1.5 Rates and Currency Conversions

When you take the ratio of two quantities that have different units of measurement (for example, money and time), you refer to the resulting ratio as a rate. As before, using the terms amount and the base:

[latex]Rate = \frac{Amount}{Base}[/latex]

A sales rate would have the amount in dollars and the base, perhaps, in days. So if sales were $18,000 in a 10-day period, you would have:

[latex]\text{Sales Rate} = \frac{$18,000}{10 \text{ days}} = $1,800\text{ per day}[/latex]

Similarly a pay rate might be in dollars per hour and an exchange rate in dollars per British pound.

One use of such rates is to find a new amount for a different base (assuming the rate is constant). You would have to do this if you wanted to find the cost of a given amount of foreign money. In this case, you would check the exchange rates, which are determined by money markets and reported regularly in the financial pages of newspapers.

Example 1.5.1

Suppose the exchange rate is $2.10 per British pound and you need £300. Then:

[latex]Rate= \frac{\text{Amount in \$}}{\text{Amount in £}}=\frac{\$2.10}{£1}[/latex]

So

[latex]\text{Amount in \$} = £ 300 ×\frac{\$2.10}{£ 1}=\frac{£ 300}{1} × \frac{\$2.10}{£ 1}[/latex]

And

[latex]\text{Amount in \$} =  300 \times \$2.10=\$630.00[/latex]

Notice that if, given the same rate, you wanted to find the number of pounds you could buy with, say $500, you could reverse the ratio:

[latex]\text{Amount in £}= \$500\times \frac{\$1}{\$2.10}£ = £238.10[/latex]

When dealing with rates, it is critical to keep track of the units; they can guide your calculations if you use them for arithmetic operations.

If, for example, you want Canadian dollars (CAD) for US dollars (USD), and if you know that the exchange rate is 0.68 USD per CAD, then your rates are

[latex]\frac{0.68 \text{ USD}}{1\text{ CAD}}=0.68 \text{ USD per CAD}[/latex]

And

[latex]\frac{1\text{ CAD}}{0.68 \text{ USD}}=1.47059\text{ CAD per USD}[/latex]

Check this by using a simple example –   say, $1,000 (Canadian). Then, by the first rate, you would have:

[latex]1,000 \text{ CAD}\times \frac{0.68 \text{ USD}}{1\text{ CAD}}=680 \text{ USD}[/latex]

By the second rate, you would find:

[latex]1,000\text{ CAD}\times \frac{1 \text{USD}}{1.47059\text{ CAD}}=680\text{ USD}[/latex]

So the rates are equivalent.

The following exchange rates were taken from a local newspaper.

Each rate gives the number of Canadian dollars required to purchase one unit of the other currency.

Example 1.5.2

To purchase Euros you would use the rate 1.532 CAD per Euro. So, if you wanted 300€ at this rate, you would need:

[latex]300€ = 300€ \times \frac{1.532\text{ CAD}}{1€}=459.60 \text{ CAD}[/latex]

To get the number of marks for, say $100.00 Canadian, you would calculate:

[latex]100\text{ CAD} = 100\text{ CAD } \times \frac{1€}{1.532 \text{ CAD}} = 65.27 €[/latex]

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