• Unit 6: Ratios and Proportions

    In this unit, we study another application of fractions, ratios, and proportions. These are mathematical concepts we use all the time, probably without even realizing it. Have you ever compared unit prices for different packages of the same type of food at the grocery store? That is a ratio.

    When driving 65 mph (miles per hour) on the highway, have you ever determined how long it will take you to get to your destination? That is a proportion. In sports, statisticians use proportions to predict an athlete's performance based on what they have accomplished in the past. In this unit, we explore how to write ratios, set up and solve proportions, and apply these skills to real-world experiences.

    Photo of a cake with chocolate frosting and raspberries on top.

    Photo from Wikipedia

    Cooking is probably where we use ratios most.

    Imagine your aunt's coffee shop is hosting a party. She and her staff are being handsomely rewarded for providing the venue and all of the food. She knows her special chocolate raspberry cake feeds 10 people and uses the following ingredients: four cups of flour, two cups of sugar, three sticks of butter, six eggs, and lots of chocolate. However, 25 people will be attending the gathering, and she needs to make a larger cake! Exactly how much flour and sugar does she need to pull this off?

    Working through the next few sections will help you use rates and ratios to answer these questions so your aunt can complete this delicious task (and others, too!).

    Completing this unit should take you approximately 2 hours.

    • 6.1: Introduction to Ratios

      What is a ratio anyway? Mathematically speaking, a ratio is nothing more than a fraction. We use the word "ratio" to signal how we will use this fraction. When we say the fraction 10/3 is a ratio, we are emphasizing how the number 10 compares to the number three. Some read this ratio as "ten compared to three". We use lots of different notation and language to express ratios.

      These expressions all express the same ratio:

      10 \text{ to } 3 \hspace{0.5cm} 10:3 \hspace{0.5cm} \frac{10}{3}

      When we use ratios, we are usually less interested in the exact numerical value. For example, we probably would not write 10/3 = 3.\overline{3} \approx 3.33. We treat the fraction as a ratio and are more interested in how the whole number values compare.

      Consider the square shapes below for a visual example. We can use various ratios to describe different aspects of this picture. For example, we can observe there are three dark blue boxes compared to two light blue ones. We can say the dark blue and light blue boxes are in a ratio of 3:2. The fraction 3/2 also expresses this comparison.

      We can use the ratio \frac{2}{3} to compare the number of dark blue to light blue boxes, but instead of only comparing "parts to parts", we can also compare "parts to whole". In this image, we have five squares, so the ratios 3:5, 3/5, and \frac{3}{5} all express how many dark blue boxes exist compared to the total number of boxes.

      This idea of ratios is also present in computing basic probabilities. For example, if someone jumbled all of our five squares in a bag and asked you to choose one at random (that is, you could not see which one you were choosing), you would use the ratio 2/5 to express your chance (or probability) of choosing a light blue square.

    • 6.2: Finding Unit Rates and Prices

      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 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:

      \dfrac{250 \text{cm}}{10 \text{m}} = \dfrac{250 \text{cm}}{1000 \text{cm}} = \dfrac{250}{1000} = \dfrac{25}{100} = \dfrac{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} = \dfrac{25 \text{ miles}}{1 \text{ hour}}

      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 \dfrac{2 \text{ apples}}{ 3 \text{ bananas}} is a RATE.

      2. \displaystyle \dfrac{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 equal one. See if you can figure out why.

      \begin{align*} \text{first package price } &= \dfrac{16}{2} = \dfrac{8}{1} \\ \\ \text{second package price } &= \dfrac{32}{4} = \dfrac{8}{1}\end{align*}.

      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.

    • 6.3: Understanding Proportions

      You have probably heard and used the words proportion, proportional, and disproportionate. How about this? When that hungry math professor was told she could not purchase 49 cannolis, screaming and throwing items seemed like a disproportionate response. Mathematically speaking, a proportion is simply an equation that relates two ratios.

      For example, the equation \dfrac{30}{9} = \dfrac{10}{3} is an example of a proportion. Ancient Greek mathematicians seriously loved proportions, especially when they applied them to similar shapes!

      Let's look at these two rectangles. The one on the right was obtained by copy-pasting the first one on the left and then resizing by a factor of 1/2.

      When comparing the corresponding side lengths for these two rectangles, we find proportions. In particular, we find:

      \dfrac{\text{Big Rectangle's Height}}{\text{Small Rectangle's Height}} = \dfrac{2}{1}= \dfrac{\text{Big Rectangle's Width}}{\text{Small Rectangle's Width}}.

      It is interesting that ancient mathematicians always viewed fractions, and the equations that involve them, as statements about ratios and proportions. For example, the expression 1/3 did not refer to a number or a quantity, such as one-third of the way between zero and one. It was necessarily and exclusively a comparison of two measurements.

      Manipulating proportions involves the multiplication and division skills we mastered in Unit 4, where we practiced multiplying and dividing fractions.

    • 6.4: Applications of Proportions

      We use proportions in so many applications of our everyday lives, often without even realizing it. For example, you may know how many calories are in one serving of ice cream according to the packaging. But how many calories will you consume if you eat two scoops instead of one? We use proportions to calculate this answer.

      Another example is our delicious chocolate-raspberry cake recipe from the start of this unit. From our description, we know our recipe calls for four cups of flour to bake a cake that feeds 10 people. But we need to make some adjustments to make a bigger cake. How many cups of flour do we need to bake a cake that feeds 25 people?

      In the language of proportions and ratios, we know that four cups of flour compare to 10 people. But how many cups of flour compare to 25 people? Let's label our unknown amount of flour x, and we can set up our equation to solve our question about proportion:

      \frac{4}{10} = \frac{x}{25}

      Do you see how solving this proportion leads to x=10? (If not, remember the strategy of cross multiplying that converts this equation into the more familiar one 25\times 4 = 10\times x, which is the same as the equation 100 = 10\times x). Let's go back to our original problem to solve a similar question. Your aunt used two cups of sugar to bake a cake that feeds 10 people. What proportion can we set up to determine how many cups of sugar we need to adjust the cake recipe to feed 25 people?