Ratios and proportions

You’ve probably used ratios and proportions before while cooking. If you’ve made pancakes from a mix you might have had to following a recipe like: 2 Cups Mix per 1.5 Cups Water results in 8 Pancakes. With this recipe if you used 4 cups of mix, you’d need 3 cups of water, and would get a yield of 16 pancakes. You could make 4 pancakes by using 1 cup of mix and 3/4 of a cup of water.

The relationship between the ingredients is a ratio. A ratio is a comparison of two or more numbers. A proportion is an equation of ratios. We solved basic proportions to calculate the correct mixtures of ingredients needed to make 4 or 16 pancakes. The proportions we solved were based on the ingredient to results ratio of 2:1.5:8 (you should read this as: 2 to 1.5 to 8).

Let’s look at another ratio. Say a particular school wants to keep a student to teacher ratio of 55:2. This means the school wants to have 2 teachers for every 55 kids. Now let’s assume 825 kids are enrolled in the school. How many teachers should the school have? To solve this we would use a proportion. Let the x represent the number of teachers needed for 825 kids to satisfy the 55:2 ratio. Then the following equation (which is a proportion) must be true.

$\frac{55}{2}=\frac{825}{x}$

We can solve the equation for x by ‘cross multiplying’ and dividing. ‘Cross multiplying’ involves multiplying the numerators of each side by the denominators of the opposing side, and then eliminating the denominators from both sides. For this equation that means multiplying 55 by x and 825 by 2, then wiping out the denominators. The next step to solving x is to isolate it on one side of the equation. We do this by diving both sides by 55, which leaves x alone on one side and gives the answer on the other side.

$\begin{align*} & \frac{55}{2}=\frac{825}{x} \\[6pt] & \rightarrow55\cdot x=825\cdot2 \\[6pt] & \rightarrow55x=1650 \\[6pt] & \rightarrow x=\frac{1650}{55}=30 \end{align*}$

Solving the proportion tells us that the school would need 30 teachers to handle the 825 students.

Here’s another example to help clarify the process: Suppose a call center likes to staff 3 employees for every 200 calls they receive in a day. The call center has been handling around 1000 calls each day with 15 employees, but the center is taking on more business and expects to soon be receiving about 1600 calls per day. How many more employees will the call center need to handle this increase in daily volume?

One way to solve this is to observe that the call center expects to be handling 600 more calls per day. We can then set up and solve the following proportion:

$\begin{align*} & \frac{x}{600}=\frac{3}{200} \\[6pt] & \rightarrow 200x=1800 \\[6pt] & \rightarrow x=9 \end{align*}$

Hence, the center would need to hire 9 more employees to handle the increased call volume.

As a final example, recall the formula: $ \text{rate}\cdot\text{time}=\text{distance} $. While driving on the freeway at a constant speed for 4 hours and you travelled 250 miles. At the same speed, how long will it take you to travel 575 miles?

Again, we solve using a proportion:

$\begin{align*} & \frac{250}{4}=\frac{575}{x} \\[6pt] & \rightarrow 250\cdot x=575\cdot4 \\[6pt] & \rightarrow x=\frac{575\cdot4}{250}=9.2 \end{align*}$

And solving for x we learn that at this speed it would take 9.2 hours to travel 575 miles. (You could have also solved that the speed is 62.5 mph, but it is not necessary to do so).

Practice questions

Solve for x, or for the given quantity

$ \frac{5}{12}=\frac{x}{108} $
$ \frac{270}{5}=\frac{450}{x} $
$ \frac{x}{80}=\frac{3}{4} $
$ \frac{10}{x}=\frac{8}{25} $
At a certain speed you travel 320 miles in 4.5 hours. At the same speed, how far would you travel in 7 hours?
If a restaurant likes to keep a ratio of 3 waiters per 42 seated guests, how many waiters should the restaurant have on staff for a night where crowds of 112 seated guests are expected?
A caterer usually bakes 4 cakes per 90 guests. How many cakes should the caterer bake for a party with 225 guests?
A recipe calls for 2 cups of sugar for every 5 cups of flour. If 12 cups of flour are used, how many cups of sugar are needed?

Solutions

Solution: $$ \begin{align*} & \frac{5}{12}=\frac{x}{108} \\[5pt] & \rightarrow 5\cdot108=x\cdot12 \\[5pt] & \rightarrow x=\frac{5\cdot108}{12}=45 \end{align*} $$
Solution: $$ \begin{align*} & \frac{270}{5}=\frac{450}{x} \\[5pt] & \rightarrow 270\cdot x=450\cdot5 \\[5pt] & \rightarrow x=\frac{450\cdot5}{270} \\[5pt] & =8\frac{1}{3}\approx8.33 \end{align*} $$
Solution: $$ \begin{align*} & \frac{x}{80}=\frac{3}{4} \\[5pt] & \rightarrow4x=3\cdot80 \\[5pt] & \rightarrow x=240/4=60 \end{align*} $$
Solution: $$ \begin{align*} & \frac{10}{x}=\frac{8}{25} \\[5pt] & \rightarrow250=8x \\[5pt] & \rightarrow x=250/8=31.25 \end{align*} $$
Solution: $$ \begin{align*} & \frac{320}{4.5}=\frac{x}{7} \\[5pt] & \rightarrow 320\cdot7=x\cdot4.5 \\[5pt] & \rightarrow x=\frac{320\cdot7}{4.5}\approx 498 \text{ miles} \end{align*} $$
Solution: $$ \begin{align*} & \frac{3}{42}=\frac{x}{112} \\[5pt] & \rightarrow 3\cdot112=x\cdot42 \\[5pt] & \rightarrow x=\frac{3\cdot112}{42}=8\text{ waiters} \end{align*} $$
Solution: $$ \begin{align*} & \frac{4}{90}=\frac{x}{225} \\[5pt] & \rightarrow 4\cdot225=x\cdot90 \\[5pt] & \rightarrow x=\frac{4\cdot225}{90}=10\text{ cakes} \end{align*} $$
Solution: $$ \begin{align*} & \frac{2}{5}=\frac{x}{12} \\[5pt] & \rightarrow 2\cdot12=x\cdot5 \\[5pt] & \rightarrow x=\frac{2\cdot12}{5}=4.8 \text{ cups of sugar} \end{align*} $$

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Ratios and proportions

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