Ratios, Rates, and Proportional Reasoning

Defining Ratios

A ratio is a way to compare two amounts. Ratios help us describe how much of one thing there is compared to another thing.

For example, if there are 2 apples and 3 bananas, we can compare apples to bananas with the ratio \(2:3\). This means 2 apples for every 3 bananas.

Ratios are useful in real life. We use them when we compare ingredients in recipes, colors in art, players on teams, and many other situations.

When we write a ratio, the order matters. The ratio apples to bananas is not the same as bananas to apples.

For example:

• Apples to bananas: \(2:3\)
• Bananas to apples: \(3:2\)

Both ratios compare the same two groups, but they answer different questions.

Ratios can be written in different ways:

• Using a colon: \(2:3\)
• Using the word to: \(2 \text{ to } 3\)
• As a fraction: \(\frac{2}{3}\)

These are all ways to show the same comparison.

A ratio compares two distinct quantities. That means the two things are different kinds of things, like red marbles and blue marbles, or boys and girls.

Let’s look at the parts of a ratio more closely:

1. Decide what two quantities you are comparing.
2. Choose the correct order.
3. Write the ratio to match the words in the question.

For example, if a class has 4 boys and 6 girls:

• Boys to girls is \(4:6\)
• Girls to boys is \(6:4\)

If the question asks for boys to girls, you must write \(4:6\), not \(6:4\).

Sometimes a ratio compares a part to another part. For example, red cubes to blue cubes.

Sometimes a ratio compares a part to the whole group. For example, red cubes to all cubes.

Suppose there are 3 red cubes and 5 blue cubes. Then there are \(3+5=8\) cubes in all.

• Red to blue: \(3:5\)
• Red to all cubes: \(3:8\)
• Blue to all cubes: \(5:8\)

Be careful: part-to-part ratios and part-to-whole ratios are not the same.

Worked Example 1

A basket has 5 oranges and 2 pears. Write the ratio of oranges to pears.

Step 1: Find the two quantities: oranges and pears.

Step 2: Put them in the correct order: oranges first, pears second.

Step 3: Write the ratio: \(5:2\).

Answer: The ratio of oranges to pears is \(5:2\).

Worked Example 2

There are 7 red balloons and 4 yellow balloons. Write:

• the ratio of red balloons to yellow balloons
• the ratio of yellow balloons to red balloons

Step 1: Count each group.

• Red balloons = 7
• Yellow balloons = 4

Step 2: Match the order to the words.

• Red to yellow = \(7:4\)
• Yellow to red = \(4:7\)

Answer:

• Red to yellow: \(7:4\)
• Yellow to red: \(4:7\)

Worked Example 3

A toy box has 6 toy cars and 3 dolls. Write the ratio of dolls to all toys.

Step 1: Find the number of dolls: \(3\).

Step 2: Find the total number of toys:

Step 3: Write dolls to all toys: \(3:9\).

Answer: The ratio of dolls to all toys is \(3:9\).

Worked Example 4

In an art set, there are 4 paintbrushes, 8 crayons, and 2 erasers. Write the ratio of crayons to paintbrushes.

Step 1: Find the two groups named in the question.

• Crayons = 8
• Paintbrushes = 4

Step 2: Write them in the correct order: crayons to paintbrushes.

So the ratio is \(8:4\).

Answer: The ratio of crayons to paintbrushes is \(8:4\).

Tips for Writing Ratios Correctly

• Read carefully. The words tell you the order.
• Compare only the quantities asked for.
• Check whether it is part-to-part or part-to-whole.
• Count the total if the question asks for a comparison to the whole.

Common Mistakes to Avoid

• Reversing the order. For example, writing \(3:5\) instead of \(5:3\).
• Using the whole group when the question asks for only two parts.
• Forgetting to add the groups when comparing to the total.

Let’s Practice Thinking About Ratios

If a fish tank has 9 goldfish and 1 black fish:

• Goldfish to black fish = \(9:1\)
• Black fish to goldfish = \(1:9\)
• Black fish to all fish = \(1:10\)

Notice how each ratio tells something different about the same fish tank.

Summary

A ratio compares two amounts. You can write ratios using a colon, the word to, or as a fraction.

To define a ratio correctly, first decide which two quantities you are comparing. Then make sure you write them in the same order as the words in the question.

Also remember to check whether the ratio is comparing one part to another part or one part to the whole group. Careful reading and correct order will help you write ratios the right way every time.

Part-to-Part and Part-to-Whole Comparisons

Sometimes in math, we compare two groups that belong to the same whole. Learning what kind of comparison we are making is very important.

There are two main ways to compare:

• Part-to-part: comparing one part of a group to another part of the same group
• Part-to-whole: comparing one part of a group to the entire group

For example, imagine a bowl with 3 red apples and 5 green apples. There are 8 apples in all.

• The comparison of red apples to green apples is part-to-part: \(3:5\)
• The comparison of red apples to all apples is part-to-whole: \(3:8\)
• The comparison of green apples to all apples is also part-to-whole: \(5:8\)

Notice the difference: in a part-to-part comparison, we compare two smaller groups. In a part-to-whole comparison, we compare one smaller group to the total number.

Step 1: Find the parts and the whole

To make the correct comparison, first ask:

• What are the parts?
• What is the whole?

The whole is the total of all the parts added together.

If a class has 12 girls and 8 boys, then:

So the whole class has \(20\) students.

Now we can make different comparisons:

• Girls to boys: \(12:8\) → part-to-part
• Girls to total students: \(12:20\) → part-to-whole
• Boys to total students: \(8:20\) → part-to-whole

Step 2: Read the words carefully

Word problems often give clues about which comparison to use.

• If the question says to, compared to, for every one group and another group, it is often part-to-part.
• If the question says out of, of the total, of all, it is often part-to-whole.

Examples of question words:

• Part-to-part: What is the ratio of cats to dogs?
• Part-to-whole: What part of the pets are cats?

Step 3: Write the comparison in the correct order

Order matters in ratios.

If there are 4 blue marbles and 7 yellow marbles:

• Blue to yellow is \(4:7\)
• Yellow to blue is \(7:4\)

These are not the same. Always match the order in the question.

Worked Example 1

A basket has 6 oranges and 4 bananas.

What is the ratio of oranges to bananas? What is the ratio of bananas to all fruit?

Step A: Identify the parts.

• Oranges = \(6\)
• Bananas = \(4\)

Step B: Find the whole.

Step C: Answer each question.

• Oranges to bananas: \(6:4\) → part-to-part
• Bananas to all fruit: \(4:10\) → part-to-whole

Worked Example 2

In a jar, there are 9 red beads, 3 blue beads, and 6 white beads.

Find:

1. the ratio of red beads to blue beads
2. the ratio of white beads to all beads

Step A: List the parts.

• Red = \(9\)
• Blue = \(3\)
• White = \(6\)

Step B: Find the whole.

Step C: Make the comparisons.

• Red to blue: \(9:3\) → part-to-part
• White to all beads: \(6:18\) → part-to-whole

Worked Example 3

A school club has 7 fifth graders and 14 fourth graders.

Which ratio shows fifth graders compared to the whole club?

Step A: Find the whole club.

Step B: Compare fifth graders to the whole.

This is part-to-whole because \(7\) is one part, and \(21\) is the total.

If we compared fifth graders to fourth graders, it would be \(7:14\), which is part-to-part.

Worked Example 4

There are 5 chocolate cupcakes, 8 vanilla cupcakes, and 2 strawberry cupcakes.

Find:

1. the ratio of vanilla cupcakes to chocolate cupcakes
2. the ratio of strawberry cupcakes to all cupcakes

Step A: Identify the parts.

• Chocolate = \(5\)
• Vanilla = \(8\)
• Strawberry = \(2\)

Step B: Find the whole.

Step C: Write each ratio.

• Vanilla to chocolate: \(8:5\) → part-to-part
• Strawberry to all cupcakes: \(2:15\) → part-to-whole

How to tell the difference quickly

• If you compare one group to another group, it is part-to-part.
• If you compare one group to the total, it is part-to-whole.

You can ask yourself:

• Am I comparing part to part?
• Or am I comparing part to whole?

Common mistakes to avoid

• Forgetting to find the whole
  For part-to-whole comparisons, add all the parts first.
• Using the wrong order
  "Cats to dogs" is different from "dogs to cats." Read carefully.
• Mixing up the comparison type
  If the question asks for a part-to-whole ratio, do not compare two parts.

Try these thinking questions

Suppose a box has 11 pencils and 9 pens.

• Pencils to pens = \(11:9\) → part-to-part
• Pencils to all items = \(11:20\) → part-to-whole
• Pens to all items = \(9:20\) → part-to-whole

See how the whole is found by adding:

Summary

A part-to-part comparison matches one part of a group with another part of the same group.

A part-to-whole comparison matches one part of a group with the total number in the group.

To solve these problems, follow these steps:

1. Identify each part.
2. Find the whole by adding the parts.
3. Read the question carefully.
4. Write the ratio in the correct order.

When you know whether the question is asking for part-to-part or part-to-whole, ratios become much easier to understand.

Lesson: Equivalent Ratios and Ratio Tables

A ratio compares two amounts. It tells how much of one thing there is compared to another thing.

For example, if there are 2 red apples and 3 green apples, the ratio of red apples to green apples is 2 to 3. We can write it as 2:3.

In this lesson, you will learn how to find equivalent ratios and how to organize them in a ratio table.

What are equivalent ratios?

Equivalent ratios are ratios that compare amounts in the same way. Even if the numbers are different, the relationship stays the same.

For example, the ratio 1:2 is equivalent to 2:4 and 3:6. In each case, the second number is still twice the first number.

We can make equivalent ratios by doing the same operation to both parts of the ratio:

• Multiply both numbers by the same number
• Divide both numbers by the same number

For example:

Here, both parts were multiplied by 2.

So, 2:5 and 4:10 are equivalent ratios.

Important rule: You must multiply or divide both parts of the ratio by the same number. If you only change one part, the ratio is no longer equivalent.

How ratio tables help

A ratio table is a chart that shows pairs of numbers that have the same ratio. It helps you organize equivalent ratios neatly.

For the ratio 2:3, a ratio table could look like this:

Each row shows an equivalent ratio. We made them by multiplying both numbers by 1, 2, 3, and 4.

How to build a ratio table

1. Start with the original ratio.
2. Choose a number to multiply by.
3. Multiply both parts of the ratio by that number.
4. Write the new pair in the table.

You can also divide both numbers by the same number when possible.

Worked Example 1: Making equivalent ratios

Find 3 equivalent ratios for 3:4.

Multiply both numbers by 2, 3, and 4.

So, 3 equivalent ratios are:

• 6:8
• 9:12
• 12:16

Worked Example 2: Filling in a ratio table

A recipe uses 2 cups of flour for every 1 cup of sugar. Complete the ratio table.

We can see:

• 2 to 4 means multiply by 2, so 1 \times 2 = 2
• 2 to 6 means multiply by 3, so 1 \times 3 = 3
• 2 to 8 means multiply by 4, so 1 \times 4 = 4

The completed table is:

Worked Example 3: Finding a missing number

The ratio of blue marbles to yellow marbles is 5:2. If there are 15 blue marbles, how many yellow marbles are there?

We ask: 5 became 15 by multiplying by what number?

So we multiply the other part by 3 too:

So the equivalent ratio is:

There are 6 yellow marbles.

Worked Example 4: Using division in a ratio table

The ratio of boys to girls is 12:8. Find a smaller equivalent ratio.

Both 12 and 8 can be divided by 4.

So:

This means 12:8 and 3:2 are equivalent ratios.

How to check if two ratios are equivalent

You can check by seeing whether both parts are multiplied or divided by the same number.

Example: Are 4:7 and 8:14 equivalent?

Yes, because:

Example: Are 4:7 and 8:15 equivalent?

No, because although 4 was multiplied by 2 to get 8, 7 was not multiplied by 2 to get 15.

Tips to remember

• A ratio compares two amounts.
• Equivalent ratios show the same comparison.
• Multiply or divide both parts by the same number.
• Ratio tables help organize equivalent ratios.
• If one part changes by a scale factor, the other part must change by the same scale factor.

Common mistake

A common mistake is changing only one number in the ratio.

For example, starting with 3:5 and changing it to 6:5 is not an equivalent ratio, because only the first number changed.

To make an equivalent ratio, both numbers must change in the same way:

Let’s review

Equivalent ratios are different number pairs that show the same comparison. You can make them by multiplying or dividing both numbers by the same number. A ratio table helps you list these pairs in an organized way and find missing values.

When you see a ratio problem, ask yourself:

• What is the original ratio?
• What number did one part get multiplied or divided by?
• Did the other part change by the same number?

If the answer is yes, then the ratios are equivalent.

Lesson: Tape Diagrams and Double Number Lines

When two amounts are compared in a ratio, we can use pictures to help us think. Two very useful pictures are tape diagrams and double number lines.

These tools help us solve ratio and rate problems by showing how amounts grow, shrink, and match each other. They are especially helpful when a word problem feels hard to picture in your head.

In this lesson, you will learn what tape diagrams and double number lines are, how they are alike, how they are different, and how to use them to solve problems.

1. What is a ratio?

A ratio compares two amounts. For example, if a recipe uses 2 cups of juice and 3 cups of water, the ratio of juice to water is \(2:3\).

This means that for every 2 parts juice, there are 3 parts water.

2. What is a tape diagram?

A tape diagram is a bar model made of equal-sized parts. It helps show how quantities are related.

If the ratio of red marbles to blue marbles is \(2:3\), we can draw:

Red: [ ] [ ]

Blue: [ ] [ ] [ ]

The boxes are equal because each box stands for the same amount. Tape diagrams are great when a problem talks about parts of a whole or compares groups.

3. What is a double number line?

A double number line has two number lines stacked on top of each other. Matching numbers line up because they belong together in the same ratio.

For example, if 2 apples cost 6 dollars, a double number line can show:

Apples:   0 --- 2 --- 4 --- 6

Cost:     0 --- 6 --- 12 --- 18

This shows that when the number of apples doubles, the cost doubles too. Double number lines are very useful for scaling up, scaling down, and finding missing values.

4. How are these models helpful?

Both models help you see multiplicative relationships. That means you can tell how many times bigger or smaller one amount is compared to another.

• Tape diagrams are helpful when you want to see equal parts.
• Double number lines are helpful when you want to skip-count or scale to new values.

5. Steps for solving with a tape diagram

1. Read the ratio in the problem.
2. Draw bars with equal parts to match the ratio.
3. Label what each part means if you can.
4. Use multiplication or division to find unknown amounts.

6. Steps for solving with a double number line

1. Write the two related quantities on two number lines.
2. Place the known pair of values.
3. Multiply or divide to find other matching pairs.
4. Find the missing value that lines up with the amount in the problem.

Worked Example 1: Basic tape diagram

The ratio of cats to dogs at a pet show is \(3:2\). If there are 6 dogs, how many cats are there?

Step 1: Draw the ratio.

Cats: [ ] [ ] [ ]

Dogs: [ ] [ ]

Step 2: Use what you know.

2 parts = 6 dogs, so 1 part = \(6 \div 2 = 3\) dogs.

Step 3: Find the cats.

Cats have 3 parts, so:

Answer: There are \(9\) cats.

Worked Example 2: Tape diagram with a total

A class has boys and girls in the ratio \(4:5\). There are 27 students in all. How many are boys?

Step 1: Draw the parts.

Boys: [ ] [ ] [ ] [ ]

Girls: [ ] [ ] [ ] [ ] [ ]

Step 2: Count total parts.

There are:

Step 3: Find the value of one part.

So each part is worth 3 students.

Step 4: Find the number of boys.

Answer: There are \(12\) boys.

Worked Example 3: Basic double number line

Four notebooks cost 8 dollars. How much do 10 notebooks cost?

Step 1: Start with the known pair.

Notebooks: 0 --- 4

Cost:      0 --- 8

Step 2: Find the cost of 1 notebook.

So 1 notebook costs \(2\) dollars.

Step 3: Scale to 10 notebooks.

On a double number line, you could show:

Notebooks: 0 --- 1 --- 4 --- 10

Cost:      0 --- 2 --- 8 --- 20

Answer: 10 notebooks cost \(20\) dollars.

Worked Example 4: Double number line with scaling

A car travels 60 miles in 2 hours. How far will it travel in 5 hours if it keeps the same speed?

Step 1: Put the known values on a double number line.

Hours: 0 --- 2

Miles: 0 --- 60

Step 2: Find 1 hour.

So the car travels 30 miles in 1 hour.

Step 3: Find 5 hours.

Double number line:

Hours: 0 --- 1 --- 2 --- 5

Miles: 0 --- 30 --- 60 --- 150

Answer: The car will travel \(150\) miles in 5 hours.

7. How to choose the right model

• Use a tape diagram when the problem is about parts, groups, or a total split into pieces.
• Use a double number line when the problem is about a rate, repeated matching values, or scaling to a new amount.

8. Common mistakes to avoid

• Do not make the parts different sizes in a tape diagram. Equal parts must look equal.
• Do not add when the problem needs multiplying or dividing.
• Make sure the numbers that line up on a double number line belong together.
• Read the question carefully so you find the amount it asks for.

9. Quick check for yourself

Ask these questions:

• What two things are being compared?
• Do I know the ratio or rate?
• Would a tape diagram or a double number line help more?
• Did I multiply or divide correctly?
• Does my answer make sense?

Summary

Tape diagrams and double number lines are tools that help you solve ratio problems with pictures. Tape diagrams show equal parts in groups, and double number lines show matching values that grow or shrink together.

When you use these models, you can see the math more clearly. That makes it easier to find missing values and solve word problems with confidence.

Unit Rates help us compare two amounts by finding the value for 1 unit. A unit rate tells us how much of one thing goes with exactly one of another thing.

For example, if 3 apples cost $6, we can ask: How much does 1 apple cost? That is a unit rate. We are finding the cost per 1 apple.

Unit rates are very useful in everyday life. We use them to compare prices at stores, find speed, and understand how much something costs or how far something goes for 1 unit.

Examples of unit rates:

• dollars per item
• miles per hour
• words per minute
• ounces per bottle

The word per is a clue. It often means for each 1.

How to find a unit rate

1. Write the rate as a comparison of two amounts.
2. Decide which amount should become 1.
3. Divide both parts by that number.
4. Write your answer using the word per.

You can think of a unit rate as a ratio with a denominator of 1.

For example:

$$\frac{12\text{ miles}}{3\text{ hours}} = \frac{4\text{ miles}}{1\text{ hour}}$$

So the unit rate is 4 miles per hour.

Why unit rates matter

Sometimes two choices are hard to compare because the numbers are different. A unit rate makes the comparison fair by changing both choices to the amount for 1 unit.

If one pack has 8 pencils for $4 and another pack has 12 pencils for $6, it may not be easy to tell which is the better deal right away. But if we find the cost for 1 pencil, then we can compare them easily.

Worked Example 1: Cost per item

4 notebooks cost $8. Find the unit rate.

We want the cost for 1 notebook.

Divide the total cost by the number of notebooks:

$$8 \div 4 = 2$$

So:

$$\frac{\$8}{4\text{ notebooks}} = \frac{\$2}{1\text{ notebook}}$$

Answer: The unit rate is $2 per notebook.

Worked Example 2: Distance per hour

A car travels 15 miles in 3 hours. Find the unit rate.

We want the distance for 1 hour.

Divide the miles by the hours:

$$15 \div 3 = 5$$

So:

$$\frac{15\text{ miles}}{3\text{ hours}} = \frac{5\text{ miles}}{1\text{ hour}}$$

Answer: The unit rate is 5 miles per hour.

Worked Example 3: Comparing prices

Which is the better buy?

• Pack A: 6 juice boxes for $12
• Pack B: 10 juice boxes for $15

Find the cost per 1 juice box for each pack.

For Pack A:

$$12 \div 6 = 2$$

So Pack A costs $2 per juice box.

For Pack B:

$$15 \div 10 = 1.5$$

So Pack B costs $1.50 per juice box.

Now compare the unit rates:

• Pack A: $2 per juice box
• Pack B: $1.50 per juice box

Answer: Pack B is the better buy because each juice box costs less.

Worked Example 4: Words per minute

Lena reads 48 words in 6 minutes. Find the unit rate.

We want the number of words for 1 minute.

Divide:

$$48 \div 6 = 8$$

So:

$$\frac{48\text{ words}}{6\text{ minutes}} = \frac{8\text{ words}}{1\text{ minute}}$$

Answer: The unit rate is 8 words per minute.

Tips for solving unit rate problems

• Read carefully to see what should be compared to 1.
• Look for the word per.
• Divide the first amount by the second amount if you want the first amount per 1 of the second.
• Include the labels, like dollars per item or miles per hour.

Watch out for these mistakes

• Mixing up the order. For example, miles per hour is different from hours per mile.
• Forgetting what the 1 stands for. Always ask, “1 what?”
• Leaving off the units. Units help explain the answer.

Try thinking about it this way

If 5 markers cost $10, you can share the $10 equally among the 5 markers.

Each marker gets:

$$10 \div 5 = 2$$

So each marker costs $2. That means the unit rate is $2 per marker.

Quick practice questions

1. 9 oranges cost $18. What is the cost per orange?
2. 20 miles are traveled in 4 hours. What is the speed in miles per hour?
3. 12 pencils cost $6. What is the cost per pencil?

Answers

1. $$18 \div 9 = 2$$ so the unit rate is $2 per orange.
2. $$20 \div 4 = 5$$ so the unit rate is 5 miles per hour.
3. $$6 \div 12 = 0.5$$ so the unit rate is $0.50 per pencil.

Summary

A unit rate is a rate that compares an amount to 1 unit. To find a unit rate, divide so that one part of the ratio becomes 1. Unit rates help us compare prices, speed, and other real-world situations in a simple and fair way.