Question

A confused bank teller transposed the rupees and paise when he cashed a cheque for Shailaja, giving her rupees instead of paise and paise instead of rupees. After buying a toffee for 50 paise, Shailaja noticed that she was left with exactly three times as much as the amount on the cheque. Which of the following is a valid statement about the cheque amount?

• A. Over Rupees 13 but less than Rupees 14
• B. Over Rupees 7 but less than Rupees 8
• C. Over Rupees 22 but less than Rupees 23
• D. Over Rupees 18 but less than Rupees 19
• E. Over Rupees 4 but less than Rupees 5

Hint:

When rupees and paise are transposed, treat both amounts in paise to set up a direct linear equation.

Answer 1

The Correct Option is A

Let's denote the original cheque amount as \( x \) Rupees and \( y \) Paise, where \( x \) and \( y \) are integers ranging from 0 to 99.

The bank teller gave Shailaja \( y \) Rupees and \( x \) Paise instead. Therefore, the amount received was \( 100y + x \) Paise or \( y + \frac{x}{100} \) Rupees.

After buying a toffee for 50 Paise, which is \( \frac{50}{100} = 0.5 \) Rupees, Shailaja was left with \( (y + \frac{x}{100}) - 0.5 \) Rupees.

According to the problem, this remaining amount equals three times the original cheque amount: \( 3(x + \frac{y}{100}) \) Rupees.

We can write the equation: 

y+x100-0.5=3(x+y100)

Simplifying further, multiply the whole equation by 100 to eliminate fractions:

100y+x-50=300x+3y

Rearrange terms to isolate variables:

97y=299x+50

Rewriting in terms of integers and solving this linear Diophantine equation:

\( 97y = 299x + 50 \)

We seek \( x \) and \( y \) such that both are less than 100. Testing values near the potential cheque amounts, let's set up test cases for constraints between Rupees 13 and Rupees 14.

Trying \( x = 13 \), the integers from \( 1300 \) to \( 1400 \):

\((97y = 299(13) + 50)\)

\(97y = 3887\)

Solve \( y = \frac{3887}{97} \approx 40.06\), which rounds to \( y = 40 \).

The recovered cheque amount \( x + \frac{y}{100} = 13 + 0.40 = 13.40 \)

The corresponding final cheque statement: 'Over Rupees 13 but less than Rupees 14'.