Henry Ernest Dudeney/536 Puzzles & Curious Problems/1 - Concerning a Cheque/Solution

by : $1$

 * Concerning a Cheque
 * A man went into a bank to cash a check.
 * In handing over the money the cashier, by mistake, gave him dollars for cents and cents for dollars.
 * He pocketed the money without examining it, and spent a nickel on his way home.
 * He then found that he possessed exactly twice the amount of the check.
 * He had no money in his pocket before going to the bank.
 * What was the exact amount of that check?

Solution
The amount on the the check was $\$ 31.63$.

After leaving the bank, he had $\$ 63.31$.

After arriving home, he had $\$ 63.26$, which is twice $\$ 31.63$.

Proof
Let $C$ be the amount written on the check.

Let $C$ consist of $C_d$ dollars and $C_c$ cents.

Let $C_1$ be the value of the check in cents.

Let $C_2$ be the money the man left the bank with in cents.

Let $C_3$ be the money the man arrived home with in cents.

We have:
 * $C_1 = 100 C_d + C_c$

After coming out of the bank, the man has:
 * $C_2 = 100 C_c + C_d$

After arriving home, the man has:
 * $C_3 = 100 C_c + C_d - 5$

But we have:
 * $C_1 \times 2 = C_3$

which leads us to:

This is a linear Diophantine equation.

Solving

we arrive at:


 * $C_d = 31$

and:
 * $C_c = 63$