Problem Solving

If 5 dollars and 35 crowns is equivalent to 7 pounds, and 4 dollars and 4 pounds is equivalent to 56 crowns, 1 pound and 28 crowns is equivalent to how many dollars?

$7.00
$7.50
$8.50
$9.00
$10.00

The OA is A.

Experts, can you help me here? What are the equations I should set here? Thanks in advanced.

Hi Vincen,
Let's take a look at your question.

Suupose that d represents the dollars, c represents the crowns and p represents the pounds, then according to the question statement, we can write the following equations.
5 dollars and 35 crowns is equivalent to 7 pounds:
$$5d+35c=7p$$
$$\frac{5}{7}d+5c=p ... (i)$$

4 dollars and 4 pounds is equivalent to 56 crowns
$$4d+4p=56c$$
$$\frac{4}{56}d+\frac{4}{56}p=c$$
$$\frac{1}{14}d+\frac{1}{14}p=c$$

Substitute p using eq(i)
$$\frac{1}{14}d+\frac{1}{14}\left(\frac{5}{7}d+5c\right)=c$$
$$\frac{1}{14}d+\frac{5}{98}d+\frac{5}{14}c=c$$
$$\frac{1}{14}d+\frac{5}{98}d=c-\frac{5}{14}c$$
$$\frac{7+5}{98}d=\frac{14-5}{14}c$$
$$\frac{12}{98}d=\frac{9}{14}c$$
$$\frac{14}{9}\times\frac{12}{98}d=c$$
$$\frac{1}{3}\times\frac{4}{7}d=c$$
$$c=\frac{4}{21}d ... (ii)$$

The last thing the question says is, 1 pound and 28 crowns is equivalent to how many dollars?
Let x represents the number of dollars we need to find out then:
$$1p+28c=xd$$
Substitute p from eq(i)
$$\frac{5}{7}d+5c+28c=xd$$
$$\frac{5}{7}d+33c=xd$$

Again substitute c using eq(ii) so that we have have the equation in terms of d,
$$\frac{5}{7}d+33\left(\frac{4}{21}d\right)=xd$$
$$\frac{5}{7}d+11\left(\frac{4}{7}d\right)=xd$$
$$\frac{5}{7}d+\frac{44}{7}d=xd$$
$$\frac{49}{7}d=xd$$
$$7d=xd$$
$$x=7$$

Therefore, Option A is correct.

Hope it helps.
I am available if you'd like any follow up.