A bill for ₹ 1,840 is paid in the denominations of ₹ 50, ₹ 20 and ₹ 10 notes. 50 notes in all are used. Consider the following statements : 1. 25 notes of ₹ 50 are used and the remaining are in the denominations of ₹ 20 and ₹ 10. 2. 35 notes of ₹ 20 are used and the remaining are in the denominations of ₹ 50 and ₹ 10. 3. 20 notes of ₹ 10 are used and the remaining are in the denominations of ₹ 50 and ₹ 20. Which of the above statements are not correct?

(a) 1 and 2 only
(b) 2 and 3 only
(c) 1 and 3 only
(d) 1, 2 and 3 ✓ UPSC's answer

Why the answer is (d)

• Let the number of ₹50, ₹20, and ₹10 notes be x, y, and z respectively. The total number of notes is 50, so x + y + z = 50. The total value is ₹1,840, so 50x + 20y + 10z = 1,840.

• Simplifying the value equation by dividing by 10 gives 5x + 2y + z = 184. Subtracting the count equation (x + y + z = 50) from this yields 4x + y = 134.

• For Statement 1: If x = 25, then 4(25) + y = 134 implies 100 + y = 134, so y = 34. Then z = 50 - 25 - 34 = -9. Since the number of notes cannot be negative, Statement 1 is incorrect.

• For Statement 2: If y = 35, then 4x + 35 = 134 implies 4x = 99, so x = 24.75. Since the number of notes must be an integer, Statement 2 is incorrect.

• For Statement 3: If z = 20, then x + y = 30. Substituting y = 30 - x into 4x + y = 134 gives 4x + 30 - x = 134, so 3x = 104, which means x = 34.67. Since x must be an integer, Statement 3 is incorrect.

• Since all three statements lead to impossible values (negative or non-integer note counts), all are not correct, making option (d) the right answer.

Why the other options are wrong

(a) 1 and 2 only: Statement 3 is also incorrect because it results in a non-integer number of ₹50 notes.
(b) 2 and 3 only: Statement 1 is also incorrect because it results in a negative number of ₹10 notes.
(c) 1 and 3 only: Statement 2 is also incorrect because it results in a non-integer number of ₹50 notes.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2022, held on 5 June 2022. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.