Each hotdog costs £0.24 and each bun costs £0.14. List the combination(s) of hotdogs and buns that can be purchased with exactly £5.
We encode the problem as follows, with $h$ the number of hotdogs, and $b$ the number of buns.
$$ 24h + 14b = 500 $$
The $\gcd(24,14)=2$, which divides 500, and so the equation has integer solutions.
By inspection, a solution is $h_0=8, b_0=22$. So a general solution is
$$ h = 8 + 7t \quad b = 22 - 12t $$
for any integer $t$.
However, the number of hotdogs and buns must be non-negative, so we have the following inequalities
$$ 8 + 7t \ge 0 $$
$$ 22 - 12t \ge 0 $$
Re-arranging,
$$ t \ge -\frac{8}{7} $$
$$ \frac{11}{6} \ge t $$
Combining,
$$ -\frac{8}{7} \le t \le \frac{11}{6} $$
The values of $t$ which generate non-negative $b$ and $h$ are -1, 0 and 1.
And so the combinations of hotdogs and buns are as follows.
| t | h=8+7t | b=22-12t | 24h+14b |
| -1 | 1 | 34 | 500 |
| 0 | 8 | 22 | 500 |
| 1 | 15 | 10 | 500 |