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If 5 dollars and 35 crowns is equivalent to 7 pounds......

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If 5 dollars and 35 crowns is equivalent to 7 pounds......

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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.

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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.

_________________
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You can run through some laborious algebra or if you're short on time you can inspect the answers.

The number 7 figures heavily in the question...I would have picked that as a wild guess

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Or algebraically:

Let X = number of crowns/pound
Y = crowns/dollar

therefore number of dollars/pound = X/Y

Since you're trying to find the number of dollars, rearrange the two given statements to equal dollars:

5 = 7*X/Y - 35/Y

4 = 56/Y - 4*X/Y

Use the first statement to solve for Y. Multiply both sides by Y: 5*Y = 7*x -35> therefore Y=(7*X-35)/5

Multiply the second statement both sides by Y: 4*Y = 56-4*X

Substitute Y=(7*X-35)/5: 4(7*X-35)/5 = 56 - 4*X

Solve for X= 105/12 = 35/4.

Substitute this value of X to solve for Y = (7*(35/4) - 35)/5 = 21/4

The question asks for how many dollars equals 1 pound and 28 crowns. Algebraically, this equals :

1*(X/Y) + 28/Y = (35/21) + 28*(4/21)

= 7(5 + 16)/21 = 7,A

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Vincen wrote:
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
To solve this problem, we need to determine the equivalent of 1 pound to dollars and 1 crown to dollars. Let d = value of 1 dollar, c = value of 1 crown and p = value of 1 pound. We are given that:

5d + 35c = 7p

4d + 4p = 56c

We can simplify the second equation as d + p = 14c. Let’s isolate d in each equation:

5d = 7p - 35c ' [1]

d = -p + 14c ' [2]

If we multiply equation [2] by 7 and add that to equation [1], we have:

12d = 63c

c = 12d/63 = 4d/21

Similarly, if we multiply equation [1] by 2 and equation [2] by 5 and add those together, we have:

15d = 9p

p = 15d/9 = 5d/3

Thus, 1 pound and 28 crowns is equivalent to:

5d/3 + 28 x 4d/21 = 5d/3 + 16d/3 = 21d/3 = 7d or 7 dollars

Answer: A

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