Question: 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?
Explanation:
Preference was given to Rs. 500 notes.
The ATM machine can dispense 10 notes of Rs. 500.
(i) 500 × 10 = Rs. 5,000
If the ATM machine dispenses 9 notes of Rs. 500;
(ii) 500 × 9 = 4500, 200 × 2 = 400 and 100 × 1= 100
(Notes of Rs. 500 denomination = 9 and notes of Rs. 200 and Rs. 100 denominations = 2 + 1 = 3, 9 > 3)
(iii) 500 × 9 = 4500, 200 × 1 = 200 and 100 × 3 = 300
(Notes of Rs. 500 denomination = 9 and notes of Rs. 200 and Rs. 100 denominations = 1 + 3 = 4, 9 > 4)
(iv) 500 × 9 = 4500, 100 × 5 = 500
(Notes of Rs. 500 denomination = 9 and notes of Rs. 200 and Rs. 100 denominations = 0 + 5 = 5, 9 > 5) If the ATM machine dispenses 8 notes of Rs. 500;
(v) 500 × 8 = 4000 and 200 × 5 = 1000
(Notes of Rs. 500 denomination = 8 and notes of Rs. 200 and Rs. 100 denominations = 5 + 0 = 5, 8 > 5)
(vi) 500 × 8 = 4000, 200 × 4 = 800 and 100 × 2 = 200
(Notes of Rs. 500 denomination = 8 and notes of Rs. 200 and Rs. 100 denominations = 4 + 2 = 6, 8 > 6)
(vii) 500 × 8 = 4000, 200 × 3 = 600 and 100 × 4 = 400
(Notes of Rs. 500 denomination = 8 and notes of Rs. 200 and Rs. 100 denominations = 3 + 4 = 7, 8 > 7)
If numbers of notes of denomination Rs. 200 are reduced further, number of notes of the customer’s preferred denomination will not exceed the total number of notes of other denominations.
The ATM machine can dispense 7 notes of Rs. 500, for the minimum number of total notes of the remaining two denominations; one can have 7 notes of Rs. 200 and one note of Rs. 100. But even then, number of notes of the customer’s preferred denomination will be less than the total number of notes of other denominations. As number of notes of denominations reduces, number of notes of the denomination Rs. 500 will be less than the total number of notes of other denominations.
if a customer’s preferred denomination is Rs. 500, the minimum number of notes of Rs. 500 the ATM machine dispenses is 8.
Hence, with fifty 500 rupee notes, the ATM machine can serve 6 customers who could have given 500 rupee notes as their preferences.
Therefore, the required answer is 6.
Answer: 6