Opening the book…
Ratios
Ratios are used to compare amounts or quantities or to describe a relationship between two amounts or quantities. For example, a ratio might be used to describe the cost of a month’s rent as compared to the income earned in one month. You may also use a ratio to compare the number of elephants to the total number of animals in a zoo, or the amount of calories per serving in two different brands of ice cream.
Ratios compare quantities using division. This means that you can set up a ratio between two quantities as a division expression between those same two quantities.
A ratio is a comparison of two quantities that shows how much of one quantity there is compared to another. It tells the relationship between the quantities using division.
A ratio that compares two different quantities with the same units of measurement.
Here is an example. If you have a platter containing \(10\) sugar cookies and \(20\) chocolate chip cookies, you can compare the cookies using a ratio.
The ratio of sugar cookies to chocolate chip cookies is:
\[\frac{\text { sugar cookies }}{\text { chocolate chip cookies }}=\frac{10}{20}\nonumber\]
The ratio of chocolate chip cookies to sugar cookies is:
\[\frac{\text { chocolate chip cookies }}{\text { sugar cookies }}=\frac{20}{10}\nonumber\]
You can write the ratio using words, a fraction, and also using a colon as shown below.
\[\text { Ratio of } {\color{red}\text{sugar cookies to}}
\quad \color{blue}\text{chocolate chip cookies}\nonumber\]
\[ \begin{align*}
&{\color{red} 10} \text{ to } \color{blue}20\quad\text{or}\\
&\frac{\color{red}10}{\color{blue}20}\quad
\text{or}\\
&{\color{red}10}:\color{blue}20\\
&{\color{red}1}:\color{blue}2\quad\text{(Simplied form)}
\end{align*}\]
Some people think about this ratio as: “For every \(10\) sugar cookies I have, I have \(20\) chocolate chip cookies.”
A ratio can be written in three different ways:
- with the word “to”: \(3\) to \(4\)
- as a fraction: \(\ \frac{3}{4}\)
- with a colon: \(3\):\(4\)
A ratio is simplified if it is equivalent to a fraction that has been simplified.
Below are two additional examples that illustrate how to compare quantities using a ratio and how to express the ratio in its simplified form.
Example \(\PageIndex{1}\): Find Ratio
A basketball player takes \(50\) jump shots during a practice. She makes \(28\) of them. What is the ratio of shots made to shots taken? Simplify the ratio.
- Answer
-
\[\begin{align*}
\dfrac{\text{shots made}}{\text{shots taken}}&=\dfrac{28}{50}~~~~~~~~\text{Unsimplified form} \\
&=\dfrac{14}{25}~~~~~~~~\text{Divide by \(2\) to write simplified form.} \\ \\
\end{align*}\]The ratio of shots made to shots taken is \(\frac{14}{25}\), \(14\):\(25\), or \(14\) to \(25\).
Often, one quantity in the ratio is greater than the second quantity. You do not have to write the ratio so that the lesser quantity comes first; the important thing is to maintain a consistent relationship.
Note that ratios can compare a part to a part or a part to a whole.
Rates
A rate is a ratio in which the units of the two quantities being compared are different.
Rates are a special type of ratio used to describe a relationship between different units of measure, such as speed, wages, or prices. A car can be described as traveling \(60\) miles per hour; a landscaper might earn \(\ $35\) per lawn mowed; gas may be sold at \(\ $3\) per gallon.
A rate is a ratio that compares two different quantities with different units of measurement. A rate is a comparison that provides information such as dollars per hour, feet per second, miles per hour, and dollars per quart, for example. The word “per” usually indicates you are dealing with a rate. Rates can be written using words, using a colon, or as a fraction. It is essential to understand which quantities are being compared.
For example, an employer wants to rent \(6\) buses to transport a group of \(300\) people on a company outing. The rate to describe the relationship can be written using words, using a colon, or as a fraction, and you must include the units.
\[\begin{align*}
&\text{six buses per \(300\) people}\\
&6\text{~~buses}:300\text{~~people}\\
&\dfrac{6\text{~~buses}}{300\text{~~people}} \\
&\dfrac{1\text{~~buses}}{50\text{~~people}}~~~~~~~~~~~~~~\text{Divide by \(6\) to reduce fraction} \\ \\
\end{align*}\]
This fraction means that the rate of buses to people is \(6\) to \(300\) or, simplified, \(1\) bus for every \(50\) people.
Example \(\PageIndex{2}\): Find Rate
Write the rate as a simplified fraction: \(8\) phone lines for \(36\) employees.
- Answer
-
\[\begin{align*}
\dfrac{\text{phone lines}}{\text{employees}}&=\dfrac{8\text{~phone lines}}{36~\text{employees}}~~~~~~~~\text{ Indentify the relationship} \\
&=\dfrac{2\text{~phone lines}}{9~\text{employees}}~~~~~~~~\text{ Divide by \(4\) to reduce fraction} \\ \\
\end{align*}\]The rate of phone lines for employees can be expressed as \(\ \frac{2 \text { phone lines }}{9\text { employees }}\).
Finding Unit Rates
A unit rate is a rate expressed so that the denominator of the corresponding ratio is one. (Recall that the "denominator" is the bottom number of the ratio when it is expressed in fraction form.)
Think about the "miles per gallon" example above. If we hear, "this car gets \(30\) miles per gallon," we know that for every 30 miles driven, the car uses 1 gallon of gas. The number 1 is implicit in the way we express most rates. Even if we are given a rate that is not a unit rate, we can convert it to a unit rate using proportions, as in the example below.
Figure \(\PageIndex{1}\): The unit ratio miles per hour is often used to express speed
"Vienna Convention road sign C14-V2 (mph)" by United Nations is licensed under Public Domain
.svg){kind=link}
A unit rate compares a quantity to one unit of measure. You often see the speed at which an object is traveling in terms of its unit rate.
For example, if you wanted to describe the speed of a boy riding his bike—and you had the measurement of the distance he traveled in miles in \(2\) hours—you would most likely express the speed by describing the distance traveled in one hour. This is a unit rate; it gives the distance traveled per one hour. The denominator of a unit rate will always be one.
Consider the example of a car that travels \(300\) miles in \(5\) hours. To find the unit rate, you find the number of miles traveled in one hour.
\[\begin{align*}
\dfrac{\text{miles}}{\text{hours}}&=\dfrac{300\text{~miles}}{5~\text{hours}}~~~~~~~~\text{Indentify the relationship} \\
&=\dfrac{60\text{~miles}}{1~\text{hour}}~~~~~~~~~~\text{Divide by \(5\) to reduce fraction} \\ \\
\end{align*}\]
A common way to write this unit rate is \(60\) miles per hour.
The word “per.” Sometimes, a slanted line / is used to mean “per.” Per means division. For example, \(60\) miles per hour can be written as \(60\) miles/hour or \(\frac{60\text{~miles}}{1\text{~hr}}\).
Example \(\PageIndex{3}\): Find Unit Rate
- A crowded subway train has \(375\) passengers distributed evenly among \(5\) cars. What is the unit rate of passengers per subway car?
- Vancouver Island has a population of \(734,860\), and a land mass of \(32,134\) square kilometres. What is the rate of number of people per square kilometre? (This is called population density.) Round your answer to the nearest whole number.
- Amy bought \(10\) lb of oranges for $\(4.99\). What is the rate in cents per pound? Find the rate.
- Answer
-
1. We need to find the passengers per subway car. Subway cars must be in the denominator.
\[\begin{align*}
\dfrac{\text{~~passengers}}{\text{~~subway cars}} &=\dfrac{375\text{~~passengers}}{5\text{~~subway cars}}\\
&=\dfrac{75\text{~~passengers}}{1\text{~~subway cars}}~~~~~~~~~~~~~~\text{Divide by \(5\) to reduce fraction} \\ \\
\end{align*}\]The unit rate of the subway car is \(75\) riders per subway car.2. We need to find the people per square kilometer. So the area must be in the denominator. \[\begin{align*}
\dfrac{\text{~~population}}{\text{~~area}} &=\dfrac{734,860\text{~~population}}{32,134\text{~~area}}\\
&=\dfrac{23\text{~~people}}{1\text{~~area}}~~~~~~~~~~~~~~\text{Divide} \\
\end{align*}\] So population density is \(23\) people/\(\rm km^2\).3. We need to price per pound. Therefore, the weight (in pounds) must be in the denominator.
\[\begin{aligned}
\frac{4.99\; \text {dollars}}{10\; \text {lb}}&=\frac{0.499\;\text{dollar}}{1\;\text{lb}}\\&=$0.449 \text{ ~per pound}\\
& =44.9 \text {~cent per pound}
\end{aligned} \nonumber \]
A unit price is a unit rate that expresses the price of something. The unit price always describes the price of one unit, so that you can easily compare prices.
You may have noticed that grocery shelves are marked with the unit price (as well as the total price) of each product. This unit price makes it easy for shoppers to compare the prices of competing brands and different package sizes.
Consider the two containers of blueberries shown below. It might be difficult to decide which is the better buy just by looking at the prices; the container on the left is cheaper, but you also get fewer blueberries. A better indicator of value is the price per single ounce of blueberries for each container.
Figure \(\PageIndex{1}\): Unit Price
"Blueberry Unit Price Comparison" by The NROC Project is licensed under CC BY-NC-SA 4.0
Look at the unit prices—the container on the right is actually a better deal, since the price per ounce is lower than the unit price of the container on the left. You pay more money for the larger container of blueberries, but you also get more blueberries than you would with the smaller container. Put simply, the container on the right is a better value than the container on the left.
So, how do you find the unit price?
Imagine a shopper wanted to use unit prices to compare a \(3\)-pack of tissue for \(\$4.98\) to a single box of tissue priced at \(\$1.60\). Which is the better deal?
Find the unit price of the \(3\)-pack: \(\ \frac{\$ 4.98}{3 \text { boxes }}\)
Since the price given is for \(3\) boxes, divide both the numerator and the denominator by \(3\) to get the price of \(1\) box, the unit price. The unit price is \(\$1.66\) per box.
The unit price of the \(3\)-pack is \(\$1.66\) per box; compare this to the price of a single box at \(\$1.60\). Surprisingly, the \(3\)-pack has a higher unit price! Purchasing the single box is the better value.
Like rates, unit prices are often described with the word “per.” Sometimes, a slanted line / is used to mean “per.” The price of the tissue might be written \(\$1.60\)/box, which is read "\(\$1.60\) per box."
The following example shows how to use the unit price to compare two products and determine which has the lower price.
Example \(\PageIndex{4}\): Compare Two Brand
Sami is trying to decide between two brands of crackers. Which brand has the lower unit price?
Brand A: $\(1.12\) for \(8\) ounces
Brand B: $\(1.56\) for \(12\) ounces
- Answer
-
Brand A: Find unit Price (price per ounce)
\[\begin{align*}
\dfrac{\text{~~dollars}}{\text{~~ounces}} &=\dfrac{1.12\text{~~dollars}}{8\text{~~ounces}}\\
&=0.14~\text{dollar per ounce} \\
&=14~\text{cent per ounce} \\
\end{align*}\]Brand B: Find unit Price (price per ounce)
\[\begin{align*}
\dfrac{\text{~~dollars}}{\text{~~ounces}} &=\dfrac{1.56\text{~~dollars}}{12\text{~~ounces}}\\
&=0.13~\text{dollar per ounce} \\
&=13~\text{cent per ounce} \\
\end{align*}\]The unit price of Brand A crackers is \(14\) cents/ounce, and the unit price of Brand B is \(13\) cents/ounce. Brand B has a lower unit price and represents the better value.
Ratios and rates are used to compare quantities and express relationships between quantities measured in the same units of measure and in different units of measure. They both can be written as a fraction, using a colon, or using the words “to” or “per”. Since rates compare two quantities measured in different units of measurement, such as dollars per hour or sick days per year, they must include their units. A unit rate or unit price is a rate that describes the rate or price for one unit of measure.
Proportion
A true proportion is an equation that states that two ratios are equal or equivalent.
If you know one ratio in a proportion, you can use that information to find values in the other equivalent ratio. Using proportions can help you solve problems such as increasing a recipe to feed a larger crowd of people, creating a design with certain consistent features, or enlarging or reducing an image to scale.
For example, imagine you want to enlarge a \(5\)-inch by \(8\)-inch photograph to fit a wood frame that you purchased. If you want the shorter edge of the enlarged photo to measure \(10\) inches, how long does the photo have to be for the image to scale correctly? You can set up a proportion to determine the length of the enlarged photo.
A proportion is usually written as two equivalent fractions. For example:
\[\ \frac{12 \text { inches }}{1 \text { foot }}=\frac{36 \text { inches }}{3 \text { feet }}\nonumber\]
Notice that the equation has a ratio on each side of the equal sign. Each ratio compares the same units, inches and feet, and the ratios are equivalent because the units are consistent, and \(\ \frac{12}{1}\) is equivalent to \(\ \frac{36}{3}\).
Proportions might also compare two ratios with the same units. For example, Juanita has two different-sized containers of lemonade mix. She wants to compare them. She could set up a proportion to compare the number of ounces in each container to the number of servings of lemonade that can be made from each container.
\[\ \frac{40 \text { ounces }}{84 \text { ounces }}=\frac{10 \text { servings }}{21 \text { servings }}\nonumber\]
Since the units for each ratio are the same, you can express the proportion without the units:
\[\ \frac{40}{84}=\frac{10}{21}\nonumber\]
When using this type of proportion, it is important that the numerators represent the same situation. In the example above, they represent 40 ounces for 10 servings. The denominators should also represent the same situation, which is ounces for servings.
Juanita could also have set up the proportion to compare the ratios of the container sizes to the number of servings of each container.
\[\ \frac{40 \text { ounces }}{10 \text { servings }}=\frac{84 \text { ounces }}{21 \text { servings }}\nonumber\]
Here are some proportions in different forms:
| Proportion | Fraction Form | Read like this… |
|---|---|---|
| \(1:2 = 2:4\) | \(\dfrac{1}{2} = \dfrac{2}{4}\) | \(1\) is to \(2\) or \(2\) is to \(4\) |
| \(18:9 = 10:5\) | \(\dfrac{18}{9} = \dfrac{10}{5}\) | \(18\) is to \(9\) as \(10\) is to \(5\) |
Finding an Unknown Quantity in a Proportion
Problems often give incomplete information; that is, one of the terms is missing. To solve such problems, you first identify the comparison or ratio given. It may be:
- A quantity of one thing that is mixed with a larger quantity of something else
- A scale of measurement given on a map, such as \(1\) cm on the map, represents \(100\) km distance on land
- Cost for a certain number of items
- Time to travel a certain distance
The problem will then give one term of the second ratio in the proportion. For example, if you have been told that \(3\) heads of lettuce cost $\(1.49\), you may be asked to find the cost of \(7\) heads of lettuce.
The missing term is the second cost. The proportion will be:
\[\dfrac{\text{number of heads of lettuce}}{\text{cost}} = \dfrac{\text{number of heads of lettuce}}{\text{cost}}\nonumber\]
\[\dfrac{3}{$1.49} = \dfrac{7}{?}\nonumber\]
\[3: $1.49 = 7: ?\nonumber\]
The most important thing to remember is to keep the order of comparison the same in the first and second ratios in a proportion. If the first ratio compares time to distance, then the second ratio in the proportion must compare time to distance.
\(\dfrac{\text{time}}{\text{distance}} = \dfrac{ \text{time} }{\text{distance}}\)
Or it could be:
\(\dfrac{\text{distance}}{\text{time}} = \dfrac{\text{distance}}{\text{time}}\)
Once you have decided on the order of comparison, it is a simple matter to write the proportion using the numbers given in the problem. Use a letter to stand for the missing term.
How would you find a missing term?
If you know that the relationship between quantities is proportional, you can use proportions to find missing quantities. Below is an example.
- Step 1: Write the first ratio using the information given.
- Step 2: Write the proportion, using a letter in place of the missing term. Be sure the order of comparison is the same in both the first and second ratios in your proportion.
- Step 3: Write the proportion in the fraction form. (Try to simplify the ratio before you do all the calculations).
- Step 4: Cross-multiply and set the cross-products equal to each other.
- Step 5: Divide both sides of the equation by the number with the unknown term.
- Step 6: Check by putting your answer back into the original proportion and cross-multiplying.
-
Example \(\PageIndex{5}\): Solve Proportion
- Solve proportion: \(\dfrac{8}{10} = \dfrac{x}{80}\)
- Solve proportion: \(\frac{n}{15}=\frac{25}{20}\)
- Answer
-
1. Given proportion \[\dfrac{8}{10} = \dfrac{x}{80}\nonumber\]
Cross multiply:
\[\begin{align*}
8 \times 80 &=10 \times x\\
640&=10x\\
\dfrac{640}{10} &= \dfrac{10x}{10}\;\;\text{Divide both side by \(10\)} \\
\dfrac{\cancel{10}x}{\cancel{10}} &= \dfrac{640}{10}\\
x &=64
\end{align*}\]2. Cross multiply
\[\begin{align*}
\ 20 \times n&=15 \times 25 \\
20 n&=375\\
\dfrac{\cancel{20}n}{\cancel{20}} &= \dfrac{375}{20}~~~~~~~~~\text{Divide both side by \(20\)} \\
n &=18.75
\end{align*}\]
The numbers in a ratio often are common fractions, decimals or mixed numbers. Follow exactly the same steps that you have been using to solve whole number proportions. The calculations will use your skills with fractions.
Example \(\PageIndex{6}\): Applying Proportion to Solve Problem
A recipe says to use \(2\) teaspoons of baking powder for every \(4\) cups of flour. If a recipe uses \(10\) cups of flour, how much baking powder is needed?
- Answer
-
The missing term is the teaspoons of baking powder for \(10\) cups of flour. Call this term \(x\).
\[\begin{align*}
\dfrac{\text{baking powder}}{\text{flour}}\text{~~(recipe)}&=\dfrac{\text{baking powder}}{\text{flour}}\text{~~(You make)}\\
\dfrac{2}{4}&=\dfrac{x}{10} \\
\end{align*}\]Cross multiply:
\[\begin{align*}
4 \times x &= 2 \times 10\\
4x&=20\\
\dfrac{4x}{4} &= \dfrac{20}{4}\;\;\text{Divide both side by \(4\)} \\
\dfrac{\cancel{4}x}{\cancel{4}} &= \dfrac{20}{4}\\
x &=5
\end{align*}\]
So \(5\) teaspoons of baking powder are needed for \(10\) cups of flour.
Now, back to the original example. Imagine you want to enlarge a \(5\)-inch-by-\(8\)-inch photograph to a length of \(10\) inches while maintaining the same width-to-length proportion. You can set up a proportion to determine the width of the enlarged photo.
\[\ \frac{\text { width }}{\text { length }}~~~\text{(First photo)}=\ \frac{\text { width }}{\text { length }} ~~~\text{(Second photo)} \nonumber \]
\[\frac{5}{8}=\frac{10}{n}\nonumber\]
\[\ 5 \cdot n=8 \cdot 10\nonumber\]
\[\dfrac{\cancel{5}n}{\cancel{5}} = \dfrac{80}{5}
\nonumber\]
\[\ n=16\nonumber\]
The length of the enlarged photograph is \(16\) inches.
Figure \(\PageIndex{2}\) - Photograph Proportions
"Scaling a Photograph" by The NROC Project is licensed under CC BY-NC-SA 4.0
Solving Application Problems Using Proportions
Setting up and solving a proportion is a helpful strategy for solving a variety of proportional reasoning problems. In these problems, it is always essential to determine the unknown value and then identify a proportional relationship that can be used to solve for the unknown value. Below are some examples.
Example \(\PageIndex{7}\): Application of Proportion
Among a species of tropical birds, \(30\) out of every \(50\) birds are female. In other words, \(30\) birds are female and \(20\) birds are male. If a certain bird sanctuary has a population of \(1,150\) of these birds, how many of them would you expect to be female?
- Answer
-
Let \(\ x=\) the number of female birds in the sanctuary.
Cross multiply:
\begin{align*}
\frac{30 \text { female birds }}{50 \text {~total birds }}&=\frac{x \text { female birds in sanctuary }}{1,150 \text { total birds in sanctuary }}\\
\frac{30}{50}&=\frac{x}{1,150}\\
50\times x &= 30 \times 1,150\\
50x&=34,500\\
\dfrac{50x}{50} &= \dfrac{34,500}{50}~~~~~~~~~\text{Divide both side by \(50\)} \\
\dfrac{\cancel{50}x}{\cancel{50}} &= \dfrac{34,500}{50}\\
x &=690
\end{align*}You would expect \(690\) birds in the sanctuary to be female.
Example \(\PageIndex{8}\): Use Proportion
A map scale indicates that \(\frac{1}{2}\) inch on the map corresponds with \(3\) real miles. How many miles apart are two cities that are \(2\dfrac{1}{4}\) inches apart on the map?
- Answer
-
We can set up a proportion by setting equal two \(\dfrac{\text{map inches}}{\text{real miles}}\) rates, and introducing a variable, \(x\), to represent the unknown quantity – the mile distance between the cities.
\[\begin{align*}
\dfrac{\text{inches}}{\text{miles }}&=\dfrac{\text{inches}}{\text{miles}}\\
\dfrac{\frac{1}{2}}{3}&=\dfrac{\frac{9}{4}}{x}\\
\end{align*}\]\[\begin{align*}
\frac{1}{2} \times x&= 3 \times \frac{9}{4}\\
\frac{1}{2} x&= \frac{27}{4}\\
\dfrac{\cancel{\frac{1}{2}}x}{\cancel{\frac{1}{2}}} &= \dfrac{\frac{27}{4}}{\frac{1}{2}}\\
x &=13.5
\end{align*}\]So \( x = 13.5= 13\dfrac{1}{2} \text{ miles} \)
Example \(\PageIndex{9}\): Use Proportion
Your car can drive \(300\) miles on a tank of \(15\) gallons. How far can it drive on \(40\) gallons? How many gallons are needed to drive \(50\) miles?
- Answer
-
We could certainly answer this question using a proportion: \(\dfrac{300 \text{ miles}}{15 \text{ gallons}} = \dfrac{x \text{ miles}}{40 \text{ gallons}}\)
Cross multiply:
\[\begin{align*}
300 \times 40 &= 15 \times x\\
15x&=12,000\\
\dfrac{15x}{15} &= \dfrac{12,000}{15}~~~~~~~~\text{Divide both side by \(4\)} \\
\dfrac{\cancel{15}x}{\cancel{15}} &= \dfrac{12,000}{15}\\
x &=800
\end{align*}\]So \(800\) miles can be driven by \(40\) gallons.
Notice if instead we were asked “how many gallons are needed to drive \(50\) miles?”
Set proportion as we did before \[\dfrac{300 \text{ miles}}{15 \text{ gallons}} = \dfrac{50 \text{ miles}}{x \text{ gallons}}\nonumber\]
Cross multiply:
\[\begin{align*}
300 \times x &= 15 \times 50\\
300x&=750\\
\dfrac{300x}{300} &= \dfrac{750}{300}~~~~~~~~~\text{Divide both side by \(300\)} \\
\dfrac{\cancel{300}x}{\cancel{300}} &= \dfrac{750}{300}\\
x &=2.5
\end{align*}\]So \(2.5\) gallons are needed to drive \(50\) miles.
A proportion is an equation comparing two ratios. If the ratios are equivalent, the proportion is true. If not, the proportion is false. Finding a cross product is another method for determining whether a proportion is true or false. Cross-multiplying is also helpful for finding an unknown quantity in a proportional relationship. Setting up and solving proportions is a valuable skill that can be applied to a wide range of problems.
If you wanted to power the city of Seattle using wind power, how many windmills would you need to install? Questions like these can be answered using rates and proportions.
1.2.2: Math Tools - Ratios, Rates, and Proportions is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts.
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