- AuthorRachel Mcconnell
Rachel is a certified math teacher for grades 6-8. She graduated from the University of Kansas with a bachelors in Middle Math Education. She has taught middle school math for four years throughout Connecticut, Georgia and Italy. She has passed Praxis tests in math, curriculum, and special education.
View bio - InstructorThomas Higginbotham
Tom has taught math / science at secondary & post-secondary, and a K-12 school administrator. He has a B.S. in Biology and a PhD in Curriculum & Instruction.
View bio - Expert ContributorKathryn Boddie
Kathryn has taught high school or university mathematics for over 10 years. She has a Ph.D. in Applied Mathematics from the University of Wisconsin-Milwaukee, an M.S. in Mathematics from Florida State University, and a B.S. in Mathematics from the University of Wisconsin-Madison.
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In this lesson, learn how to solve ratio problems and find ratios. Understand how to approach ratio word problems and practice with real-world example equations.Updated: 11/21/2023
Table of Contents
- What Is a Ratio?
- Ratio Problem Solving
- How to Find Ratios in Word Problems
- How to Solve Ratios
- How to Do Ratios with Unknown Variables
- Ratio Example Problems
- Lesson Summary
- FAQs
- Activities
Ratio Word Problems in Real Life
Imagine you are going to host a large dinner party for 20 people, including yourself. You have a lot of favorite recipes you want to make, but none of them are written to serve 20 people. Can you use your skills in solving ratio word problems to figure out the quantities you would need for the ingredients listed below?
Problems
1) Your recipe for chocolate chip cookies uses 1 cup of flour for 12 cookies. How much flour will you need in order to make 20 cookies?
2) Your recipe for beef and barley soup serves 6 people and requires 4 cups of beef broth. How many cups of beef broth will you need to serve 20 people?
3) Your spaghetti sauce recipe for your world-famous spaghetti and meatballs requires 3.5 pounds of tomatoes. The sauce recipe makes enough sauce for 8 servings. How many pounds of tomatoes will you need to make enough sauce for 20 servings?
4) Your Caesar salad recipe uses 4 pounds of romaine lettuce for 15 servings. How much romaine lettuce do you need to serve all 20 people?
Solutions
1) 1 cup of flour is used for 12 cookies, and we can write that ratio as 1/12. We need to find how many cups of flour are needed for 20 cookies. We can set up a proportion and solve by cross multiplying.
1/12 = x/20
1 * 20 = 12x
20/12 = x
5/3 = x
You will need 5/3 cups of flour - or 1 2/3 cups of flour.
2) 4 cups of beef broth are needed for 6 people, which gives a ratio of 4/6. To find how many cups of beef broth are needed to make enough soup for 20 people, set up a proportion and solve by cross multiplying.
4/6 = x/20
4 * 20 = 6x
80 = 6x
80/6 = x
40/3 = x
You will need 40/3, or 13 1/3, cups of beef broth.
3) 3.5 pounds of tomatoes makes enough sauce for 8 servings. This is the ratio 3.5/8. To find how many pounds of tomatoes are needed for 20 servings, set up a proportion and solve with cross multiplication.
3.5/8 = x/20
3.5 * 20 = 8x
70 = 8x
70/8 = x
8.75 = x
You will need 8.75 pounds of tomatoes.
4) 4 pounds of romaine lettuce for 15 servings will result in the ratio 4/15. To find how many pounds are needed for 20 servings, set up a proportion and cross multiply.
4/15 = x/20
4 * 20 = 15x
80 = 15x
80/15 = x
16/3 = x
You will need 16/3 pounds, or 5 1/3 pounds of romaine lettuce.
How do we find a missing value in two equivalent ratios?
Express each ratio of two quantities by writing its numerical values as a fraction. Remember, that's one of the three ways to write a ratio. Find an equivalent ratio, and then set these fractions equal in a proportion and determine the missing value by using cross-multiplication.
How can a ratio be expressed?
A ratio can be represented in three ways. For example, the same comparison could be expressed as 4 boys to 7 girls, 4 boys : 7 girls, or as the fraction (4 boys) / (7 girls). Be sure to express the ratio in its simplest form by dividing both its numerator and its denominator by their greatest common factor.
How can the ratio of a mixture be expressed?
Describe the ratio of a mixture by comparing two of its ingredients. This could be 1 cup of butter to 2 cups of flour, for example, or 1 egg to 1 cup of sugar.
Table of Contents
- What Is a Ratio?
- Ratio Problem Solving
- How to Find Ratios in Word Problems
- How to Solve Ratios
- How to Do Ratios with Unknown Variables
- Ratio Example Problems
- Lesson Summary
Let's say a family is driving in a car at 60 miles per hour. What does that actually mean? The phrase "60 miles per hour" is a ratio that means the car would travel 60 miles in one hour. A mathematical ratio is a comparison between two numbers, often numbers of objects or measuring units. They surround us in our daily lives. For example, one method of cooking rice uses a ratio of one quantity of rice to every two equivalent quantities of water. Ratios can be created from almost anything, from as grand a comparison as the size of earth to that of the sun, to as small as the weight an ant can carry as compared to the ant's own weight. There are two different types of ratios. The first are ratios that compare part-to-part, like three bananas to seven apples. The other is part-to-whole, as in three bananas to ten fruits. In the latter example, imagine the ten comes from the addition problem {eq}3+7=10 {/eq}. Perhaps the other seven fruits were papayas, or a combination of several types. Most ratios might be explained one way, for example in part-to-part, but then rewritten in the other type, part-to-whole, by adding parts together from the first ratio.
There are three different ways to write a ratio. As an example, use the ratio 5 girls to 8 boys in a classroom of 5 + 8 = 13 students.. Here are the three ways we could express that ratio:
- 5 girls : 8 boys
- {eq}\frac{5 girls}{8 boys} {/eq}
- 5 girls to 8 boys
The units, in this case girls and boys, are important, but they don't need to be used throughout a problem once they're known. Make sure, however, to include units in the answer once the problem has been solved and simplified.
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Ratio problem solving means solving for missing values or even creating a new ratio from the information given. Keep in mind, the units are very important. They explain what the numbers mean and are necessary to compare the correct values. As we proceed, we'll use the ratio problem below to clarify understanding and solving for ratios.
- A recipe for lemonade calls for 6 cups of water to every 2 cups of mix. How much mix would be needed if 24 cups of water were used?
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Word problems in math are common but sometimes confusing. Let's examine how to find ratios in a ratio word problem.
Use the example from above:
- A recipe for lemonade calls for 6 cups of water to every 2 cups of mix. How much mix would be needed if 24 cups of water were used?
Read through the word problem and highlight, or rewrite, the important information. Note that sometimes word problems throw in extra numbers that can confuse the reader.
The important information found in this problem is 6 cups of water, 2 cups of lemonade mix and the answer to look for, how much mix for 24 cups of water. Now that the information is found, the next step will be to write the ratios for this word problem.
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From the important information found in the word problem, write the ratios in their three different forms. Remember, the ratios just compare two numbers.
Part-to-Part: These ratios compare a part of the lemonade to another part of the lemonade.
- Water to mix: {eq}\frac{6}{2} {/eq}
- 6 cups of water : 2 cups of mix
- 6 cups of water to 2 cups of mix
Part-to-Whole: These ratios were calculated by combining the ingredients in the lemonade, 6 cups of water plus 2 cups of mix, to get the total amount of 8 cups.
Water to materials: {eq}\frac{6}{8} {/eq}
- 6 cups of water : 8 cups of materials
- 6 cups of water to 8 cups of materials
Mix to materials: {eq}\frac{2}{8} {/eq}
- 2 cups of mix : 8 cups of materials
- 2 cups of mix to 8 cups of materials
Those are the ratios created from important information in the word problem. Most of those ratios, however, are not in simplified form. Using simplified form makes a problem easier to handle, and answers should always be in the simplest form anyway. Simplify a ratio the same way any fraction is simplified: Divide the numerator and denominator by the greatest common factor. Use the fraction example to simplify the ratios.
Water to mix: {eq}\frac{6}{2} {/eq}
In this fraction, the greatest common factor is 2, as it can be divided cleanly from both the numerator and the denominator. {eq}\frac{6}{2} = 3 {/eq} and {eq}\frac{2}{2} = 1 {/eq}.
The simplified ratio, then, is {eq}\frac{3}{1} {/eq}. In other words, it adds 3 cups of water to 1 cup of mix.
The other two ratios can also be simplified by dividing by 2, the greatest common factor for both. The results are the simplified ratios of 3 cups of water to 4 cups of lemonade and 1 cup of mix to 4 cups of lemonade.
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Now that all the ratios have been found and simplified, it's time to answer the question asked. We want to know how much mix would be needed if 24 cups of water were used. To solve this, set up a proportion. A proportion states that two ratios are equal to each other. They need to be equal, because if the ratios aren't equal then their outcome, in this case the lemonade, will not be, or in this example taste, the same. When setting up a proportion, compare units that are the same. In this case, compare water to water and mix to mix. We can do so by making water the numerators of the ratiios, and the mix their denominators. Set the ratios equal to each other by listing one ratio on the left of an equal sign and the other on the right. We don't know the amount of mix needed for 24 cups of water, so we'll set that amount as the variable x. Our equation will state that 3 cups of water to 1 cup of mix is an equal ratio to 24 cups of water to x cups of mix. Notice we're using the simplified form of the ratio 6 to 2, which is 3 to 1.
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To solve the equation, use a method called cross-multiplication. This is specifically used to solve proportions with a missing value. To cross-multiply, first multiply diagonally across the equal sign in an X pattern, then set the resulting two products equal to each other. In this case, 3 multiplied by x is 3x, and 24 multiplied by 1 is 24. Our cross-multiplied equation looks like this:
{eq}3x=24 {/eq}
Now, solve for x. Divide both sides by 3 to leave only x on the left side. The right side will also have to be multiplied by 3, of course, and 24 divided by 3 yields the solution of 8.
{eq}x=8 {/eq}
This means the amount of mix needed, x, would be 6 cups. (Remember, the unit must be included in the answer.)
On paper, the finished problem looks like the image below. Note that the blue-green X shape over the equal sign indicates only the two "directions" of multiplication, not a variable x or the elimination of the equal sign.
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Let's try some more word problems. Check your answers with the solutions below.
Ratio Word Problem 1
Lisa needs to make bread for a class of 20 people. Her recipe yields enough bread for 10 people and uses 3.5 cups of flour. How much flour will she need to make bread for her entire class?
- Solution
Start by listing all important information from the problem.
- Lisa needs bread for 20 people.
- Her recipe makes bread for only 10 people.
- That recipe uses 3.5 cups of flour for those 10 people.
Now, write the ratios needed for the problem, which asks for the amount of flour needed to make bread for 20 people. That, then, will be the missing value. The ratio to help find that missing value is 3.5 cups of flour to 10 people.
It's time to set up a proportion to solve. The cups of flour will be in the numerators, while the number of people served will be in the denominators.
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Cross-multiply to find the missing value, which we'll call x. {eq}10 \cdot x = 10x {/eq} and {eq}3.5 \cdot 20 = 70 {/eq}.
Set the two diagonal products equal to each other.
{eq}10x = 70 {/eq}
Finally, solve for x by dividing each side by 10. {eq}\frac{10x}{10}=x {/eq} and {eq}\frac{70}{10} = 7 {/eq}.
The answer, then, is {eq}x=7 {/eq}, but don't forget the units: Lisa needs 7 cups of flour to make bread for 20 people.
Ratio Word Problem 2
Randy and Katie have the same ratio of girls to boys in their classes. Randy's class has 12 girls and 8 boys. Katie's class has 30 students. How many girls are in Katie's class?
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- Solution
Start by pulling out important information.
- Randy's class: 12 girls and 8 boys.
- Katie's class: 30 students in total.
- We're looking for the number of girls in Katie's class.
Now, arrange the information into a proportion of two equal ratios. We're looking for the number of girls in Katie's class. We know the total number of students in Katie's class is 30 students, and can easily find the number of students in Randy's class by adding: {eq}12+8=20 {/eq}. Therefore, the total numbers of students are 30 students in Katie's class and 20 students in Randy's. We're looking for the number of girls in Katie's class, however, not the total number, so our variable will represent the number of girls. Use the number of girls in Randy's class and the total amount of students in both classes to set up a proportion and find the number of girls in Katie's class.
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The proportion can be set up with Randy's class on the left and Katie's on the right. Cross-multiply to calculate the number of girls in Katie's class.
{eq}20 \cdot x = 20x {/eq} and {eq}12 \cdot 30 = 360 {/eq}.
Set the diagonal products equal to each other to get:
{eq}20x=360 {/eq}
Finally, divide each side by 20 to isolate the value of x. {eq}\frac{20x}{20} = x {/eq} and {eq}\frac{360}{20}=18 {/eq}.
Therefore, {eq}x=18 {/eq}, which means there are 18 girls in Katie's class.
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Ratios are a great tool for comparing things in the world. A ratio is a comparison between two numbers. Ratios can be written in three different ways. For example, a single ratio could be written as 4 to 7, 4:7 or {eq}\frac{4}{7} {/eq}. Ratios can compare two things as a part-to-part ratio, as in 5 chickens to 2 pigeons, or as a part-to-whole ratio, as in 2 pigeons to 7 birds. To solve for a missing value in a ratio, use a proportion that sets that ratio equal to a corresponding ratio. Use cross-multiplication to solve for that proportion's missing value to quickly calculate the missing information.
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Video Transcript
Where Are Ratios?
Ratios are everywhere around us. Try these on for size:
- A 5 oz. bag of gummy bears is $1.49. Is it a better deal to get the 144 oz. bag for $15.99?
- You've got 60 homework problems to do and it took you 10 minutes to do eight of them. At that rate, how long will it take?
- Your favorite painting in the museum is 5 feet by 8 feet. How big will the eyes in that painting be on your smart phone's 4.3-inch screen?
We could go on and on; and while each of these appear to be different problems - dealing with money, time, and size - they are, at their core, the same. They all involve ratios.
Let's break down ratios a little more and see how they can help us solve these types of problems.
What Is a Ratio?
A ratio is a comparison between two numbers. To keep it simple, we'll ignore the units (e.g., cost in dollars or weight in ounces) and focus just on the number part for a bit. For example, how does 3 compare to 6? Well, three is half of six. We can write ratios in one of three ways:
- 3:6
- 3/6
- 3 to 6
Because we'll be using ratios mathematically, we'll use the format '/' for the rest of the lesson.
What Is a Proportion?
By itself, a ratio is limited to how useful it is. However, when two ratios are set equal to each other, they are called a proportion. For example, 1/2 is a ratio and 3/6 is also a ratio. If we write 1/2 = 3/6, we have written a proportion. We can also say that 1/2 is proportional to 3/6. In math, a ratio without a proportion is a little like peanut butter without jelly or bread.
How Proportions Can Help
In math problems and in real life, if we have a known ratio comparing two quantities, we can use that ratio to predict another ratio, if given one half of that second ratio. In the example 1/2 = 3/?, the known ratio is 1/2. We know both terms of the known ratio. The unknown ratio is 3/?, since we know one term, but not the other (thus, it's not yet a comparison between two ratios). We only know one of the two terms in the unknown ratio. However, if we set them as a proportion, we can use that proportion to find the missing number.
Solving Proportions with an Unknown Ratio
There are a few different methods we can use to solve proportions with an unknown ratio. However, the easiest and most fail-safe method is to cross-multiply and solve the resulting equation. For the last example, we would have:
1 * x = 2 * 3
1x = 6
x = 6 / 1
x = 6
To check the accuracy of our answer, simply divide the two sides of the equation and compare the decimal that results. In the example, 1/2 = 0.5 and 3/6 = 0.5. That was the correct result.
Solving Ratio Word Problems
To use proportions to solve ratio word problems, we need to follow these steps:
- Identify the known ratio and the unknown ratio.
- Set up the proportion.
- Cross-multiply and solve.
- Check the answer by plugging the result into the unknown ratio.
Your favorite store says it will donate to your soccer team $3 for every $50 that anyone wearing a soccer shirt spends at the store. Your team needs at least $1,200 donated to be able to travel to a tournament. How much money needs to be spent at the store by people wearing soccer shirts?
Our known ratio is $3 donated / $50 spent, and the unknown ratio is $1,200 donated / ? spent. The proportion would look like this:
Now let's do the math.
3 * x = 50 * 1,200
3x = 60,000
x = 60,000 / 3
x = $20,000
Checking this, we get:
3 / 50 = 1,200 / 20,000
0.06 = 0.06
This checks out!
Your friends and family will need to spend $20,000 at the store. No problem, right?
Is it really this easy? You betcha! You can use this process to solve any ratio word problem. The trickiest part is often identifying the known ratio and the unknown ratio. Once you've done that, make sure you are careful with tracking your calculations accurately, and you should have no trouble with these kinds of problems.
Lesson Summary
Ratios are found all around us every day and are simply a comparison between two numbers (e.g., red jellybeans to yellow jellybeans). A proportion is a statement that allows you to find an unknown ratio from a known ratio. In the known ratio, you know both of the numbers. In the unknown ratio, you only know one of the numbers. To solve for the unknown number, set up a proportion with the known ratio on one side and the unknown ratio on the other, cross multiply, and solve the resulting equation. This method works every single time, so long as you have identified the known and unknown ratios correctly.
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