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5.5: Ratios and Proportions

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Learning Objectives

After completing this section, you should be able to:

• Construct ratios to express comparison of two quantities.
• Use and apply proportional relationships to solve problems.
• Determine and apply a constant of proportionality.
• Use proportions to solve scaling problems.

Ratios and proportions are used in a wide variety of situations to make comparisons. For example, using the information from Figure 5.15, we can see that the number of Facebook users compared to the number of Twitter users is 2,006 M to 328 M. Note that the "M" stands for million, so 2,006 million is actually 2,006,000,000 and 328 million is 328,000,000. Similarly, the number of Qzone users compared to the number of Pinterest users is in a ratio of 632 million to 175 million. These types of comparisons are ratios.

Ratio problems: Three-term Ratios

Example 1: A special cereal mixture contains rice, wheat and corn in the ratio of 2:3:5. If a bag of the mixture contains 3 pounds of rice, how much corn does it contain?

Solution: Step 1: Assign variables: Let x = amount of corn

Step 2: Solve the equation Cross Multiply 2 × x = 3 × 5 2 x = 15

Answer: The mixture contains 7.5 pounds of corn.

Example 2: Clothing store A sells T-shirts in only three colors: red, blue and green. The colors are in the ratio of 3 to 4 to 5. If the store has 20 blue T-shirts, how many T-shirts does it have altogether?

Solution: Step 1: Assign variables: Let x = number of red shirts and y = number of green shirts

Step 2: Solve the equation Cross Multiply 3 × 20 = x × 4 60 = 4 x x = 15

5 × 20 = y × 4 100 = 4 y y = 25

The total number of shirts would be 15 + 25 + 20 = 60

Algebra And Ratios With Three Terms

Let’s study how algebra can help us think about ratios with more than two terms.

Example: There are a total of 42 computers. Each computer runs one of three operating systems: OSX, Windows, Linux. The ratio of the computers running OSX, Windows, Linux is 2:5:7. Find the number of computers that are running each of the operating systems.

A ratio compares values .

A ratio says how much of one thing there is compared to another thing.

Ratios can be shown in different ways:

A ratio can be scaled up:

Try it Yourself

Using ratios.

The trick with ratios is to always multiply or divide the numbers by the same value .

Example: A Recipe for pancakes uses 3 cups of flour and 2 cups of milk.

So the ratio of flour to milk is 3 : 2

To make pancakes for a LOT of people we might need 4 times the quantity, so we multiply the numbers by 4:

3 ×4 : 2 ×4 = 12 : 8

In other words, 12 cups of flour and 8 cups of milk .

The ratio is still the same, so the pancakes should be just as yummy.

"Part-to-Part" and "Part-to-Whole" Ratios

The examples so far have been "part-to-part" (comparing one part to another part).

But a ratio can also show a part compared to the whole lot .

Example: There are 5 pups, 2 are boys, and 3 are girls

Part-to-Part:

The ratio of boys to girls is 2:3 or 2 / 3

The ratio of girls to boys is 3:2 or 3 / 2

Part-to-Whole:

The ratio of boys to all pups is 2:5 or 2 / 5

The ratio of girls to all pups is 3:5 or 3 / 5

Try It Yourself

We can use ratios to scale drawings up or down (by multiplying or dividing).

Example: To draw a horse at 1/10th normal size, multiply all sizes by 1/10th

This horse in real life is 1500 mm high and 2000 mm long, so the ratio of its height to length is

1500 : 2000

What is that ratio when we draw it at 1/10th normal size?

We can make any reduction/enlargement we want that way.

"I must have big feet, my foot is nearly as long as my Mom's!"

But then she thought to measure heights, and found she is 133cm tall, and her Mom is 152cm tall.

In a table this is:

The "foot-to-height" ratio in fraction style is:

We can simplify those fractions like this:

And we get this (please check that the calcs are correct):

"Oh!" she said, "the Ratios are the same".

"So my foot is only as big as it should be for my height, and is not really too big."

You can practice your ratio skills by Making Some Chocolate Crispies

Ratios and proportional relationships

Ratios and proportions are similar figures and concepts that are as easily confused as toads and frogs (all toads are frogs, but not all frogs are toads). Ratios compare values, while proportions compare ratios.

What are ratios?

Ratios  compare values. You can compare the number of brown-haired boys to the number of blond-haired boys, or to the number of pencils in the classroom, or to the number of brown-haired girls, or … well, you get the idea. Ratios compare values of the same things or things that are different.

Say you have 10 brown-haired girls in a class, and 6 blonde-haired girls in the same class. You can set up  six   different ratios:

10 16 \frac{10}{16} 16 10 ​ : Brown-haired girls to all girls

6 10 \frac{6}{10} 10 6 ​ : Blonde-haired girls to brown-haired girls

6 16 \frac{6}{16} 16 6 ​ : Blonde-haired girls to all girls

10 6 \frac{10}{6} 6 10 ​ : Brown-haired girls to blonde-haired girls

16 10 \frac{16}{10} 10 16 ​ : All girls to brown-haired girls

16 6 \frac{16}{6} 6 16 ​ : All girls to blonde-haired girls

Three of those ratios are improper fractions; that is okay! Ratios can be written as proper or improper fractions. They can also be written with a semicolon, like this:

What are proportions?

When you compare two ratios, you use  proportions . You are asking if the first ratio is the same, less than, or more than the second ratio. Compare the ratios of brown-to-all girls and blonde-to-all girls:

You can see these two   ratios are  not  equal, so they are  not  proportional :

How to solve ratios and proportions

What would proportional fractions look like? Let's add eight class pets to the classroom: 5 hamsters and 3 frogs . The  ratios  you can create are:

5:3  (hamsters to frogs)

3:5  (frogs to hamsters)

5:8  (hamsters to all pets)

3:8  (frogs to all pets)

8:5  (all pets to hamsters)

8:3  (all pets to frogs)

Proportions can tell us if two ratios are equal or not. Compare the ratio of hamsters to all pets and the ratio of brown-haired girls to all girls:

You can check these fractions in a few ways, such as simplifying  10 16 \frac{10}{16} 16 10 ​  to  10 16 \frac{10}{16} 16 10 ​ , or by cross-multiplying and dividing: 5 × 16 10 = 8 \frac{5\times 16}{10}=8 10 5 × 16 ​ = 8

These two ratios  are  proportional to each other. The ratio of hamsters to all class pets is the same as the ratio of brown-haired girls to all girls in the class.

Ratios and proportions word problems

Cooking, comparing prices, driving, engineering, construction and finance are just some areas where ratios and proportions work every day.

Here is a recipe for hamster food to feed one hamster:

20g of five-cereal blend

10g small seed blend

10g rolled oats

10g dried vegetables

5g dried fruit

One hamster gets 60 grams of hamster chow. How much should you mix for five hamsters?

Whatever you multiply 1 times to get 5 , multiply 60 times the same number. You need 300 grams.

How much of each ingredient should you mix?

For every 60 grams of hamster chow for one hamster, 20 grams is five-cereal blend, a ratio of  20:60 or  1:3 .

If you want to feed five hamsters, you have to mix more of everything in the right proportions. How many grams of five-cereal blend will you need?

Say you did your calculations and mixed the five-cereal blend at a ratio of  50:300 . Is that correct? Check: Is  50 300 \frac{50}{300} 300 50 ​  proportional to  20 60 \frac{20}{60} 60 20 ​ or  20 60 \frac{20}{60} 60 20 ​ ?

You can cross-multiply and divide to check:  50 × 3 300 \frac{50\times 3}{300} 300 50 × 3 ​ .

You see that  150 300 = 1 2 \frac{150}{300}=\frac{1}{2} 300 150 ​ = 2 1 ​ , not 1 . So your mix is  not  in the right proportion because 50 is not one third of 300 .

You needed 100 g of five-cereal blend to maintain the right proportions.

Ratios and proportions examples

Perhaps you have a part-time job in a grocery store, assembling gift baskets of fruit. Your manager tells you to maintain a ratio of  2:3 of pears to apples for every size of basket. A small basket gets 2 pears and 3 apples . An extra-large basket must have the same ratio,  2:3 , but be five times larger.

The ratio of  pears:apples  is  2:3 , so multiply both parts of the ratio times 5 to get the new ratio:  10:15.

Ratios and proportions practice

The class of 10 brown-haired and 6 blonde-haired girls also has boys in it. Of the 12 boys in the class, 4 have blond hair and 8 have brown hair.

Write three ratios using this new information.

Many ratios can be written from the information. See if you can figure out what these ratios describe:

4:12 (Blond-haired boys to all boys)

8:12  (Brown-haired boys to all boys)

4:28 (Blond-haired boys to all students)

8:10 (Brown-haired boys to brown-haired girls)

28:8 (All students to brown-haired boys)

Lesson summary

You have learned that ratios compare values, while proportions compare ratios. Proportions are most often used to ensure ratios are equal when they increase or decrease. You can write ratios as either a fraction or with a colon between them, like this:  10 16 \frac{10}{16} 16 10 ​ or  10 : 16 10:16 10 : 16 . Ratios can compare like and unlike things. Both ratios and proportions are useful in many aspects of everyday life.

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Ratios and proportions and how to solve them

Let's talk about ratios and proportions. When we talk about the speed of a car or an airplane we measure it in miles per hour. This is called a rate and is a type of ratio. A ratio is a way to compare two quantities by using division as in miles per hour where we compare miles and hours.

A ratio can be written in three different ways and all are read as "the ratio of x to y"

$$x\: to\: y$$

$$\frac{x}{y}$$

A proportion on the other hand is an equation that says that two ratios are equivalent. For instance if one package of cookie mix results in 20 cookies than that would be the same as to say that two packages will result in 40 cookies.

$$\frac{20}{1}=\frac{40}{2}$$

A proportion is read as "x is to y as z is to w"

$$\frac{x}{y}=\frac{z}{w} \: where\: y,w\neq 0$$

If one number in a proportion is unknown you can find that number by solving the proportion.

You know that to make 20 pancakes you have to use 2 eggs. How many eggs are needed to make 100 pancakes?

$$\frac{eggs}{pancakes}=\frac{eggs}{pancakes}\: \: or\: \: \frac{pancakes}{eggs}=\frac{pancakes}{eggs}$$

If we write the unknown number in the nominator then we can solve this as any other equation

$$\frac{x}{100}=\frac{2}{20}$$

Multiply both sides with 100

$${\color{green} {100\, \cdot }}\, \frac{x}{100}={\color{green} {100\, \cdot }}\, \frac{2}{20}$$

$$x=\frac{200}{20}$$

If the unknown number is in the denominator we can use another method that involves the cross product. The cross product is the product of the numerator of one of the ratios and the denominator of the second ratio. The cross products of a proportion is always equal

If we again use the example with the cookie mix used above

$$\frac{{\color{green} {20}}}{{\color{blue} {1}}}=\frac{{\color{blue} {40}}}{{\color{green} {2}}}$$

$${\color{blue} {1}}\cdot {\color{blue} {40}}={\color{green} {2}}\cdot {\color{green} {20}}=40$$

It is said that in a proportion if

If you look at a map it always tells you in one of the corners that 1 inch of the map correspond to a much bigger distance in reality. This is called a scaling. We often use scaling in order to depict various objects. Scaling involves recreating a model of the object and sharing its proportions, but where the size differs. One may scale up (enlarge) or scale down (reduce).  For example, the scale of 1:4 represents a fourth. Thus any measurement we see in the model would be 1/4 of the real measurement. If we wish to calculate the inverse, where we have a 20ft high wall and wish to reproduce it in the scale of 1:4, we simply calculate:

$$20\cdot 1:4=20\cdot \frac{1}{4}=5$$

In a scale model of 1:X where X is a constant, all measurements become 1/X - of the real measurement. The same mathematics applies when we wish to enlarge. Depicting something in the scale of 2:1 all measurements then become twice as large as in reality. We divide by 2 when we wish to find the actual measurement.

Video lesson

$$\frac{x}{x + 20} = \frac{24}{54}$$

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ACT Math: How to Solve Ratio Problems

There are a few key concepts to get down in order to ace ACT Math ratios. Let’s go right into how the ACT will test you on ratios and break it down for you.

Ratio Basics

A ratio tells you the proportional quantity of one thing relative to another.

Make sure not to get ratios confused with fractions. Fractions tell you the proportional quantity of something relative to its whole. Ratios expressed as fractions do not tell you the whole. One instance where you need to use the concept of ratios involves baking. If you want to make double the amount of cookies that a recipe will yield, then you need to double the quantity of each ingredient.

ACT Math: Dealing With Ratios

You might see ACT ratios written in fraction form, colon form, or in plain English. Whatever the case may be, you can treat them all the same way. In the case of the fraction form, do not get it confused with a regular fraction! The denominator of a ratio is not necessarily equivalent to the denominator of a ratio.

For example, the ratio 12/8, 12:8, and 12 to 8 are all the same. Like fractions, you should reduce ratios down to simplest terms – in this case, it is 3/2. Keep your numbers manageable, especially when you need to look for the lowest common multiple later on in the multi-step ratio section.

On the test, ratios will be clearly spelled out for you. If you are looking at a ratio problem, you’ll know it because the test makers will make it obvious.

The important part lies in knowing how to manipulate ratios to get to your answer. The two main things you need to know are proportions and multi-step ratios.

Proportions

You’ll find that these are very common on the ACT. Thankfully, they are also easy to solve.

You will usually be given a ratio along with a hypothetical quantity of one of the things on the original ratio. The key is to set up two ratios and cross-multiply as you would two fractions to solve for the missing fourth quantity.

If you have a ratio of 3 cats to 2 dogs, how many cats do you have if you have 20 dogs? You could use mental math or set up two fractions to get 30 cats as your answer.

Multi-Step Ratios

These are a little bit more involved, but shouldn’t pose much of a threat to your ACT Math score one you learn about how to go about solving them.

Here you’ll be given two ratios and three different types of quantities: a, b, and c. The two ratios given compare a to b and b to c. You’ll then be asked to figure out the ratio of a to c.

In order to solve, you need to figure out the least common multiple of the two b’s and multiply the respective a’s accordingly. Now with your b’s equal to each other, simply take the values of your a and c and create a new ratio.

For example, if a:b is 2:3 and b:c is 6:9, then what is a:c? Here we need to multiply our first ratio by 2 in order to get 4:6. Since our b from the first and second ratio match, we can take our a and c and form a new ratio.

Minh’s passion for helping students succeed grew during his time as a career counselor at the University of California, Irvine. Now, he’s helping students all over the world by spilling SAT/ACT secrets through blog posts on Magoosh. When he’s not busy tutoring or writing, he enjoys playing guitar, traveling, and talking about himself in third-person.

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Course: 6th grade   >   Unit 1

• Ratios with tape diagrams
• Ratios with tape diagrams (part:whole)

Equivalent ratio word problems

• Simplify a ratio from a tape diagram
• Equivalent ratios with equal groups
• Ratios and double number lines
• Create double number lines
• Ratios with double number lines
• Relate double number lines and ratio tables

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Video transcript

1. Ratio Problem Solving

40 \div 8=5 40 ÷ 8 = 5. Then you multiply each part of the ratio by 5. 5. 3\times 5:5\times 5=15 : 25 3 × 5: 5 × 5 = 15: 25. This means that Charlie will get 15 15 sweets and David will get 25 25 sweets. There can be ratio word problems involving different operations and types of numbers.

2. Ratios and proportions

It compares the amount of one ingredient to the sum of all ingredients. part: whole = part: sum of all parts. To write a ratio: Determine whether the ratio is part to part or part to whole. Calculate the parts and the whole if needed. Plug values into the ratio. Simplify the ratio if needed.

3. Ratios and rates

Do you want to learn how to compare and measure different quantities using ratios and rates? Khan Academy's pre-algebra course offers you a comprehensive introduction to these concepts, with interactive exercises and videos. You will also learn how to use proportions to solve word problems and graph proportional relationships. Join Khan Academy and start your journey to master ratios and rates!

4. Ratio Math Problems

Two-Term Ratio Word Problems More Ratio Word Problems Math Word Problems More Algebra Lessons. Ratio problems are word problems that use ratios to relate the different items in the question. Ratio problems: Three-term Ratios. Example 1: A special cereal mixture contains rice, wheat and corn in the ratio of 2:3:5.

5. 1.3: Ratios

In general, if you find a way to solve a problem that works for you, as long as you get the same numerical answer, you can use that method. We will cover proportions in the next section. In this section, we are choosing to focus on equivalent ratios as a method for problem-solving. Let's see one more example.

6. 5.5: Ratios and Proportions

Determine and apply a constant of proportionality. Use proportions to solve scaling problems. Ratios and proportions are used in a wide variety of situations to make comparisons. For example, using the information from Figure 5.15, we can see that the number of Facebook users compared to the number of Twitter users is 2,006 M to 328 M. Note ...

7. Ratio Problem Solving

Ratio problem solving is a collection of word problems that link together aspects of ratio and proportion into more real life questions. This requires you to be able to take key information from a question and use your knowledge of ratios (and other areas of the curriculum) to solve the problem.

8. Algebra: Ratio Word Problems

Ratio Word Problems: relating different things using ratios and algebra, how to solve ratio word problems that have two-term ratios or three-term ratios, How to solve proportion word problems, questions and answers, with video lessons, examples and step-by-step solutions.

9. Solving ratio problems with tables (video)

The ratio 54:3 shows us that the simplified form is 18:1. This is because 54/3 = 18. Let's just call the blank part x. To find 36: x, you have to divide 36 by 18 to get 2 (so the ratio is 36:2). To find x:5, you have to multiply 18*5, which is 90 (the ratio is 90:5) Hope this helps! :) 4 comments.

10. Ratios

Using Ratios. The trick with ratios is to always multiply or divide the numbers by the same value. Example: A Recipe for pancakes uses 3 cups of flour and 2 cups of milk. So the ratio of flour to milk is 3 : 2. To make pancakes for a LOT of people we might need 4 times the quantity, so we multiply the numbers by 4: 3 ×4 : 2 ×4 = 12 : 8.

11. Art of Problem Solving: Introducing Ratios

12. Ratios and Proportions

The ratio of pears:apples is 2:3, so multiply both parts of the ratio times 5 to get the new ratio: 10:15. Your extra-large gift basket needs 10 pears and 15 apples.. Ratios and proportions practice. The class of 10 brown-haired and 6 blonde-haired girls also has boys in it. Of the 12 boys in the class, 4 have blond hair and 8 have brown hair.. Write three ratios using this new information.

13. Ratios and proportions and how to solve them

If we wish to calculate the inverse, where we have a 20ft high wall and wish to reproduce it in the scale of 1:4, we simply calculate: 20 ⋅ 1: 4 = 20 ⋅ 1 4 = 5 20 ⋅ 1: 4 = 20 ⋅ 1 4 = 5. In a scale model of 1:X where X is a constant, all measurements become 1/X - of the real measurement. The same mathematics applies when we wish to enlarge.

14. 100% of Math Students Need to UNDERSTAND! Rates, Ratios and ...

How to solve math problems with rates, ratios and proportions.For more in-depth math help check out my catalog of courses. Every course includes over 275 vid...

15. Part to whole ratio word problem using tables

What you need to do in any word problem involving the ratios is exactly the same. Take the entire amount and divide it by the sum of the ratios. This will give you the number you need to multiply both ratios by. So the entire amount of playtimes is 30, and the sum of the ratios is 2+3, which is 5.

16. Solving ratio problems

Divide the total amount in the initial ratio. Find the value of one part by dividing the total amount by the sum of the parts. Multiply the value of one part by the number of parts for each share ...

17. Ratios with tape diagrams (video)

Tape diagrams are visual models that use rectangles to represent the parts of a ratio. Since they are a visual model, drawing them requires attention to detail in the setup. In this problem David and Jason have numbers of marbles in a ratio of 2:3. This ratio is modeled here by drawing 2 rectangles to represent David's portion, and 3 ...

18. Solving Ratios, Practice with SAT Problems

Definition: A comparison between quantities using division. Examples : 3:2 , 3:2:88, 3 to 2, 3 to 2 to 88. A 2 to 5 ratio can be represented as 2:5. A ration between X and Y can be written. X/Y. X:Y. X to Y. MEDIUM SAT PROBLEM#8 out of a 25 problem section. A bucket holds 4 quarts of popcorn.

19. ACT Math: How to Solve Ratio Problems

The two ratios given compare a to b and b to c. You'll then be asked to figure out the ratio of a to c. In order to solve, you need to figure out the least common multiple of the two b's and multiply the respective a's accordingly. Now with your b's equal to each other, simply take the values of your a and c and create a new ratio.

20. Equivalent ratio word problems (practice)

Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere. ... Solving ratio problems with tables. Ratio tables. Equivalent ratios. Equivalent ratios: recipe ...

21. Equivalent ratio word problems (video)

Equivalent ratio word problems. Google Classroom. About. Transcript. This video teaches solving ratio word problems, using examples like Yoda Soda for guests, fish ratios in a tank, ice cream sundae ingredients, and dog color ratios at a park. Mastering these techniques helps students tackle real-world math challenges.