A useful application of ratios is finding unit rates and prices. However, let's clarify the distinction between rate and ratio before we go any further. The difference plays an important role in real-world applications.

  • A ratio compares two quantities that are expressed in the same units of measurement.
  • A rate is a ratio that compares two quantities that cannot be expressed in the same units of measurement.

For example, the five squares we discussed in Section 6.1 above allow us to form ratios and not rates: the fractions we set up compare numbers of squares in the numerators and denominators. On the other hand, if we want to use a ratio to compare the two lengths, 250 cm and 10 m, we need to rewrite the measurements in the same units. After we convert 10 m into 1000 cm, we can then write the ratio:

\frac{250 cm}{10 m} = \frac{250 cm}{1000 cm} = \frac{250}{1000} = \frac{25}{100} = \frac{1}{4}.

Again, this fraction is a ratio and not a rate. On the other hand, when your car odometer says you are driving at a speed of 25 mph, you see a rate. The acronym "mph" stands for "miles per hour" and indicates the quantities you are measuring or comparing (miles and hours) are not in the same units. That is:

25 \text {mph} = \frac{25 \text{ miles}}{1 \text{ hour}}

This is a rate. See if you can distinguish the following fractions as ratios or rates (highlight just after the phrase "is a" to see answers).

  1. \displaystyle \frac{2 \text{ apples}}{ 3 \text{ bananas}} is a RATE.
  2. \displaystyle \frac{2 \text{ cups}}{1 \text{ cup}} is a RATIO.
  3. 10 \text{ days}: 24 \text{ hours} is a RATE (although the units need to be converted).

Observe that you do not need to make this distinction between rates and ratios for purely mathematical questions. You can call the fraction 2/3 a rational number, a rate, a fraction, a ratio, or just a ""real number"" without anything going wrong. This distinction is only useful when you attach real-world units to them.

Now, let's return to prices and unit rates. Imagine going to the grocery store to buy pasta. There are so many different brands, and each one has a slightly different price for a slightly different size of package. How do you decide which box of pasta is the best deal?

We can determine the best deal by using unit prices. Remember: a unit rate is a fraction or rate whose denominator equals one. Let's compare two packages of pasta – one 16 oz package costs $2.00, and another 2 lb (pound) package costs $4.00. We should form a ratio to compare the amount per given price and then reduce these fractions so their denominators are equal. See if you can figure out why:

\text{first package price } = \frac{16}{2} = \frac{8}{1} \\ \text{second package price } = \frac{32}{4} = \frac{8}{1}

Notice that for the second package, first, we need to convert pounds to ounces: 1 lb (pound) = 16 ounces. These unit rates make it easy to compare the amount of pasta per dollar. Both packages charge one dollar for every 8 ounces of pasta.

By using ratios and rates, we can calculate the price per ounce of pasta in each box to determine the best deal.
Back to '6.2 Finding Unit Rates and Prices'