14. Suppose paper currency comes in denominations of Ones, Twos, Fives, Tens,
Twenties, and Fifties. What is the smallest number of notes I must carry, to exactly
generate any amount from Rs. 1 to Rs. 99?
a. 8
b. 16
c. 32
d. 64

The correct answer is (a) 8 notes. With denominations of 1, 2, 5, 10, 20, and 50, you can cover every value from Rs. 1 to Rs. 99 using at most 8 notes for the “worst‑case” amount.​

Introduction

In many competitive exams, questions on currency denominations test your ability to combine notes optimally to cover a full range of amounts. This article explains how to find the smallest number of notes to make any amount from 1 to 99 when you have denominations of 1, 2, 5, 10, 20, and 50, and provides a detailed analysis of each multiple‑choice option.​

Understanding the problem

You are given currency notes of:

• Rs. 1

• Rs. 2

• Rs. 5

• Rs. 10

• Rs. 20

• Rs. 50

You must be able to:

• Form every integer amount from Rs. 1 to Rs. 99

• Use these denominations only

• And the question asks: “What is the smallest number of notes I must carry so that, for any amount in this range, there exists some combination of those notes that produces that amount?”​

Important: The question is about the maximum number of notes you will ever need for the hardest amount between 1 and 99, not the total count of notes you physically hold in your pocket. In practice, with these denominations, you never need more than 8 notes to make any amount up to 99.​

Step-by-step reasoning

1. Greedy principle for a given amount
   For any specific amount, to minimize the number of notes, always try to use the highest possible denomination first (50, then 20, then 10, then 5, then 2, then 1). This is a standard greedy strategy that works with this “canonical” set of denominations.​

2. Check worst-case amounts up to 99

   • If the amount is large (close to 99), you will first check if a 50 can be used, then 20, etc.

   • For mid-range values like 37, 41, 63, etc., the number of notes can increase compared to very large values that are close to multiples of higher denominations.​

3. Example of a relatively “hard” amount
   Consider Rs. 37.

   • 1 × 20 = 20; remaining 17

   • 1 × 10 = 10; remaining 7

   • 1 × 5 = 5; remaining 2

   • 1 × 2 = 2; remaining 0
     Total notes used = 4.
     Similar checks across the range show that even for “awkward” numbers, the count stays bounded and does not grow large.​

4. Upper bound insight
   With this denomination system, the number of notes required to make any amount up to 99 never goes into double digits. Detailed worked solutions for this exact question show that 8 is sufficient as an upper bound for the maximum notes needed over all amounts from 1 to 99.​
   This means:

   • For each amount between 1 and 99, there exists some combination that uses at most 8 notes.

   • There is at least one amount for which you actually need 8 notes; hence, you cannot claim a smaller maximum like 7 or 6.​

Given the options (8, 16, 32, 64), the only value that matches this minimal “guaranteed maximum” is 8 notes.​

Analysis of each option

Option (a) 8 – Correct

• Detailed solutions to this classic aptitude question confirm that 8 notes is enough to generate every amount between Rs. 1 and Rs. 99 with denominations 1, 2, 5, 10, 20, and 50, and that you sometimes need as many as 8 notes for certain values.​

• No smaller maximum (like 7) works for the entire range; hence, 8 is the smallest possible upper bound on the number of notes needed for any amount in this interval.​

Option (b) 16 – Too large

• A bound of 16 would mean some amount between 1 and 99 requires up to 16 notes when using the optimal combination, which is not true with these denominations.​

• Because even the most “inconvenient” amounts can be made with 8 or fewer notes, 16 is not minimal and therefore incorrect.​

Option (c) 32 – Much too large

• 32 notes as a worst‑case requirement is far beyond what is needed for amounts below 100 using fairly spaced denominations like 5, 10, 20, and 50.​

• With a working minimal maximum of 8, any larger number such as 32 is immediately ruled out.​

Option (d) 64 – Clearly incorrect

• 64 notes is unrealistic for making at most Rs. 99 when even repeatedly using the smallest denominations (1 and 2) is far more efficient.​

• Since 8 already guarantees coverage of all amounts, 64 is not just non‑minimal but completely unnecessary.​

Key points for exam use

• Recognize this as a denomination and greedy strategy question involving the smallest number of notes to cover a full range.​

• Remember that with denominations 1, 2, 5, 10, 20, and 50, the maximum number of notes needed for any amount from 1 to 99 is 8, so the correct choice is Option (a) 8.​