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

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

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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?

A. $7.00
B. $7.50
C. $8.50
D. $9.00
E. $10.00

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nonplus2 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?

A. $7.00
B. $7.50
C. $8.50
D. $9.00
E. $10.00
$$?\,\,\,:\,\,\,p + 28c = f\left( d \right)$$
$$\left( {\text{I}} \right)\,\,5d + 35c = 7p$$
$$4d + 4p = 56c\,\,\,\, \Rightarrow \,\,\,\,\left( {{\text{II}}} \right)\,\,d + p = 14c$$
$$\left\{ \matrix{
2 \cdot \left( {\rm{I}} \right)\,\,\,\,10d + 70c = 14p\,\,\, \hfill \cr
5 \cdot \,\left( {{\rm{II}}} \right)\,\,\,5d + 5p = 70c \hfill \cr} \right.\,\,\,\mathop \Rightarrow \limits^{\left( + \right)} \,\,\,\,\,\,15d = 9p\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,p = {5 \over 3}d\,\,\,\,\left( {{\rm{III}}} \right)$$
$$\left\{ \matrix{
\left( {\rm{I}} \right)\,\,\,\,5d + 35c = 7p\,\,\, \hfill \cr
7 \cdot \,\left( {{\rm{II}}} \right)\,\,\,7d + 7p = 98c \hfill \cr} \right.\,\,\,\mathop \Rightarrow \limits^{\left( + \right)} \,\,\,\,\,\,12d = 63p\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,c = {4 \over {21}}d\,\,\,\,\left( {{\rm{IV}}} \right)$$
$$?\,\,:\,\,\,p + 28c\,\,\,\mathop = \limits^{{\text{III}}\,,\,\,{\text{IV}}} \,\,\,\,\frac{5}{3}d + 28\left( {\frac{4}{{21}}d} \right) = \frac{5}{3}d + \frac{{16}}{3}d = 7d\,\,\,\,\, \Rightarrow \,\,\,\,\boxed{\,\,? = \$ 7\,\,}$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

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nonplus2 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?

A. $7.00
B. $7.50
C. $8.50
D. $9.00
E. $10.00
To solve this problem, we need to determine the equivalent of 1 pound to dollars and 1 crown to dollars. Let c = value of 1 crown in dollars and p = value of 1 pound in dollars. So we have:

5 + 35c = 7p

4 + 4p = 56c

We can simplify the second equation as 1 + p = 14c. Isolating p, we have p = 14c - 1. Substituting this back in the first equation, we have:

5 + 35c = 7(14c - 1)

5 + 35c = 98c - 7

12 = 63c

c = 12/63 = 4/21 dollars

Since c = 4/21, p = 14 x 4/21 - 1 = 2 x 4/3 - 1 = 8/3 - 3/3 = 5/3 dollars

Therefore, 1 pound and 28 crowns = 1 x 5/3 + 28 x 4/21 = 5/3 + 4 x 4/3 = 5/3 + 16/3 = 21/3 = 7 dollars.

Answer: A

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Scott Woodbury-Stewart Founder and CEO

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