 × ₹100 + 1 × ₹200 = 3 notes
All valid combinations as they involve fewer notes than ₹500 notes ⇒ Valid ✅
So, this case gives at least 1 valid way.
Case 3: ₹5000 dispensed as ₹500 notes
₹5000 using 10 × ₹500 notes ⇒ No other denominations needed.
This is also valid ✅
Total valid ways in this case = 1
Final Calculation:
ATM has 50 notes of ₹500.
We want to find how many customers can be served, assuming each uses the least number of ₹500 notes to meet the preference condition.
From case analysis:
- The minimum number of ₹500 notes needed to fulfill the condition is 8.
So, maximum number of customers = $\left\lfloor \frac{50}{8} \right\rfloor = 6$
Answer: 6 customers
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?
- 10
- 16
- 12
- 13
The Correct Option is C
Solution and Explanation
The correct answer is (C):
Since there are a limited number of 500 rupee notes, we can minimize the number of 500 rupee notes dispensed to each customer, while ensuring that each customer is served at most 20 notes.
If no 500 rupee notes is dispensed, the minimum number of notes that must be dispensed is 25 (all 200 rupee notes). This is not possible.
If one 500 rupee note is dispensed, the minimum number of notes is 14 (one 500 rupee note, twelve 200 rupee notes and one 100 rupee note). This is also not possible.
If two 500 rupee notes are dispensed, the minimum number of notes is 22 (two 500 rupee notes and twenty 200 rupee notes).
If three 500 rupee notes are dispensed, the minimum number of notes is 21 (three 500 rupee notes, seventeen 200 rupee notes and one 100 rupee note). If four 500 rupee notes are dispensed, the minimum number of notes is 19 (four 500 rupee notes and fifteen 200 rupee notes). Hence, the minimum number of 500 rupee notes that can be dispensed to any person is 4. With fifty 500 rupee notes, a maximum of 12 persons can be served. Ans : 12
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?
- 800
- 750
900
1400
The Correct Option is C
Solution and Explanation
An ATM dispenses exactly Rs 5000 per withdrawal using notes of denominations Rs 100, Rs 200, and Rs 500. Each customer chooses a preferred denomination, and the number of notes dispensed in their preferred denomination must exceed the total number of all other notes dispensed. We need to determine the minimum number of Rs 500 notes required for 50 customers preferring Rs 500 notes and another 50 customers preferring Rs 100 notes.
Analysis:
1. For customers preferring Rs 500 notes:
To ensure Rs 500 notes are in the majority, one must dispense Rs 5000 using mostly Rs 500 notes. The best way is to give 9 notes of Rs 500 and 1 of Rs 100 (Rs 4500 + Rs 500 = Rs 5000).
Total: 10 notes.
2. For customers preferring Rs 100 notes:
To ensure Rs 100 notes are in the majority, use the combination 31 Rs 100 notes and 3 Rs 300 (from Rs 500) to make Rs 5000 (Rs 3100 + Rs 600 = Rs 5000).
Total: 34 notes.
Calculations:
For 50 customers preferring Rs 500 notes, each gets 9 Rs 500 notes.
Total Rs 500 notes needed = 50 customers × 9 notes = 450 notes.
For 50 customers preferring Rs 100 notes, each gets 3 Rs 500 notes.
Total Rs 500 notes needed = 50 customers × 3 notes = 150 notes.
Total Rs 500 Notes:
Total Rs 500 notes needed = 450 (for Rs 500 preference) + 150 (for Rs 100 preference) = 600 notes.
Conclusion:
The minimum number of Rs 500 notes required for both preferences is:
600
Top Questions on Conclusion
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