Q8·CSAT · Prelims 2022

Linear equations and currency denomination constraints

NumericalArithmetic Word ProblemsStatement-based● Medium

Question

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:

125 notes of 50 are used and the remaining are in the denominations of 20 and 10.

235 notes of 20 are used and the remaining are in the denominations of 50 and 10.

320 notes of 10 are used and the remaining are in the denominations of 50 and 20.

Which of the above statements are not correct?

Options

1 and 2 only

2 and 3 only

1 and 3 only

1, 2 and 3

Answer

Explanation

Let x, y, z be the count of ₹50, ₹20, and ₹10 notes respectively. Total notes: x + y + z = 50. Total value: 50x + 20y + 10z = 1840 \implies 5x + 2y + z = 184.

Substitute z = 50 - x - y into the value equation: 5x + 2y + (50 - x - y) = 184 \implies 4x + y = 134.

Test the statements using this reduced linear constraint:

Statement 1 (x = 25): 4(25) + y = 134 \implies 100 + y = 134 \implies y = 34. The sum of notes is x+y = 59. This violates the hard cap of 50 total notes. (Not Correct)

Statement 2 (y = 35): 4x + 35 = 134 \implies 4x = 99. Since 99 is not divisible by 4, x is not an integer. (Not Correct)

Statement 3 (z = 20): This means x + y = 30. Subtracting this from 4x + y = 134 yields 3x = 104. Since 104 is not divisible by 3, x is not an integer. (Not Correct)

All three statements are mathematically impossible.

For 3-variable linear Diophantine systems with a fixed total count, immediately substitute the substitution constraint (z = Total - x - y) to reduce the system to 2 variables.

Answer: (d).

Question details

Year

2022

Paper

CSAT

Question

Section

Numerical Ability

Sub-topic

Arithmetic Word Problems

Type

Statement-based

Difficulty

Medium

Source hint

Bill of 1840 paid in 50, 20, 10 notes

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