Question

In how many ways can $5$ pounds be paid in half-crowns and florins?

• A. $9$
• B. $10$
• C. $11$
• D. $12$

Answer 1

Answer: (C)

$11$

$1$ pound $=$ $10$ florins

$1$ pound $=$ $8$ half crowns

Let $x$ be the number of florins and $y$ be the number of half crowns

According to question, we have

$\frac{x}{10}+\frac{y}{8}=5\Rightarrow \frac{4x+5y}{40}=5\Rightarrow 4x+5y=200$ ........(i)

We will approach the problem by finding the number of positive integral solution of the equation (i) as the number of coins can only be a positive integer.

$x+\frac{5y}{4}=50\Rightarrow x+y+\frac{y}{4}=50$

As $x$ and $y$ are integers

$\Rightarrow \frac{y}{4}=$ integer

Let the integer be $p$

$\frac{y}{4}=py=4p$ .........(ii)

Substituting $y$ in (i), we get

$4x+20p=200\Rightarrow 4x=200-20p\Rightarrow x=50-5p$ ......(ii)

We can see from (ii) that $y<0$ for integer $p<0$ and from (iii) $x<0$ for integer $p>11$ , which is not possible.

So, $p$ can take all integer values from $[0,10]$.

So, the number of possible solutions are $11$.