Foundational Math Skills

# 2 Ratios and Proportions

# Lesson

Learning Outcomes

By the end of this chapter, learners will be able to:

- define the terms ratio and proportion,
- write ratios and proportions in numerical format using a colon and using fractions, and
- describe how to solve for an unknown amount in a proportion equation.

## Ratios

You have likely used ratios in many ways in everyday life, but perhaps you are not sure how to define a ratio. A ratio is a numerical expression which shows the connection between two or more numbers. These numbers must be related in some way; we do not use ratios to compare unrelated numbers. Most often, you will see a ratio using just two whole numbers.

When writing a ratio, the numbers are written in a particular order. For instance, if I was writing a ratio of how many nursing students were in one clinical practice group, I would write [latex]8:1[/latex].

In healthcare, we use ratios in a variety of situations. For instance, when describing safe staffing levels, reviewing morbidity and mortality data, or while carrying out medication administration.

### Examples of Ratios in Healthcare

When identifying staffing levels, a ratio is used to clearly indicate the number of patients a nurse can safely provide nursing care for on a particular unit:

- In an intensive care unit, you may see a ratio of one nurse for one intubated, acute patient. This is written in numerical form as [latex]1:1[/latex].
- In a surgical unit, you may see a ratio of one nurse for four patients. This is written as [latex]1:4[/latex].
- On a medical unit where care is provided by teams of two nurses (a licensed practical nurse and a registered nurse) for every ten patients, you may see a ratio written as [latex]2:10[/latex], or [latex]1:1:10[/latex] if they are separating the types of nurses within this example.

Mortality rates may also be expressed as ratios:

- An infant mortality rate of four deaths in every one thousand births is written as [latex]4:1000[/latex].

The quantity of medication in a unit measure can also be expressed as a ratio:

- A tablet of acetaminophen containing 325 mg of drug is written as [latex]1:325[/latex].

Sample Exercise 2.1

- How would you write the ratio of thirty two students in one classroom?

**Answer:**

[latex]32:1[/latex]

A ratio can also be expressed as a fraction, with the two numbers acting as the numerator and denominator. The numerator is always from the left of the colon, the denominator is from the right. For example, an infant mortality rate of [latex]4:1000[/latex] can be expressed as:

[latex]\dfrac{4}{1000}[/latex]

Sample Exercise 2.2

- How would you write the ratio of the amount of one tablet contains 325 mg of acetaminophen, or [latex]1:325[/latex] as a fraction?

**Answer:**

[latex]\dfrac{1}{325}[/latex]

## Proportions

A proportion is an equation of two ratios of equivalent amounts. The terms of the first ratio are related to the terms of the second ratio. This means you should ensure the numbers of each ratio are written in the same order.

Example of a Proportion

[latex]1:325 = 2:650[/latex]

In this case, the proportion is comparing the amount of 1 tablet containing 325 mg of acetaminophen with 2 tablets containing 650 mg of acetaminophen. Both ratios have equivalent amounts of acetaminophen per tablet.

As with ratios, you can write proportions in fraction form:

[latex]\dfrac{1}{325}[/latex] = [latex]\dfrac{2}{650}[/latex]

Both ways mean the same thing: 1 tablet is to 325 mg, just as 2 tablets are to 650 mg.

Sample Exercise 2.3

How would you write a proportion equation in fraction format which compares how 1 cup of apple juice (of a particular brand) contains 24 g of sugar to 4 cups of the same brand apple juice containing 96 g of sugar?

**Answer:**

[latex]\dfrac{1}{24}[/latex] = [latex]\dfrac{4}{96}[/latex]

You can confirm if two ratios make a proportion equation by reducing the fractions to see if they reduce to the same amount. It is not likely you will come across a problem like this very often in the practice setting, but you may use it to confirm there is the same amount of drug in different brands of tablets when engaging in medication reconciliation with patients.

Verifying Proportions

If 4 tablets contain a total of 200 mg of prednisone, do 8 tablets containing a total of 400 mg reduce to the same amount?

[latex]\dfrac{4}{200}[/latex] = [latex]\dfrac{8}{400}[/latex]

Recall to reduce a fraction, you need to divide each numerator and denominator by the same amount. In this example, we can divide by 4.

[latex]\dfrac{1}{50}[/latex] = [latex]\dfrac{2}{100}[/latex]

A fraction is not reduced if it can still be simplified further. In this case, the fraction to the right can be reduced further by dividing the numerator and denominator by 2.

[latex]\dfrac{1}{50}[/latex] = [latex]\dfrac{1}{50}[/latex]

Now you can see these ratios do indeed have the same proportions because the fractions are equal once they have been reduced.

### Unknown Amounts Within a Proportion Equation

Sometimes we are presented with an unknown amount in one of the ratios in a proportion equation. This is where the short cut of “cross multiply and divide” is often used to solve for the unknown amount. We can do this when we know that the ratios presented are related to each other and have equivalent proportions. For instance, we can solve for an unknown amount when comparing two ratios with units of minutes and hours.

Solving for an Unknown Amount in a Proportion Equation

Here is an example of two ratios depicting the relationship between minutes and hours, one with an unknown amount of minutes. In this case the ratio reads as 120 minutes to 2 hours and [latex]\text{x}[/latex] minutes in 8 hours.

[latex]120:2[/latex] and [latex]\text{x}:8[/latex]

**Step 1: Set up your equation**

To solve for x, the unknown amount, you first need to set up the formula. It is helpful to write it as a fraction to see how the process works.

[latex]\dfrac{\text{x}}{8}[/latex] = [latex]\dfrac{120}{2}[/latex]

**Step 2: Cross multiply opposing numerators and denominators**

[latex]\text{(2)(x) = (8)(120)}[/latex]

*Numbers next to each other in brackets indicates they are being multiplied, so the letter [latex]\text{x}[/latex] is not confused with x as a multiplication symbol.

You can see [latex]\text{x}[/latex] was the numerator of the first ratio and 2 the denominator from the other ratio. Likewise, eight was the denominator and 120 the numerator.

[latex]\text{2x = 960}[/latex]

**Step 3: Simplify the equation (get [latex]\text{x}[/latex] by itself!)**

To simplify the equation, divide both sides by the same amount. In this case, dividing by 2[latex]\text{x}[/latex] by 2 simplifies to 1[latex]\text{x}[/latex], or just [latex]\text{x}[/latex]. Now we know what [latex]\text{x}[/latex] equals.

[latex]\dfrac{\text{(x)(2)}}{2}[/latex] = [latex]\dfrac{960}{2}[/latex]

[latex]\text{x = 480}[/latex]

Key Takeaways

- A ratio is used to express the relationship between numbers.
- A ratio can be written as numbers separated by a colon or as a fraction.
- A proportion is an equation of two ratios of equal values.
- A proportion can be written with ratios written as numbers separated by a colon or as a fraction.
- Use the “cross multiply and divide” process to solve for an unknown amount in a proportion equation.

# Practice Set 2.1: Ratios

Practice Set 2.1: Ratios

Complete the following questions and click on the word **Answers** to check your work.

- Write a ratio using a colon expressing the relationship between three computers to a nursing station on a hospital ward.
- Write a ratio using a colon expressing the relationship between one hundred ibuprofen tablets per bottle.
- Write a ratio using a colon expressing how one tablet contains fifty mg of dimenhydrinate.
- Write a ratio using a colon expressing the staffing ratio of one pediatric nurse to four stable patients.
- Write a ratio as a fraction expressing four out of five people of a particular country have blue eyes.
- Write a ratio as a fraction expressing in two nebules there are 500 mcg of ipratropium.
- Write a ratio as a fraction expressing there is one dean representing each nursing school in Canada.
- If one cup of water is the same as 250 mL, how would you write the ratio?
- If there are 500 mg of ciprofloxacin in one tablet, how would you write the ratio?
- If two out of every thirty nursing students receive an entrance scholarship over $1,000, how would you write this ratio?

**Answers:**

- [latex]3:1[/latex]
- [latex]100:1[/latex]
- [latex]1:50[/latex]
- [latex]1:4[/latex]
- [latex]\tfrac{4}{5}[/latex]
- [latex]\tfrac{2}{500}[/latex]
- [latex]\tfrac{1}{1}[/latex]
- [latex]1:250[/latex] or [latex]\tfrac{1}{250}[/latex]
- [latex]1:500[/latex] or [latex]\tfrac{1}{500}[/latex]
- [latex]2:30[/latex] or [latex]\tfrac{2}{30}[/latex], or [latex]1:15[/latex] or [latex]\tfrac{1}{15}[/latex]

# Practice Set 2.2: Proportions

Practice Set 2.2: Proportions

Complete the following questions and click on the word **Answers** to check your work.

- Write a proportion equation comparing the ratios of 500mcg ipratropium per two nebules to 250 mcg ipratropium per one nebule in fraction form.
- Write a proportion equation comparing the ratios of one crash cart per nursing ward to five crash carts in five wards.
- Write a proportion equation comparing the ratio of twenty five mg of sertraline per capsule to seventy five mg of sertraline per three capsules in fraction form.
- Write a proportion equation comparing the ratios of sixty minutes in one hour to 320 minutes in four hours.
- Write a proportion equation expressing the rate of five infant births per month is equivalent to sixty infant births per year at a small hospital. Hint: Ensure the units are the same.

**Answers:**

- [latex]\tfrac{500}{2}=\tfrac{250}{1}[/latex]
- [latex]1:1 = 5:5[/latex]
- [latex]\tfrac{25}{1}=\tfrac{75}{3}[/latex] or [latex]25:1 = 75:3[/latex]
- [latex]\tfrac{60}{1}=\tfrac{320}{4}[/latex] or [latex]60:1 = 320:4[/latex]
- [latex]5:1 = 60:12[/latex] (Convert units first: One year = 12 months)

# Practice Set 2.3: Solving for Unknown Amounts in Proportions

Practice Set 2.3: Solving for Unknown Amounts in Proportions

Use the cross multiply and divide method to solve for the unknown amount in each of the following proportions. Click on the word **Answers** to check your work.

- [latex]{3:4 = x:12}[/latex]
- [latex]{x:10 = 1:2}[/latex]
- [latex]\tfrac{x}{6}[/latex] = [latex]\tfrac{75}{3}[/latex]
- [latex]\tfrac{225}{1}[/latex] = [latex]\tfrac{x}{3}[/latex]
- [latex]\tfrac{800}{4}[/latex] = [latex]\tfrac{200}{x}[/latex]
- [latex]\tfrac{4}{13} = \tfrac{12}{x}[/latex]
- [latex]\tfrac{24}{x} = \tfrac{8}{3}[/latex]
- [latex]{4:8 = x:6}[/latex]
- [latex]\tfrac{x}{150} = \tfrac{5}{2}[/latex]
- [latex]{x:12 = 18:3}[/latex]

**Answers:**

- x = 9

[latex]\begin{align*} \dfrac{3}{4} &=\dfrac{x}{12} \\ \\ (3)(12) &=(4)(x) \\ \\ 36 &=4x \\ \\ 9 &=x\end{align*}[/latex] - x = 5

[latex]\begin{align*} \dfrac{x}{10} &=\dfrac{1}{2} \\ \\ (2)(x) &=(10)(1) \\ \\ 2x &=10 \\ \\ x &=5\end{align*}[/latex] - x = 150

[latex]\begin{align*} (3)(x) &=(6)(75) \\ \\ 3x &=450 \\ \\ x &=150\end{align*}[/latex] - x= 675

[latex]\begin{align*} (1)(x) &=(225)(3) \\ \\ x &=675\end{align*}[/latex] - x = 1

[latex]\begin{align*} (800)(x) &=(4)(200) \\ \\ 800x &= 800 \\ \\ x &=1\end{align*}[/latex] - x = 39

[latex]\begin{align*} 4x &=156 \\ \\ x &=39\end{align*}[/latex] - x = 9

[latex]\begin{align*} 72 &=8x \\ \\ 9 &=x\end{align*}[/latex] - x = 3

[latex]\begin{align*} \dfrac{4}{8} &=\dfrac{x}{6} \\ \\ (4)(6) &=(8)(x) \\ \\ 24 &=8x \\ \\ 3 &=x\end{align*}[/latex] - x = 375

[latex]\begin{align*} 2x &=750 \\ \\ x &=375\end{align*}[/latex] - x = 12

[latex]\begin{align*} \dfrac{x}{2} &=\dfrac{18}{3} \\ \\ (x)(2) &=(18)(3) \\ \\ 3x &=36 \\ \\ x &=12\end{align*}[/latex]

# Practice Set 2.4: Solving for Unknown Amounts in Proportions

Practice Set 2.4: Solving for Unknown Amounts in Proportions

Use the cross multiply and divide method to solve for the unknown amount in each of the following proportions. Click on the word **Answers** to check your work.

- [latex]6:x = 9:32[/latex]
- [latex]\tfrac{135}{x} = \tfrac{20}{5}[/latex]
- [latex]\tfrac{x}{13} = \tfrac{17}{2}[/latex]
- [latex]13:7 = x:3[/latex]
- [latex]x:2 = 14:7[/latex]
- [latex]\tfrac{200}{12} = \tfrac{x}{4}[/latex]
- [latex]\tfrac{28}{8} = \tfrac{3}{x}[/latex]
- [latex]\tfrac{x}{3} = \tfrac{12}{72}[/latex]
- [latex]6:4 = 22:x[/latex]
- [latex]\tfrac{2}{19} = \tfrac{x}{27}[/latex]

**Answers:**

- [latex]x = 21.3[/latex]

[latex]\begin{align*} \dfrac{6}{x} = \dfrac{9}{32} &= \dfrac{9}{32} \\ \\ 9x &=192 \\ \\ x &=21.3\end{align*}[/latex] - [latex]x = 33.75[/latex]

[latex]\begin{align*} 675 &=(20)(x) \\ \\ 33.75 &=x\end{align*}[/latex] - [latex]x = 110.5[/latex]

[latex]\begin{align*} (2)(x) &=221 \\ \\ x &=110.5\end{align*}[/latex] - [latex]x = 5.6[/latex]

[latex]\begin{align*} \dfrac{13}{7} &=\dfrac{x}{3} \\ \\ 39 &=(7)(x) \\ \\ 5.6 &=x\end{align*}[/latex] - [latex]x = 4[/latex]

[latex]\begin{align*} \dfrac{x}{2} &=\dfrac{14}{7} \\ \\ (7)(x) &=28 \\ \\ x &=4\end{align*}[/latex] - [latex]x = 66.7[/latex]

[latex]\begin{align*} 800 &=(12)(x) \\ \\ 66.7 &=x\end{align*}[/latex] - [latex]x = 0.86[/latex]

[latex]\begin{align*} (28)(x) &=24 \\ \\ x &=0.86\end{align*}[/latex] - [latex]x = 0.5[/latex]

[latex]\begin{align*} (72)(x) &=36 \\ \\ x &=0.5\end{align*}[/latex] - [latex]x = 14.7[/latex]

[latex]\begin{align*} \dfrac{6}{4} &=\dfrac{22}{x} \\ \\ (6)(x) &=88 \\ \\ x &=14.7\end{align*}[/latex] - [latex]x = 2.8[/latex]

[latex]\begin{align*} 54 &=(19)(x) \\ \\ 2.8 &=x\end{align*}[/latex]

# Practice Set 2.5: Solving for Unknown Amounts in Proportions

Practice Set 2.5: Solving for Unknown Amounts in Proportions

Use the cross multiply and divide method to solve for the unknown amount in each of the following proportions. Unless the answer is a whole number, round to the second decimal place. Click on the word **Answers** to check your work.

- [latex]\tfrac{4}{x} = \tfrac{125}{20}[/latex]
- [latex]\tfrac{x}{17} = \tfrac{7}{2}[/latex]
- [latex]3:x = 8:12[/latex]
- [latex]1200:2 = x:20[/latex]
- [latex]\tfrac{490}{14} = \tfrac{72}{x}[/latex]
- [latex]397:17 = 3:x[/latex]
- [latex]\tfrac{x}{19} = \tfrac{7}{325}[/latex]
- [latex]\tfrac{13}{3} = \tfrac{x}{2}[/latex]
- [latex]12:x = 17:19[/latex]
- [latex]x:4 = 2:5[/latex]

**Answers:**

- [latex]x = 0.64[/latex]

[latex]\begin{align*} 80 &=(125)(x) \\ \\ 0.64 &=x\end{align*}[/latex] - [latex]x = 59.5[/latex]

[latex]\begin{align*} (2)(x) &=119 \\ \\ x &=59.5\end{align*}[/latex] - [latex]x = 4.5[/latex]

[latex]\begin{align*} \dfrac{3}{x} &=\dfrac{8}{12} \\ \\ 36 &=(8)(x) \\ \\ 4.5 &=x\end{align*}[/latex] - [latex]x = 12000[/latex]

[latex]\begin{align*} \dfrac{1200}{2} &=\dfrac{x}{20} \\ \\ 24000 &=(2)(x) \\ \\ 12000 &=x\end{align*}[/latex] - [latex]x = 2.06[/latex]

[latex]\begin{align*} (490)(x) &=1008 \\ \\ x &=2.06\end{align*}[/latex] - [latex]x = 0.13[/latex]

[latex]\begin{align*} \dfrac{397}{17} &=\dfrac{3}{x} \\ \\ (397)(x) &=51 \\ \\ x &=0.13\end{align*}[/latex] - [latex]x = 0.41[/latex]

[latex]\begin{align*} (325)(x) &=133 \\ \\ x &=0.41\end{align*}[/latex] - [latex]x = 8.67[/latex]

[latex]\begin{align*} 26 &=(3)(x) \\ \\ 8.67 &=x\end{align*}[/latex] - [latex]x = 13.41[/latex]

[latex]\begin{align*} \dfrac{12}{x} &=\dfrac{17}{19} \\ \\ 228 &=(17)(x) \\ \\ 13.41 &=x\end{align*}[/latex] - [latex]x = 1.60[/latex]

[latex]\begin{align*} \dfrac{x}{4} &=\dfrac{2}{5} \\ \\ (5)(x) &=8 \\ \\ x &=1.6\end{align*}[/latex]

A ratio is a numerical expression which shows the connection between two or more related numbers.

The number on top of a fraction.

The number on the bottom of a fraction.