# While vacationing, john has dinner at a restaurant in Japan and his tab comes to 3,428 yen. The...

## Question:

While vacationing, john has dinner at a restaurant in Japan and his tab comes to 3,428 yen. The exchange rate at the time of his visit is 1 yen = 1.25 cents. What was the cost of the meal in US dollars?

## Proportions and Variation:

In diverse applications in the study of arithmetic and algebra, problems are achieved that imply a proportional relationship that exists between two values. A typical example would be when the price of conversion between two different currencies is established. If we know the conversion value of one currency with respect to another, then we can find the price of a product in both currencies.

{eq}\eqalign{ & {\text{In this specific case }}{\text{,we have two proportional values }}\,x\,\left( {yen} \right){\text{ and }} \cr & y\,\left( {{\text{dollar cost}}} \right){\text{ that have a variation in directly proportional form}}{\text{. }} \cr & {\text{We can begin by taking into consideration the following:}} \cr & \,\,\,\,1\,cent = \$\,0.01\,\,\,\, \Rightarrow 1.25\,cents = \$ \,0.0125 \cr & {\text{So we have:}} \cr & \,\,\,\,{x_1} = 1\,\,yen \cr & \,\,\,\,{y_1} = \$\,0.0125 \cr & \,\,\,\,{x_2} = 3428\,\,yen \cr & \,\,\,\,{y_2} = \$ \,\,? \cr & {\text{Since}}{\text{, }}x{\text{ and }}y{\text{ vary directly}}{\text{, then}}{\text{, when }}x{\text{ increases it also }} \cr & {\text{increases }}y{\text{. For this reason}}{\text{, it must be satisfied that:}} \cr & \,\,\,\,\frac{{{y_2}}}{{{x_2}}} = \frac{{{y_1}}}{{{x_1}}} \cr & {\text{So if we do cross - multiplying:}} \cr & \,\,\,\,{y_2} \cdot {x_1} = {y_1} \cdot {x_2} \cr & {\text{Now}}{\text{, solving for }}\,{y_2}{\text{:}} \cr & \,\,\,\,{y_2} = \frac{{{y_1} \cdot {x_2}}}{{{x_1}}} \cr & {\text{So}}{\text{, substituting the given values:}} \cr & \,\,\,\,{y_2} = \frac{{0.0125 \times 3428}}{1} = \$\,42.85 \cr & {\text{Therefore}}{\text{, the cost of the meal was }}\boxed{{\text{\$ }}\,{\text{42}}{\text{.85}}}. \cr} {/eq} 