Question

Nandini paid for an article using currency notes of denominations Re. 1, Rs. 2, Rs. 5, and Rs. 10 using at least one note of each denomination. The total number of five and ten rupee notes used was one more than the total number of one and two rupee notes used. What was the price of the article?

A. Nandini used a total of 13 currency notes.

B. The price of the article was a multiple of Rs. 10.

• A. The question can be answered by using one of the statements alone but not by using the other statement alone.
• B. The question can be answered by using either of the statements alone.
• C. The question cannot be answered on the basis of the two statements.
• D. The question can be answered by using both statements together but not by either statement alone.

Hint:

Let the numbers of Re. 1, Rs. 2, Rs. 5, Rs. 10 notes be a, b, c, d (each at least 1) and write the price in terms of them. Test whether each statement, alone or together, forces a single price.

Answer 1

The Correct Option is C

Step 1: Set up.
Let a, b, c, d be the numbers of Re. 1, Rs. 2, Rs. 5 and Rs. 10 notes, each at least 1. The given condition is \(c+d=a+b+1\) and the price is \(P=a+2b+5c+10d\).

Step 2: Statement A alone.
Total notes = 13 gives \((a+b)+(c+d)=13\), so \(2(a+b)+1=13\), i.e. \(a+b=6,\ c+d=7\). Then \(P=(6+b)+(35+5d)=41+b+5d\) with \(1\le b\le 5,\ 1\le d\le 6\). For example \(b=1,d=1\) gives \(P=47\) and \(b=2,d=1\) gives \(P=48\). Not unique, so A alone is insufficient.

Step 3: Statement B alone.
Take \((a,b,c,d)=(1,2,1,3)\): \(c+d=4=a+b+1\), and \(P=1+4+5+30=40\). Take \((a,b,c,d)=(2,4,6,1)\): \(c+d=7=a+b+1\), and \(P=2+8+30+10=50\). Both prices are multiples of Rs. 10, so B alone does not fix the price.

Step 4: Both together.
We need \(41+b+5d\equiv 0 \pmod{10}\), i.e. \(b+5d\equiv 9 \pmod{10}\). Since \(5d\) ends in 0 or 5 and \(1\le b\le5\), we need \(b=4\) and \(d\) odd, so \(d=1,3,5\). That gives \(P=50,\,60,\,70\) (for \((a,b,c,d)=(2,4,6,1),(2,4,4,3),(2,4,2,5)\)). The price is still not unique.

Final Answer:
The correct option is (3): The question cannot be answered on the basis of the two statements.