Ratios, Rates, and Proportional Reasoning

Lesson: Understanding Ratios and Ratio Language

In math, we often compare quantities. Sometimes we compare by finding how many more, and sometimes we compare by finding how many times as much. A ratio helps us compare two quantities using multiplication.

A ratio tells how much of one thing there is compared to another thing. Ratios can compare:

• part to part — one part of a group to another part of the same group
• part to whole — one part of a group to the total group

Learning to use ratio language clearly is very important. The order matters. For example, saying “the ratio of cats to dogs” is not the same as “the ratio of dogs to cats.”

Ratios can be written in three common ways:

• using the word to: 3 to 2
• using a colon: 3:2
• as a fraction: \(\frac{3}{2}\)

These all mean the same ratio, as long as the order stays the same.

Main Idea 1: A ratio compares two quantities

Suppose a basket has 4 red apples and 6 green apples. We can compare these amounts in different ways.

• Ratio of red apples to green apples: \(4:6\)
• Ratio of green apples to red apples: \(6:4\)

Notice that these are different because the order is different.

We can also compare a part to the whole. There are \(4 + 6 = 10\) apples total.

• Ratio of red apples to total apples: \(4:10\)
• Ratio of green apples to total apples: \(6:10\)

Main Idea 2: Part-to-part and part-to-whole ratios

A part-to-part ratio compares one part of a group to another part of the group.

Example: In a class, there are 8 girls and 12 boys.

• Girls to boys is \(8:12\)
• Boys to girls is \(12:8\)

A part-to-whole ratio compares one part of a group to the total number in the group.

The total number of students is:

• Girls to total students is \(8:20\)
• Boys to total students is \(12:20\)

Be careful: if a question asks for a ratio to the whole, you must first find the total.

Main Idea 3: Ratio language must be precise

When you describe a ratio, always say exactly what is being compared.

For example, if there are 5 blue marbles and 3 yellow marbles, you can say:

• “The ratio of blue marbles to yellow marbles is \(5:3\).”
• “For every 5 blue marbles, there are 3 yellow marbles.”

Both sentences describe the same ratio.

The words for every are often helpful in ratio language. They show the multiplicative relationship clearly.

For example, the ratio \(2:7\) can be read as:

• “2 to 7”
• “the ratio of 2 to 7”
• “for every 2, there are 7”

Main Idea 4: Ratios show multiplicative thinking

A ratio is not just about the difference between two numbers. It is about how the numbers compare by multiplication.

For example, if the ratio of pencils to erasers is \(6:2\), that means:

• for every 6 pencils, there are 2 erasers
• the number of pencils is 3 times the number of erasers because \(6 \div 2 = 3\)

This is different from saying there are 4 more pencils than erasers. The ratio focuses on multiplication, not just subtraction.

Worked Example 1: Writing a part-to-part ratio

A fish tank has 7 goldfish and 5 angelfish. Write the ratio of goldfish to angelfish in three ways.

Step 1: Identify the order asked for: goldfish to angelfish.

Step 2: Use the numbers in that same order: 7 and 5.

Answer:

• \(7\) to \(5\)
• \(7:5\)
• \(\frac{7}{5}\)

We can also say, “For every 7 goldfish, there are 5 angelfish.”

Worked Example 2: Writing a part-to-whole ratio

A jar contains 9 red beads and 6 blue beads. What is the ratio of blue beads to all beads?

Step 1: Find the total number of beads.

Step 2: Compare blue beads to total beads.

Answer: The ratio of blue beads to all beads is \(6:15\).

We can say, “For every 15 beads in the jar, 6 are blue.”

Worked Example 3: Understanding ratio language carefully

At a pet shelter, there are 4 cats and 10 dogs.

1. What is the ratio of cats to dogs?
2. What is the ratio of dogs to cats?
3. What is the ratio of cats to all animals?

Step 1: Use the order from each question.

• Cats to dogs: \(4:10\)
• Dogs to cats: \(10:4\)

Step 2: Find the total animals for part-to-whole.

• Cats to all animals: \(4:14\)

Answer:

• Cats to dogs = \(4:10\)
• Dogs to cats = \(10:4\)
• Cats to all animals = \(4:14\)

This example shows why reading carefully matters. Each ratio compares different quantities.

Worked Example 4: Describing a ratio with words

A recipe uses 3 cups of flour and 2 cups of sugar.

Describe the ratio of flour to sugar using ratio language.

Step 1: Identify the quantities in order: flour, then sugar.

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

Step 3: Say it in words.

Answer: The ratio of flour to sugar is \(3:2\). For every 3 cups of flour, there are 2 cups of sugar.

Common Mistakes to Avoid

• Reversing the order: “Cats to dogs” is not the same as “dogs to cats.”
• Forgetting the whole: In part-to-whole ratios, add all parts first.
• Using subtraction instead of comparison: Ratios compare using multiplication.
• Being too vague: Always name what each number stands for.

Helpful Tips

• Read the ratio question slowly.
• Underline the order of the words being compared.
• Ask yourself: Is this part to part or part to whole?
• Use the phrase for every to check if your ratio makes sense.

Quick Practice Ideas

Try these on your own:

• A team has 11 girls and 9 boys. Write the ratio of girls to boys.
• A box has 8 chocolate cookies and 4 vanilla cookies. Write the ratio of vanilla cookies to total cookies.
• There are 6 white flowers and 12 red flowers. Say the ratio of white flowers to red flowers in words.

Summary

A ratio compares two quantities using multiplication. Ratios can compare part to part or part to whole. You can write ratios as “to,” with a colon, or as a fraction. Good ratio language tells exactly what is being compared and keeps the order correct.

When you see a ratio problem, always ask:

• What two quantities am I comparing?
• In what order?
• Is it part-to-part or part-to-whole?

If you can answer those questions, you are using ratio language correctly.

Equivalent Ratios and Ratio Tables

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

For example, if a class has 2 red markers and 3 blue markers, the ratio of red markers to blue markers is \(2:3\).

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

What does equivalent mean?

Equivalent ratios are ratios that name the same comparison, even though the numbers are different.

For example, \(2:3\), \(4:6\), and \(6:9\) are all equivalent ratios. They all compare the two amounts in the same way.

You can make equivalent ratios by multiplying or dividing both parts of the ratio by the same nonzero number.

Here is the idea with math:

From \(2:3\) to \(4:6\), both numbers were multiplied by \(2\).

From \(2:3\) to \(6:9\), both numbers were multiplied by \(3\).

If you only multiply one part of a ratio, the comparison changes, so the ratios are not equivalent.

For example, \(2:3\) and \(2:6\) are not equivalent, because only one number changed.

Why ratio tables are useful

A ratio table is a table that shows a list of equivalent ratios.

Ratio tables help you:

• see patterns,
• find missing values,
• solve real-world problems,
• think multiplicatively instead of additively.

Important idea: use multiplication, not addition

When working with equivalent ratios, we scale both parts by the same factor. This is multiplicative thinking.

That means we ask, “What number do I multiply by?”

We do not keep the ratios equivalent by adding the same amount over and over unless that addition comes from a matching multiplication pattern already shown in the table.

For example, starting with \(3:5\):

• Multiply both parts by \(2\): \(6:10\)
• Multiply both parts by \(3\): \(9:15\)
• Multiply both parts by \(4\): \(12:20\)

Each new ratio is equivalent to \(3:5\).

How to build a ratio table

Start with one ratio. Then multiply or divide both numbers by the same value.

Example starting ratio: \(2:5\)

In this table:

• \(2:5\) was multiplied by \(2\) to get \(4:10\)
• \(2:5\) was multiplied by \(3\) to get \(6:15\)
• \(2:5\) was multiplied by \(4\) to get \(8:20\)
• \(2:5\) was multiplied by \(5\) to get \(10:25\)

You can also move from one row to another by multiplying or dividing.

For example, from \(4:10\) to \(8:20\), both numbers are multiplied by \(2\).

How to find a missing value in a ratio table

To find a missing value, look for the scale factor. Ask yourself:

• What happened to the known number?
• Did it get multiplied or divided?
• Do I do the same thing to the other number?

Worked Example 1: Fill in a simple ratio table

The ratio of apples to baskets is \(3:1\). Complete the ratio table.

Step 1: Compare each number of apples to 3.

• \(6 = 3 \times 2\)
• \(9 = 3 \times 3\)
• \(12 = 3 \times 4\)

Step 2: Multiply the number of baskets by the same factors.

• \(1 \times 2 = 2\)
• \(1 \times 3 = 3\)
• \(1 \times 4 = 4\)

Completed table:

So the equivalent ratios are \(3:1\), \(6:2\), \(9:3\), and \(12:4\).

Worked Example 2: Find one missing number

The ratio of blue beads to white beads is \(4:7\). If there are 20 blue beads, how many white beads are there?

Step 1: Find the scale factor from 4 to 20.

So both parts of the ratio were multiplied by \(5\).

Step 2: Multiply 7 by 5.

Answer: There are \(35\) white beads.

We can show it in a ratio table:

Worked Example 3: Use repeated scaling in a ratio table

A recipe uses 2 cups of juice concentrate for every 3 cups of water. Complete the ratio table.

Step 1: Look at the row with 9 cups of water.

Since \(9 = 3 \times 3\), multiply the concentrate by \(3\) too.

So the missing number is \(6\).

Step 2: Look at the row with 8 cups of concentrate.

Since \(8 = 2 \times 4\), multiply the water by \(4\).

So the missing number is \(12\).

Completed table:

Worked Example 4: Solve a real-world problem

At a school event, the ratio of teachers to students is \(1:15\). If there are 6 teachers, how many students are there?

Step 1: Find how the number of teachers changed.

So the ratio was multiplied by \(6\).

Step 2: Multiply the number of students by \(6\).

Answer: There are \(90\) students.

Ratio table:

Tips for checking your work

• Make sure both parts of the ratio were multiplied or divided by the same number.
• Check that the comparison stays the same.
• If one value got bigger, the other value should grow by the same scale factor.
• If one value was divided, divide the other value by the same number.

Common mistakes to avoid

• Changing only one number: For example, turning \(3:4\) into \(6:4\) is not equivalent.
• Adding instead of scaling: Equivalent ratios come from multiplication or division by the same number.
• Using different operations on each part: If one side is multiplied by \(3\), the other side must also be multiplied by \(3\).

Quick practice thinking

Are these equivalent to \(5:8\)?

• \(10:16\) — yes, both parts were multiplied by \(2\)
• \(15:24\) — yes, both parts were multiplied by \(3\)
• \(10:18\) — no, the scale factor is not the same for both parts

Summary

A ratio compares two amounts. Equivalent ratios show the same comparison using different numbers.

You can make equivalent ratios by multiplying or dividing both parts of the ratio by the same number.

A ratio table is a helpful way to organize equivalent ratios and find missing values.

When solving ratio table problems, always look for the scale factor and use the same operation on both parts of the ratio.

Lesson: Tape Diagrams and Double Number Lines

When we compare two amounts using a ratio, we are showing how much of one thing there is compared to another. In 6th grade, two very helpful tools for ratio problems are tape diagrams and double number lines.

These tools help us see the relationship between amounts. They are especially useful when we need to find a missing part, a whole, or the total of two combined amounts.

In this lesson, you will learn how to use both models to solve problems step by step.

1. What is a tape diagram?

A tape diagram is a drawing made of equal-sized boxes or parts. Each part stands for the same amount. Tape diagrams help us show ratios clearly.

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

Red: [ ] [ ]

Blue: [ ] [ ] [ ]

This shows that for every 2 equal parts of red, there are 3 equal parts of blue.

If each part were worth 4 marbles, then:

• Red would have \(2 \times 4 = 8\) marbles
• Blue would have \(3 \times 4 = 12\) marbles

2. What is a double number line?

A double number line uses two number lines stacked on top of each other. Matching points line up to show equal ratios.

For the same ratio \(2:3\), a double number line could show:

Red: 2    4    6    8

Blue: 3   6   9   12

This means:

• 2 red matches 3 blue
• 4 red matches 6 blue
• 6 red matches 9 blue
• 8 red matches 12 blue

Both tape diagrams and double number lines show equivalent ratios. That means the relationship stays the same even when the amounts grow.

3. When should you use these models?

You can use tape diagrams and double number lines when:

• you know a ratio and need to find a missing amount
• you know the total and need to split it into parts
• you need to compare two related amounts
• you want to solve ratio problems in a visual way

4. How to solve using a tape diagram

1. Write the ratio.
2. Draw equal parts to match the ratio.
3. Use the given information to find the value of one part.
4. Multiply to find the missing amount.

5. How to solve using a double number line

1. Write the starting ratio on the two lines.
2. Count up by equal groups or scale both amounts by the same number.
3. Find the amount that matches the number you need.

Worked Example 1: Find a missing part with a tape diagram

The ratio of apples to oranges is \(3:2\). If there are 12 apples, how many oranges are there?

Step 1: Draw the ratio.

Apples: [ ] [ ] [ ]

Oranges: [ ] [ ]

Step 2: Use the known amount.

3 parts = 12 apples, so 1 part is:

Each part is 4.

Step 3: Find oranges.

Oranges have 2 parts, so:

Answer: There are \(8\) oranges.

Worked Example 2: Find a missing amount with a double number line

The ratio of juice mix to water is \(1:4\). If you use 3 cups of juice mix, how many cups of water do you need?

Start with the ratio:

Juice mix: 1   2   3

Water:      4   8   12

When juice mix is \(3\), water is \(12\).

Answer: You need \(12\) cups of water.

Worked Example 3: Find the total using a tape diagram

The ratio of boys to girls in a club is \(4:5\). There are 36 students in all. How many boys and how many girls are there?

Step 1: Draw the ratio.

Boys: [ ] [ ] [ ] [ ]

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

Step 2: Count the total parts.

There are 9 equal parts total.

Step 3: Find the value of one part.

Each part is 4 students.

Step 4: Find each group.

Answer: There are \(16\) boys and \(20\) girls.

Worked Example 4: Use a double number line to solve a bigger ratio problem

A recipe uses rice and beans in the ratio \(2:3\). If a cook uses 18 cups of beans, how many cups of rice are needed?

Step 1: Set up the ratio.

Rice:    2   4   6   8   10   12

Beans: 3   6   9   12   15   18

When beans are \(18\), rice is \(12\).

Answer: The cook needs \(12\) cups of rice.

6. Finding a whole from one part

Sometimes you know only one part of the ratio, and you need to find the total.

Example: The ratio of cats to dogs at a shelter is \(3:5\). If there are 15 cats, how many animals are there in all?

Step 1: 3 parts = 15, so 1 part is:

Step 2: Dogs have 5 parts, so dogs are:

Step 3: Total animals:

Answer: There are \(40\) animals in all.

7. Important idea: Equal groups

In both tape diagrams and double number lines, the parts must stay equal. If one amount is multiplied by a number, the other amount must be multiplied by the same number.

For example, if the ratio is \(2:5\), then these are equivalent ratios:

• \(2:5\)
• \(4:10\)
• \(6:15\)
• \(8:20\)

Each pair keeps the same relationship.

8. Common mistakes to avoid

• Do not add the ratio numbers to scale them. Ratios grow by multiplying.
• Do not use unequal boxes in a tape diagram. Each part must represent the same amount.
• Match the order carefully. If the ratio is apples to oranges, do not switch it to oranges to apples.
• Check the total parts before dividing when finding a whole.

9. Quick check for yourself

Ask these questions:

• What two things are being compared?
• What is the ratio?
• How many equal parts are there?
• Do I know the value of one part?
• Am I multiplying both parts by the same number?

Summary

Tape diagrams and double number lines are visual tools for solving ratio problems. A tape diagram uses equal boxes to show parts of a ratio, and a double number line shows matching values that grow in equal steps.

To solve problems, first write the ratio, then use the known amount to find the value of one part or scale both amounts equally. These models can help you find a missing part, a whole, or a total when two amounts are combined.

Unit Rates and Per-Unit Comparisons

Sometimes we compare two different kinds of quantities, like miles and hours, dollars and ounces, or pages and minutes. These comparisons are called rates.

A unit rate is a rate that compares a quantity to 1 unit of another quantity. For example:

• 60 miles in 1 hour
• $2 for 1 pound
• 15 words in 1 minute

Unit rates are helpful because they make comparisons easy. If two things have different sizes or times, finding the amount for 1 lets us see which is faster, cheaper, or more efficient.

How to find a unit rate

To find a unit rate, divide so that one quantity becomes 1.

If a car travels 180 miles in 3 hours, divide both numbers by 3:

So the unit rate is 60 miles per hour.

You can think of it as:

• total amount divided by number of units
• Then write the answer with the word per

For example:

• 12 cookies in 4 bags means $$12 \div 4 = 3$$ cookies per bag
• $15 for 5 notebooks means $$15 \div 5 = 3$$ dollars per notebook

Rates and unit rates in real life

We use unit rates all the time:

• At the store to compare prices
• On trips to compare speed
• In sports to compare points per game
• In reading to compare pages per day

If one box has 8 granola bars for $4 and another has 15 granola bars for $6, the box with more bars is not always the better deal. We need to find the cost for 1 bar or the bars for $1.

Two ways to compare

You can compare using either of these unit rates:

• cost per 1 item
• items per $1

Both ways work. Just be careful to use the same kind of unit rate for both choices.

For example, if you are comparing prices, you might use:

• dollars per ounce
• dollars per pound
• dollars per item

Worked Example 1: Finding a simple unit rate

A student reads 24 pages in 3 days. How many pages does the student read per day?

Step 1: Identify the total pages and total days.

• 24 pages
• 3 days

Step 2: Divide pages by days.

Answer: The student reads 8 pages per day.

Worked Example 2: Comparing speed

Bike A travels 36 miles in 3 hours. Bike B travels 50 miles in 5 hours. Which bike is faster?

Step 1: Find the unit rate for Bike A.

Bike A travels 12 miles per hour.

Step 2: Find the unit rate for Bike B.

Bike B travels 10 miles per hour.

Step 3: Compare the unit rates.

Since 12 is greater than 10, Bike A is faster.

Worked Example 3: Finding the better buy

Store A sells 6 juice boxes for $3. Store B sells 10 juice boxes for $4. Which is the better buy?

We will compare cost per juice box.

Store A:

Store A costs $0.50 per juice box.

Store B:

Store B costs $0.40 per juice box.

Since $0.40 is less than $0.50, Store B is the better buy.

Worked Example 4: Using a unit rate to solve a problem

A faucet fills 18 cups of water in 3 minutes. At this rate, how many cups will it fill in 1 minute? How many cups will it fill in 5 minutes?

Step 1: Find the unit rate.

The faucet fills 6 cups per minute.

Step 2: Use the unit rate.

In 1 minute, it fills:

In 5 minutes, it fills:

Answer: The faucet fills 6 cups in 1 minute and 30 cups in 5 minutes.

How to decide which answer is better

When comparing unit rates, think about the situation:

• For speed, a greater unit rate is usually better because it means faster.
• For price per item, a smaller unit rate is usually better because it means cheaper.
• For work done per hour, a greater unit rate may mean more done in less time.

Common mistakes to avoid

• Mixing up what to divide: Be sure you divide the total amount by the number of units you want to make 1.
• Comparing different units: Do not compare dollars per pound with dollars per ounce unless you change them to the same unit.
• Forgetting what the number means: Always include the label, like miles per hour or dollars per item.
• Choosing the wrong “better” answer: Bigger is not always better. For cost, smaller is better.

Helpful steps for solving unit rate problems

1. Read the problem carefully.
2. Identify the two quantities being compared.
3. Decide which quantity should be compared to 1.
4. Divide to find the unit rate.
5. Write the answer with units.
6. Compare unit rates if the problem asks which is better.

Quick check

• 20 dollars for 5 tickets means $$20 \div 5 = 4$$ dollars per ticket
• 42 miles in 6 hours means $$42 \div 6 = 7$$ miles per hour
• 16 ounces for $4 means $$4 \div 16 = 0.25$$ dollars per ounce

Summary

A rate compares two different kinds of quantities. A unit rate compares a quantity to 1 unit of another quantity.

To find a unit rate, divide the total amount by the number of units. Then use that per-unit value to compare prices, speeds, and other real-world situations.

Unit rates help you make fair comparisons. They show which choice is faster, cheaper, or more efficient.

Graphing Proportional Relationships

Sometimes two quantities change together in a special way. When they do, they can form a proportional relationship. In a proportional relationship, the ratio between the two quantities stays the same.

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Solving Basic Proportions

Sometimes two ratios compare things in the same way. When that happens, they form a proportion.

A proportion is an equation that says two ratios are equal. For example, if 2 apples cost $4, then 4 apples would cost $8 if the price stays the same. We can write that as:

\(\frac{2}{4} = \frac{4}{8}\)

Both ratios show the same relationship, so they are proportional.

Learning to solve basic proportions helps you find a missing number when two ratios match. This is useful when working with prices, recipes, maps, and groups of objects.

What does a proportion look like?

A basic proportion often looks like this:

$$\frac{a}{b} = \frac{c}{d}$$

One of the numbers may be missing. Your job is to figure it out.

For example:

$$\frac{3}{6} = \frac{x}{12}$$

This means: 3 is to 6 in the same way that \(x\) is to 12.

Important idea: keep the ratios equivalent

To solve a proportion, think about how one ratio changes into the other. If the bottom number is multiplied by 2, then the top number must also be multiplied by 2.

In equivalent ratios:

• multiply the top and bottom by the same number, or
• divide the top and bottom by the same number.

That keeps the relationship the same.

Method 1: Use scaling

Scaling means finding how one ratio grows or shrinks.

Look at this proportion:

$$\frac{2}{5} = \frac{x}{15}$$

Ask yourself: how did 5 become 15?

Since \(5 \times 3 = 15\), multiply the top by 3 too.

$$x = 2 \times 3 = 6$$

So:

$$\frac{2}{5} = \frac{6}{15}$$

Method 2: Find the unit rate

A unit rate tells the amount for 1 unit. This can help when the scaling is not easy to see right away.

Example: If 4 notebooks cost $12, how much do 8 notebooks cost?

First find the cost of 1 notebook:

$$12 \div 4 = 3$$

So 1 notebook costs $3.

Now find the cost of 8 notebooks:

$$8 \times 3 = 24$$

So 8 notebooks cost $24.

We can also write the proportion as:

$$\frac{4}{12} = \frac{8}{x}$$

Then \(x = 24\).

Method 3: Use the cross-products idea

There is also a shortcut for solving proportions. In a true proportion, the products across the diagonals are equal.

For example:

$$\frac{3}{4} = \frac{6}{8}$$

Check the diagonals:

$$3 \times 8 = 24$$

$$4 \times 6 = 24$$

Since both products are equal, the ratios form a proportion.

You can use this idea to find a missing value too. For 5th Grade, it is best to think of this as a pattern that helps check your answer.

Steps for solving a basic proportion

1. Write the two equal ratios.
2. Find what happened from one ratio to the other.
3. Use the same change on the missing number.
4. Check that the new ratios are equivalent.

Worked Example 1: Easy scaling

Solve:

$$\frac{1}{3} = \frac{x}{9}$$

Look at the bottom numbers: \(3\) became \(9\).

$$3 \times 3 = 9$$

So multiply the top by 3 too:

$$x = 1 \times 3 = 3$$

Answer:

$$x = 3$$

Check:

$$\frac{1}{3} = \frac{3}{9}$$

Both ratios are equal, so the answer is correct.

Worked Example 2: Missing number in the first ratio

Solve:

$$\frac{x}{4} = \frac{6}{12}$$

Look at the second ratio first. Can it be simplified?

$$\frac{6}{12} = \frac{1}{2}$$

Now think: what number over 4 equals \(\frac{1}{2}\)?

Half of 4 is 2, so:

$$x = 2$$

Check:

$$\frac{2}{4} = \frac{6}{12}$$

Both equal \(\frac{1}{2}\).

Worked Example 3: Use a real-world problem

3 juice boxes cost $6. How much do 9 juice boxes cost?

Write a proportion:

$$\frac{3}{6} = \frac{9}{x}$$

You can solve by scaling.

The number of juice boxes went from 3 to 9.

$$3 \times 3 = 9$$

So the cost must also be multiplied by 3.

$$x = 6 \times 3 = 18$$

Answer: $18

You can also use the unit rate:

$$6 \div 3 = 2$$

Each juice box costs $2.

$$9 \times 2 = 18$$

Same answer.

Worked Example 4: A proportion with a less obvious scale

Solve:

$$\frac{4}{6} = \frac{x}{9}$$

Ask: how did 6 become 9?

It is not a whole-number multiplication fact you may know right away, so simplify the first ratio first.

$$\frac{4}{6} = \frac{2}{3}$$

Now solve:

$$\frac{2}{3} = \frac{x}{9}$$

Since \(3 \times 3 = 9\), multiply the top by 3.

$$x = 2 \times 3 = 6$$

Answer:

$$x = 6$$

Check:

$$\frac{4}{6} = \frac{6}{9}$$

Both simplify to \(\frac{2}{3}\).

Helpful strategies

• Look for multiplication or division: See how one part of the ratio changes.
• Simplify first: Sometimes reducing a ratio makes the pattern easier to see.
• Use a unit rate: Find the amount for 1 unit, then build back up.
• Check your answer: Make sure both ratios are equivalent.

Common mistakes to avoid

• Changing only one number: If you multiply or divide, do it to both parts of the ratio.
• Mixing up the order: Keep the compared quantities in the same order.
• Forgetting to check: Your answer should make both sides equal.

Quick practice to think about

• \(\frac{2}{7} = \frac{x}{14}\)
• \(\frac{5}{10} = \frac{x}{6}\)
• 4 pencils cost $8. How much do 10 pencils cost?

Brief summary

A proportion shows that two ratios are equal. To solve a basic proportion, use scaling, simplify ratios, or find a unit rate.

Always keep the relationship the same by changing both parts of the ratio in the same way. When you finish, check that the two ratios are still equivalent.

Percent as a Rate per Hundred

Have you seen a store sign that says 25% off or a test score of 90%? These numbers use percent.

A percent means “per hundred”. The word cent means 100, so percent tells us how many parts out of 100 we are talking about.

This means that:

• \(1\%\) means \(1\) out of \(100\)

• \(25\%\) means \(25\) out of \(100\)

• \(100\%\) means \(100\) out of \(100\), or the whole thing

So, percent is a special kind of ratio. It compares a number to 100.

We can write percent in different ways:

• As a percent: \(45\%\)

• As a fraction: \(\frac{45}{100}\)

• As words: 45 per hundred

These all mean the same thing.

Using a 10 by 10 grid

A 10 by 10 grid has:

That means the grid has 100 small squares.

Each small square stands for 1 out of 100, so each square is worth:

This makes a 10 by 10 grid a great model for percents.

For example:

• If 12 squares are shaded, that is \(12\%\).

• If 50 squares are shaded, that is \(50\%\).

• If all 100 squares are shaded, that is \(100\%\).

Connecting percent and fractions

Since percent means “out of 100,” we can write any percent as a fraction with denominator 100.

For example:

This fraction can sometimes be simplified.

Both fractions still represent the same amount.

Important benchmark percents

Some percents are especially useful to know.

• \(10\% = \frac{10}{100} = \frac{1}{10}\)

• \(25\% = \frac{25}{100} = \frac{1}{4}\)

• \(50\% = \frac{50}{100} = \frac{1}{2}\)

• \(75\% = \frac{75}{100} = \frac{3}{4}\)

• \(100\% = \frac{100}{100} = 1\)

Knowing these can help you quickly understand parts of a whole.

Worked Example 1: Reading a percent from a grid

A 10 by 10 grid has 18 shaded squares. What percent of the grid is shaded?

Step 1: Count the shaded squares.

There are \(18\) shaded squares.

Step 2: Compare to 100 total squares.

Step 3: Write as a percent.

Answer: \(18\%\) of the grid is shaded.

Worked Example 2: Writing a percent as a fraction

Write \(64\%\) as a fraction.

Step 1: Use the meaning of percent: per hundred.

Step 2: Simplify if possible.

Answer: \(64\%\) is \(\frac{64}{100}\), which simplifies to \(\frac{16}{25}\).

Worked Example 3: Writing a fraction as a percent

Write \(\frac{37}{100}\) as a percent.

Step 1: Notice the denominator is already 100.

This means the fraction already shows “per hundred.”

Step 2: Write the numerator with the percent sign.

Answer: \(\frac{37}{100}\) is \(37\%\).

Worked Example 4: Thinking about part of a whole

In a class survey, 100 students were asked whether they liked reading. If 73 students said yes, what percent liked reading?

Step 1: Write the ratio out of 100.

Step 2: Change it to a percent.

Answer: \(73\%\) of the students liked reading.

How to think about percents

• Percent always compares to \(100\).

• A 10 by 10 grid helps you see that each small square is \(1\%\).

• If half of the grid is shaded, that is \(50\%\).

• If one fourth of the grid is shaded, that is \(25\%\).

Common mistakes to avoid

• Do not forget that percent means out of 100.

• Do not write \(30\%\) as \(\frac{30}{10}\). It should be \(\frac{30}{100}\).

• Do not think \(100\%\) means more than the whole. It means the entire whole.

• Remember that each square in a 10 by 10 grid is only \(1\%\), not \(10\%\).

Quick check

1. What does \(15\%\) mean? Answer: \(15\) out of \(100\)

2. If 42 squares are shaded on a 10 by 10 grid, what percent is shaded? Answer: \(42\%\)

3. Write \(9\%\) as a fraction. Answer: \(\frac{9}{100}\)

4. Write \(\frac{88}{100}\) as a percent. Answer: \(88\%\)

Summary

A percent is a rate per hundred. It tells how many parts there are out of \(100\).

A 10 by 10 grid helps us see percents clearly because it has exactly \(100\) squares. Each square represents \(1\%\).

When you see a percent, you can think of it as a fraction with denominator \(100\). This helps you connect ratios, fractions, and percents.

Converting Among Fractions, Decimals, and Percents

Fractions, decimals, and percents are three different ways to show the same amount. In 6th grade math, it is important to move easily between these forms so you can choose the one that makes a problem easier to solve.

For example, the numbers \(\frac{1}{2}\), \(0.5\), and \(50\%\) all represent the same value. They just look different.

In this lesson, you will learn how to:

• convert a fraction to a decimal,
• convert a decimal to a percent,
• convert a percent to a fraction,
• and check whether different forms are equal.

1. What each form means

A fraction shows a part of a whole using a numerator and denominator, like \(\frac{3}{4}\).

A decimal shows a part of a whole using place value, like \(0.75\).

A percent means “out of 100.” The symbol \(%\) means per hundred.

So:

• \(\frac{3}{4}\)
• \(0.75\)
• \(75\%\)

all describe the same amount.

2. Converting a fraction to a decimal

To change a fraction into a decimal, divide the numerator by the denominator.

For a fraction \(\frac{a}{b}\), compute:

$$\frac{a}{b} = a \div b$$

Example: To convert \(\frac{3}{4}\) to a decimal, divide:

$$3 \div 4 = 0.75$$

So:

$$\frac{3}{4} = 0.75$$

Some fractions make decimals that end, like \(0.2\) or \(0.75\). Others make decimals that continue, like \(0.333\ldots\).

3. Converting a decimal to a percent

To change a decimal into a percent, multiply by 100. This is the same as moving the decimal point two places to the right.

Then add the percent symbol.

Example:

$$0.75 \times 100 = 75$$

So:

$$0.75 = 75\%$$

Another example:

$$0.4 \times 100 = 40$$

So:

$$0.4 = 40\%$$

4. Converting a percent to a decimal

To change a percent into a decimal, divide by 100. This is the same as moving the decimal point two places to the left.

Do not keep the percent symbol when you write the decimal.

Example:

$$65\% = \frac{65}{100} = 0.65$$

Another example:

$$8\% = \frac{8}{100} = 0.08$$

5. Converting a fraction to a percent

There are two good methods for changing a fraction to a percent.

Method A: Fraction to decimal, then decimal to percent

1. Divide numerator by denominator.
2. Multiply the decimal by 100.

Example with \(\frac{1}{5}\):

$$\frac{1}{5} = 1 \div 5 = 0.2$$

$$0.2 = 20\%$$

Method B: Make an equivalent fraction with denominator 100

If possible, rewrite the fraction so the denominator is 100.

Example with \(\frac{3}{4}\):

$$\frac{3}{4} = \frac{75}{100} = 75\%$$

This works because \(4 \times 25 = 100\), so multiply both numerator and denominator by 25.

6. Converting a percent to a fraction

To change a percent into a fraction, write the percent over 100, then simplify if possible.

Example:

$$45\% = \frac{45}{100}$$

Now simplify by dividing top and bottom by 5:

$$\frac{45}{100} = \frac{9}{20}$$

So:

$$45\% = \frac{9}{20}$$

Another example:

$$25\% = \frac{25}{100} = \frac{1}{4}$$

7. Converting a decimal to a fraction

To change a decimal to a fraction, use place value.

If the decimal has:

• 1 digit after the decimal point, write it over 10,
• 2 digits after the decimal point, write it over 100,
• 3 digits after the decimal point, write it over 1000.

Then simplify if possible.

Example:

$$0.6 = \frac{6}{10} = \frac{3}{5}$$

Another example:

$$0.35 = \frac{35}{100} = \frac{7}{20}$$

8. A helpful conversion chart

• Fraction to Decimal: divide
• Decimal to Fraction: write using place value, then simplify
• Decimal to Percent: multiply by 100
• Percent to Decimal: divide by 100
• Percent to Fraction: write over 100, then simplify
• Fraction to Percent: divide to get a decimal, then multiply by 100

9. Worked Examples

Example 1: Convert \(\frac{1}{2}\) to a decimal and a percent.

First, divide the numerator by the denominator:

$$1 \div 2 = 0.5$$

So the decimal is \(0.5\).

Now change the decimal to a percent:

$$0.5 \times 100 = 50$$

So the percent is \(50\%\).

Final answer:

$$\frac{1}{2} = 0.5 = 50\%$$

Example 2: Convert \(0.32\) to a fraction and a percent.

Since \(0.32\) has two digits after the decimal point, write it over 100:

$$0.32 = \frac{32}{100}$$

Simplify the fraction by dividing by 4:

$$\frac{32}{100} = \frac{8}{25}$$

Now convert the decimal to a percent:

$$0.32 \times 100 = 32$$

So:

$$0.32 = \frac{8}{25} = 32\%$$

Example 3: Convert \(60\%\) to a decimal and a fraction.

To make a decimal, divide by 100:

$$60\% = 0.60 = 0.6$$

To make a fraction, write it over 100:

$$60\% = \frac{60}{100}$$

Simplify by dividing by 20:

$$\frac{60}{100} = \frac{3}{5}$$

Final answer:

$$60\% = 0.6 = \frac{3}{5}$$

Example 4: Convert \(\frac{7}{8}\) to a decimal and a percent.

Divide numerator by denominator:

$$7 \div 8 = 0.875$$

Now change the decimal to a percent:

$$0.875 \times 100 = 87.5$$

So:

$$\frac{7}{8} = 0.875 = 87.5\%$$

This example shows that percents do not always have to be whole numbers.

10. Common mistakes to avoid

• Do not forget the percent symbol. Writing \(45\) is not the same as writing \(45\%\).
• Move the decimal the correct way. Decimal to percent: move right 2 places. Percent to decimal: move left 2 places.
• Simplify fractions when needed. For example, \(\frac{50}{100}\) should be simplified to \(\frac{1}{2}\).
• Remember that percent means out of 100. This helps when turning percents into fractions.

11. How to choose the best form

Different forms are useful in different situations.

• Use fractions when comparing parts of a whole exactly, like \(\frac{3}{4}\).
• Use decimals when working with money or doing calculator work, like \(0.75\).
• Use percents when talking about parts out of 100, like discounts, grades, and surveys.

The more you practice, the easier it becomes to switch between them.

Summary

Fractions, decimals, and percents can all name the same value. To convert between them, remember these key ideas: divide to change a fraction to a decimal, use place value to change a decimal to a fraction, multiply by 100 to change a decimal to a percent, and write percents over 100 to make fractions.

If you can recognize that \(\frac{1}{4}\), \(0.25\), and \(25\%\) are equal, you are building strong number sense. This skill will help you with ratios, rates, and many real-world math problems.

Calculating the Percent of a Quantity

Percent means "out of 100". When we see a percent, we can think of it as a part of a whole.

For example, 25% means 25 out of 100, and 50% means 50 out of 100.

In this lesson, you will learn how to find a percent of a number by changing the percent into a decimal or a fraction, and then multiplying.

Why do we learn this?

Finding a percent of a quantity is useful in everyday life. We use it to calculate discounts in stores, tips at restaurants, test scores, and parts of groups.

Main Idea

To find a percent of a quantity:

1. Change the percent to a decimal or fraction.
2. Multiply by the whole amount.

So the basic rule is:

$$\text{percent of a quantity} = \text{percent} \times \text{whole}$$

When using a decimal, remember:

• Divide the percent by 100.
• Or move the decimal point in the percent two places left.

Here are some common percent forms:

• \(10\% = 0.10 = \frac{10}{100} = \frac{1}{10}\)
• \(25\% = 0.25 = \frac{25}{100} = \frac{1}{4}\)
• \(50\% = 0.50 = \frac{50}{100} = \frac{1}{2}\)
• \(75\% = 0.75 = \frac{75}{100} = \frac{3}{4}\)
• \(20\% = 0.20 = \frac{20}{100} = \frac{1}{5}\)

Method 1: Use a Decimal

If you want to find \(30\%\) of \(50\), first change \(30\%\) to a decimal.

$$30\% = 0.30$$

Then multiply:

$$0.30 \times 50 = 15$$

So, \(30\%\) of \(50\) is 15.

Method 2: Use a Fraction

If you want to find \(25\%\) of \(36\), change \(25\%\) to a fraction.

$$25\% = \frac{25}{100} = \frac{1}{4}$$

Then multiply:

$$\frac{1}{4} \times 36 = 9$$

So, \(25\%\) of \(36\) is 9.

Worked Examples

Example 1: Find \(10\%\) of \(80\)

Step 1: Change the percent to a decimal.

$$10\% = 0.10$$

Step 2: Multiply by the whole number.

$$0.10 \times 80 = 8$$

Answer: \(10\%\) of \(80\) is 8.

Example 2: Find \(50\%\) of \(34\)

Since \(50\% = \frac{1}{2}\), we can find half of 34.

$$\frac{1}{2} \times 34 = 17$$

Answer: \(50\%\) of \(34\) is 17.

Example 3: Find \(25\%\) of \(64\)

Step 1: Change \(25\%\) to a fraction.

$$25\% = \frac{1}{4}$$

Step 2: Multiply.

$$\frac{1}{4} \times 64 = 16$$

Answer: \(25\%\) of \(64\) is 16.

Example 4: Find \(15\%\) of \(120\)

Step 1: Change the percent to a decimal.

$$15\% = 0.15$$

Step 2: Multiply.

$$0.15 \times 120 = 18$$

Answer: \(15\%\) of \(120\) is 18.

Thinking About Reasonableness

It is always smart to ask, "Does my answer make sense?"

• If the percent is less than \(100\%\), the answer should usually be less than the whole amount.
• If the percent is \(50\%\), the answer should be half of the whole.
• If the percent is small, like \(5\%\), the answer should be much smaller than the whole.

For example, \(10\%\) of \(200\) should be much less than \(200\). Since \(0.10 \times 200 = 20\), the answer makes sense.

Helpful Shortcuts

Some percents are easy to find using what you already know:

• \(50\%\) means half
• \(25\%\) means one-fourth
• \(10\%\) means divide by 10
• \(1\%\) means divide by 100

These shortcuts can help you solve problems quickly.

For example, to find \(20\%\) of \(60\):

First find \(10\%\) of \(60\), which is \(6\). Then double it.

$$20\% \text{ of } 60 = 12$$

Common Mistakes to Avoid

• Do not leave the percent as a whole number when multiplying. For example, use \(0.25\), not \(25\).
• Be careful when moving the decimal point. \(7\% = 0.07\), not \(0.7\).
• Make sure you multiply the percent by the whole quantity.

Try These in Your Head

• \(50\%\) of \(20\) is \(10\)
• \(25\%\) of \(20\) is \(5\)
• \(10\%\) of \(90\) is \(9\)
• \(5\%\) of \(80\) is \(4\)

Summary

To calculate the percent of a quantity, change the percent into a decimal or fraction and multiply by the whole amount.

Remember:

$$\text{part} = \text{percent} \times \text{whole}$$

With practice, you will get faster at recognizing common percents like \(10\%\), \(25\%\), \(50\%\), and \(75\%\). These are very helpful when solving real-life problems.

Finding the Whole Given a Part and Percent means working backward.

Sometimes a problem tells you a part of something and what percent that part is. Your job is to find the whole, or the full amount.

For example, if 15 is 30% of a number, what is the number? We know the part, and we know the percent, but we do not know the whole.

This lesson will show you how to find the whole by using percent ideas, equations, and ratio tables.

First, remember what percent means.

The word percent means out of 100. So:

• 20% means 20 out of 100
• 50% means 50 out of 100
• 100% means the whole amount

If you know that a part is some percent of the whole, you can think like this:

$$\text{part} = \text{percent} \times \text{whole}$$

To find the whole, work backward:

$$\text{whole} = \frac{\text{part}}{\text{percent as a decimal or fraction}}$$

Important: Before dividing, make sure the percent is written in a useful form.

• As a decimal: 25% = \(0.25\)
• As a fraction: 25% = \(\frac{25}{100} = \frac{1}{4}\)

You may use either form, depending on which feels easier.

Method 1: Use an equation

If you do not know the whole, use a letter such as \(w\) for the whole.

Then write:

$$\text{part} = \text{percent} \times w$$

After that, solve for \(w\) by dividing.

Method 2: Use a ratio table

A ratio table helps you connect the percent to the amount.

If you know one percent-amount pair, you can scale up or down until you reach 100%.

For example:

• If 25% is 10
• Then 100% is 4 times as much
• So the whole is \(10 \times 4 = 40\)

This works because \(100\%\) is the full amount.

Worked Example 1: A simple benchmark percent

12 is 50% of what number?

Step 1: Let the whole be \(w\).

$$12 = 50\% \times w$$

Step 2: Change 50% to a decimal or fraction.

$$50\% = 0.5 = \frac{1}{2}$$

Step 3: Solve.

Using decimals:

$$12 = 0.5w$$

$$w = \frac{12}{0.5} = 24$$

Or using fractions:

$$12 = \frac{1}{2}w$$

If 12 is half of the whole, the whole is 24.

Answer: 12 is 50% of 24.

Check: Is 50% of 24 equal to 12?

$$0.5 \times 24 = 12$$

Yes.

Worked Example 2: A percent that is not a benchmark

18 is 30% of what number?

Step 1: Write an equation.

$$18 = 30\% \times w$$

Step 2: Change 30% to a decimal.

$$30\% = 0.30$$

Step 3: Solve.

$$18 = 0.30w$$

$$w = \frac{18}{0.30} = 60$$

Answer: 18 is 30% of 60.

Check:

$$0.30 \times 60 = 18$$

It works.

Worked Example 3: Using a ratio table

35 is 70% of what number?

You can solve this with a ratio table.

Start with what you know:

• 70% → 35

Now think: how do we get from 70% to 100%?

One easy way is to first find 10%.

If 70% is 35, then divide by 7:

• 10% → 5

Then multiply by 10 to get 100%:

• 100% → 50

Answer: 35 is 70% of 50.

Check:

$$0.70 \times 50 = 35$$

Worked Example 4: A word problem

A student finished 24 problems, which was 80% of the homework. How many problems were in the whole homework assignment?

Step 1: Identify the part and the percent.

• Part = 24
• Percent = 80%

Step 2: Write an equation.

$$24 = 80\% \times w$$

Step 3: Change the percent to a decimal.

$$80\% = 0.8$$

Step 4: Solve.

$$24 = 0.8w$$

$$w = \frac{24}{0.8} = 30$$

Answer: There were 30 problems in the whole assignment.

Check:

$$0.8 \times 30 = 24$$

A helpful way to think about it

When the percent is less than 100%, the part is smaller than the whole.

So your answer for the whole should be greater than the part.

For example, if 18 is 30% of the whole, the whole must be bigger than 18. A whole of 60 makes sense.

Common mistakes to avoid

• Do not treat the percent like a whole number. For example, 25% is not 25 in the equation. It is \(0.25\) or \(\frac{25}{100}\).
• Do not forget that 100% means the whole.
• Do not multiply the part by the percent when you are trying to find the whole. Usually, you need to divide by the percent.
• Check whether your whole is bigger than the part when the percent is less than 100%.

Steps you can always use

1. Find the part.
2. Find the percent.
3. Let the whole be a variable, like \(w\).
4. Write the equation: $$\text{part} = \text{percent} \times \text{whole}$$
5. Change the percent to a decimal or fraction.
6. Divide to solve for the whole.
7. Check your answer by multiplying the whole by the percent.

Quick practice thinking

If 9 is 25% of a number, then the whole is 4 times as much, because 25% is \(\frac{1}{4}\) of the whole.

So the whole is:

$$9 \times 4 = 36$$

If 40 is 20% of a number, then 20% is one fifth of the whole.

So the whole is:

$$40 \times 5 = 200$$

Summary

To find the whole when you know a part and a percent, remember that the part equals the percent times the whole.

Write an equation, change the percent to a decimal or fraction, and divide to find the whole.

You can also use a ratio table to scale up to 100%.

Always check your answer by seeing whether the percent of your whole gives the part you started with.

Real-World Percent Applications help us use math in everyday life. Percents appear when we shop, eat at restaurants, save money, or borrow money. In this lesson, you will learn how to use percents to solve problems about sales tax, tips, markups, discounts, and simple interest.

A percent means “out of 100.” For example, \(25\%\) means \(25\) out of \(100\), or \(\frac{25}{100}\). It can also be written as the decimal \(0.25\).

When solving real-world percent problems, it helps to follow the same plan every time.

1. Find the percent amount by multiplying the original amount by the percent written as a decimal.
2. Decide whether to add or subtract that amount.
3. Check if the problem has more than one step.

Here is the basic percent formula:

$$\text{percent amount} = \text{original amount} \times \text{percent as a decimal}$$

To change a percent to a decimal, divide by \(100\), or move the decimal point two places left.

• \(8\% = 0.08\)
• \(15\% = 0.15\)
• \(25\% = 0.25\)
• \(125\% = 1.25\)

Let’s look at the most common real-world percent situations.

1. Sales Tax

Sales tax is an extra amount added to the price of an item. If an item costs \(\$20\) and the sales tax is \(5\%\), you do not find \(5\%\) of the new total. You find \(5\%\) of the original price.

Then you add the tax to the original price.

$$\text{total cost} = \text{original price} + \text{sales tax}$$

2. Tips

A tip is extra money paid for service, such as at a restaurant. A tip is found by taking a percent of the bill before the tip is added.

Then you add the tip to the bill.

$$\text{total bill} = \text{meal cost} + \text{tip}$$

3. Discounts

A discount means the price goes down. If a store gives \(20\%\) off, first find \(20\%\) of the original price. Then subtract that amount.

$$\text{sale price} = \text{original price} - \text{discount}$$

4. Markups

A markup means the price goes up. Stores often increase a price by a certain percent to make a profit.

First find the markup amount, then add it to the original cost.

$$\text{selling price} = \text{original cost} + \text{markup}$$

5. Simple Interest

Simple interest is money earned or paid based on the original amount of money. The original amount is called the principal.

The simple interest formula is:

$$I = P \times r \times t$$

• \(I\) = interest
• \(P\) = principal
• \(r\) = yearly interest rate written as a decimal
• \(t\) = time in years

After finding the interest, add it to the principal to find the total amount.

$$\text{total amount} = P + I$$

Important idea: use the original amount unless the problem says otherwise. In 6th Grade percent problems, tax, tips, discounts, and simple interest are usually based on the starting amount.

Worked Example 1: Finding sales tax

A backpack costs \(\$36\). The sales tax is \(8\%\). What is the total cost?

Step 1: Find the tax.

Change \(8\%\) to a decimal: \(0.08\)

$$36 \times 0.08 = 2.88$$

The tax is \(\$2.88\).

Step 2: Add the tax to the original price.

$$36 + 2.88 = 38.88$$

Answer: The total cost is \(\$38.88\).

Worked Example 2: Finding a tip

A family’s restaurant bill is \(\$24\). They want to leave a \(15\%\) tip. How much tip should they leave, and what will the total be?

Step 1: Find the tip.

Change \(15\%\) to \(0.15\).

$$24 \times 0.15 = 3.60$$

The tip is \(\$3.60\).

Step 2: Add the tip to the bill.

$$24 + 3.60 = 27.60$$

Answer: The tip is \(\$3.60\), and the total bill is \(\$27.60\).

Worked Example 3: Discount and sales tax in one problem

A jacket costs \(\$50\). It is on sale for \(20\%\) off. After the discount, a \(6\%\) sales tax is added. What is the final price?

This is a multi-step problem. Work in order.

Step 1: Find the discount.

Change \(20\%\) to \(0.20\).

$$50 \times 0.20 = 10$$

The discount is \(\$10\).

Step 2: Find the sale price.

$$50 - 10 = 40$$

The sale price is \(\$40\).

Step 3: Find the sales tax on the sale price.

Change \(6\%\) to \(0.06\).

$$40 \times 0.06 = 2.40$$

The tax is \(\$2.40\).

Step 4: Add the tax.

$$40 + 2.40 = 42.40$$

Answer: The final price is \(\$42.40\).

Notice: The tax is found after the discount, not from the original \(\$50\). Always pay attention to the order of the steps.

Worked Example 4: Simple interest

Lena puts \(\$200\) in a savings account that earns \(4\%\) simple interest each year. How much interest will she earn in \(3\) years? What will the total amount be?

Step 1: Use the simple interest formula.

$$I = P \times r \times t$$

Here, \(P = 200\), \(r = 0.04\), and \(t = 3\).

$$I = 200 \times 0.04 \times 3$$

$$I = 8 \times 3 = 24$$

The interest is \(\$24\).

Step 2: Add the interest to the principal.

$$200 + 24 = 224$$

Answer: Lena earns \(\$24\) in interest, and the total amount is \(\$224\).

How to tell whether to add or subtract

• Add for sales tax, tips, markups, and interest.
• Subtract for discounts.

Common mistakes to avoid

• Forgetting to change the percent to a decimal.
• Adding when you should subtract, or subtracting when you should add.
• Using the wrong starting amount in a multi-step problem.
• Stopping after finding only the percent amount instead of the final total.

Helpful strategy

Ask yourself these questions:

1. What is the original amount?
2. What percent am I finding?
3. What is that percent as a decimal?
4. Do I add or subtract?
5. Is there another step after this one?

Quick Practice Ideas

• A bike helmet costs \(\$30\) and has \(10\%\) tax. Find the total cost.
• A haircut costs \(\$18\) and you leave a \(20\%\) tip. Find the tip and total.
• A game costs \(\$40\) and is \(25\%\) off. Find the sale price.
• You borrow \(\$100\) at \(5\%\) simple interest for \(2\) years. Find the interest.

Summary

Real-world percent problems often use the same idea: first find the percent amount by multiplying the original amount by the percent as a decimal. Then decide whether the situation means add, subtract, or continue to another step.

Use addition for tax, tips, markups, and interest. Use subtraction for discounts. In multi-step problems, be careful about the order, because each step can change the amount you use next.