A ratio expresses a fixed relationship between two quantities. This section illustrates using ratios to calculate amounts subject to a ratio and presents how to recognize what a ratio does and cannot tell us. Note, percents are simply ratios scaled to parts per 100, that is, what we know about percents is applicable here.
Simple syrup consists of one cup of sugar and one cup of water which is heated until the sugar is disolved. There are multiple ratios that express this combination. \(\frac{1 \text{ cup sugar}}{1 \text{ cup water}}\text{,}\)\(\frac{7.05 \text{ oz}}{8 \text{ oz}}\) ratio of sugar to water by weight, \(\frac{7.05 \text{ oz}}{15.05 \text{ oz}}\) ratio of sugar to simple syrup.
Each of these ratios indicates that there is a fixed amount of sugar relative to the water or resulting syrup. That is, is if we made simple syrup with 2 cups of sugar we would need 2 cups of water because
\begin{equation*}
\frac{2 \text{ cups sugar}}{2 \text{ cups water}} = \frac{1 \text{ cup sugar}}{1 \text{ cup water}}
\end{equation*}
Note that \(\frac{7}{12} \approx 0.58 \lt 1\text{.}\) Because it is less than one the ratio tells us that there are fewer dogs than cats in this neighborhood.
We could also write \(\frac{12 \text{ cats}}{7 \text{ dogs}}\) to express the exact same relationship. Note that \(\frac{12}{7} \approx 1.7 \gt 1\text{.}\) Because it is greater than one the ratio tells us that there are more cats than dogs in the neighborhood (same result as above).
Notice that these (and most rates) are expressed with the denominator being one. This makes it easier to calculate as will be seen below. Frequently we will skip writing the one in the denominator, e.g., \(\frac{35 \text{ miles}}{\text{hour}}\text{.}\)
Just like percents (which are ratios written in decimal form) if we know a ratio and one of the amounts we can calculate the other amount. The method is the same as with percents, namely multiplying the correct number or solving an equation.
We notice that the speed units (nautical miles per hour) match with the hours. If the speed and time are multiplied we will end up with a distance. Thus the units suggest we can multiply.
Because this ratio remains the same during cruise we can break the problem down into pieces. Based on the ratio (speed) in the first hour the plane will fly 110 nm. In the second hour it will fly another 110 nm. In the last half hour it will fly half of the distance which is \(110/2=55\) nm. Thus in 2.5 hours it will fly \(110+110+55=275\) nm.
Some manuals provide tables to make this method convenient. See Table 2.3.6 for a table of this sort. How far can the DA-20 cruise in 2.7 hours? We need to write 2.7 as the sum of numbers in the table. One way is \(2.7 = 2+0.5+0.2\text{.}\) From the table we know it will fly 220 nm in 2 hours, 55 nm in an additional 0.5 hours, and 22 nm in the final 0.2 hours. Thus it will fly \(220+55+22=297\) nm.
We can set this up like the calculation in Example 2.3.4. In that example multiplied the ratio (nm/hr) by the number of hours which gave us nm. Here we multiply the ratio (gal/min) by the number of minutes which will give us gallons.
Based on data from the FDA the average amount of mercury found in fresh or frozen salmon is 0.022 ppm (parts per million). This means there are 0.022 mg of mercury in one liter of salmon. If a meal portion of salmon is 0.0020 liters how much mercury is consumed?
A saline solution intended for nasal rinsing has a ratio of 2.5 g of salt (sodium chloride) per 240 mL of pure water. How much salt is needed to make a half liter of this saline solution? Your scale is precise to a gram. Round appropriately.
We can apply the given ratio (2.5 g/240 mL) to the given amount (0.5 L). First it will be convenient to convert a half liter to milliliters. This is also a ratio problem. See Table 1.1.8 for the conversion ratio.
Like percents ratios tell us a relationship between two quantities but do not tell us how much. For example if a cookie recipe calls for 2 cups of milk for every 3 eggs, we do not know how many eggs are needed for a dozen cookies. We would also need to know either how many cups of milk per dozen or how many eggs per dozen.
Ratios may be based on rounded numbers. For example the unit convesion \(\frac{8 \text{ kilometers}}{5 \text{ miles}}\) is convenient for quick calculations. However, \(8/5 = 1.6 \approx 1.609344\) which is a more accurate conversion rate. If we are trying to convert “Tempo 30” (a speed limit of 30 kph) to mph this ratio is fine. If we are sending a satellite to another planet, we will need a more accurate conversion.
Rounding may occur to make the ratio easier to comprehend. For example according to an article by the U.S. Census Bureau 1 in 6 people in the U.S. was aged 65 and over. Because this ratio uses small numbers it is easy to understand. It is much easier to read and use than a more precise estimate of \(\frac{57822315}{333287562}\text{.}\)
