Ratios, Rates, and Proportions

Ratio Concepts and Notation

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

For example, if there are 3 red apples and 5 green apples, we can compare red apples to green apples with the ratio 3 to 5.

Ratios are important because they help us describe groups, recipes, teams, maps, mixtures, and many real-life situations.

Ways to write a ratio

The same ratio can be written in 3 different ways:

• with words: 3 to 5
• with a colon: 3:5
• as a fraction: \(\frac{3}{5}\)

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

Order matters in ratios

When writing a ratio, the order tells what is being compared first and what is being compared second.

For example, if a class has 12 boys and 15 girls:

• boys to girls = 12:15
• girls to boys = 15:12

These are not the same ratio, because they compare different things in different order.

Part-to-part and part-to-whole ratios

A ratio can compare:

• part to part: one part of a group to another part of the group
• part to whole: one part of a group to the total number in the group

Suppose a bag has 4 blue marbles and 6 yellow marbles. There are 10 marbles in all.

• blue to yellow = 4:6 (part to part)
• blue to total = 4:10 (part to whole)
• yellow to total = 6:10 (part to whole)

It is very important to read the question carefully so you know which comparison is being asked.

Simplifying ratios

Just like fractions, ratios can often be simplified. To simplify a ratio, divide both numbers by the same greatest common factor.

For example, simplify 8:12.

Both 8 and 12 can be divided by 4:

So the simplified ratio is 2:3.

If a ratio is written as a fraction, we simplify it the same way:

Important note: simplifying a ratio does not change the comparison. It only writes it in a simpler form.

Worked Example 1: Writing a ratio in different forms

A basket has 7 oranges and 9 bananas. Write the ratio of oranges to bananas in three forms.

Step 1: Identify the comparison asked for: oranges to bananas.

There are 7 oranges and 9 bananas, so the ratio is:

• in words: 7 to 9
• with a colon: 7:9
• as a fraction: \(\frac{7}{9}\)

Answer: 7 to 9, 7:9, and \(\frac{7}{9}\)

Worked Example 2: Part-to-part and part-to-whole

A classroom has 8 desks in one row and 12 desks in another row. There are 20 desks total.

Find:

1. the ratio of first-row desks to second-row desks
2. the ratio of first-row desks to total desks

Step 1: Find the first comparison.

First-row desks to second-row desks is 8:12.

Simplify by dividing both numbers by 4:

Step 2: Find the second comparison.

First-row desks to total desks is 8:20.

Simplify by dividing both numbers by 4:

Answer:

• first row to second row = 2:3
• first row to total = 2:5

Worked Example 3: Finding the correct order

A pet store has 6 cats and 14 fish.

Write each ratio in simplest form:

1. cats to fish
2. fish to cats
3. cats to total animals

Step 1: Find the total number of animals.

Step 2: Write each ratio.

1. Cats to fish

Simplify by dividing by 2:

2. Fish to cats

Simplify by dividing by 2:

3. Cats to total animals

Simplify by dividing by 2:

Answer:

• cats to fish = 3:7
• fish to cats = 7:3
• cats to total = 3:10

Worked Example 4: Using words to understand a ratio

A recipe uses 2 cups of juice and 3 cups of water.

What is the ratio of juice to water? What is the ratio of juice to total liquid?

Step 1: Compare juice to water.

Juice to water is 2:3.

Step 2: Find the total liquid.

Step 3: Compare juice to total liquid.

Juice to total liquid is 2:5.

Answer:

• juice to water = 2:3
• juice to total liquid = 2:5

Common mistakes to avoid

• Mixing up the order. If the question says boys to girls, do not write girls to boys.
• Using the total when not asked. A part-to-part ratio and a part-to-whole ratio are different.
• Forgetting to simplify. Always check whether both numbers can be divided by the same number.
• Adding when you should compare. A ratio compares amounts; it does not always ask for the total.

Tips for solving ratio questions

1. Read the question carefully.
2. Decide which quantities are being compared.
3. Write the ratio in the correct order.
4. Simplify if possible.
5. Check whether the question asks for part-to-part or part-to-whole.

Quick practice ideas

• If there are 5 pencils and 11 pens, pencils to pens is 5:11.
• If there are 9 soccer balls and 3 basketballs, soccer balls to basketballs is 9:3 = 3:1.
• If a box has 4 red markers, 6 blue markers, and 10 total markers, red to total is 4:10 = 2:5.

Summary

A ratio compares two quantities. Ratios can be written with words, a colon, or as a fraction.

The order in a ratio matters. Ratios can compare part to part or part to whole, and they should be simplified when possible.

When solving ratio problems, always ask: What two quantities am I comparing, and in what order?

Unit Rates and Price Optimization

Have you ever wondered which snack pack is the better deal, or which runner is faster if they ran different distances? To answer questions like these, we use unit rates.

A rate compares two quantities with different units, such as miles per hour, dollars per pound, or words per minute. A unit rate is a rate with a denominator of 1. That means we compare the amount for one unit.

Unit rates help us make fair comparisons. When we turn different rates into “per 1,” it becomes easier to see which option is faster, cheaper, or more efficient.

1. What is a unit rate?

If 3 notebooks cost $6, the rate is 3 notebooks for $6. To find the cost for 1 notebook, divide both parts of the rate by 3:

So the unit rate is $2 per notebook.

We can write unit rates in different ways:

• $2 per notebook
• \(2\) dollars for 1 notebook
• \(\$2/\text{notebook}\)

2. How to find a unit rate

To find a unit rate:

1. Identify the two quantities being compared.
2. Decide which quantity should become 1.
3. Divide to make that quantity 1.
4. Write the answer with units.

For example, if a car travels 180 miles in 3 hours, we want to know how far it travels in 1 hour:

The unit rate is 60 miles per hour.

3. Why units matter

Always pay attention to the units in a rate. The numbers alone are not enough. For example, 60 miles per hour is not the same as 60 hours per mile.

In a unit rate, the word per tells us what the denominator is 1 of. So:

• miles per hour means 1 hour
• dollars per item means 1 item
• ounces per bottle means 1 bottle

4. Using division with rates

Most unit rates are found by dividing:

For example, if 24 ounces of juice cost $3, the cost per ounce is:

So the unit rate is $0.125 per ounce. That is also 12.5 cents per ounce.

Sometimes it is more useful to flip the rate. Instead of cost per ounce, you might want ounces per dollar:

That means you get 8 ounces per dollar.

5. Price optimization: finding the best buy

Price optimization means finding which choice gives the best value for the money. To do this, compare unit prices.

A unit price is the cost for 1 unit of something, such as:

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

Usually, the lower unit price is the better deal.

For example, if one bag of rice costs $4 for 2 pounds, the unit price is:

That bag costs $2 per pound.

If another bag costs $7.50 for 5 pounds, the unit price is:

That bag costs $1.50 per pound.

Since \(1.50 < 2.00\), the 5-pound bag is the better buy.

6. Worked Example 1: Finding a simple unit rate

A student reads 45 pages in 3 hours. What is the unit rate in pages per hour?

Step 1: Write the rate.

45 pages in 3 hours

Step 2: Divide by 3 to find 1 hour.

Answer: The unit rate is 15 pages per hour.

7. Worked Example 2: Comparing speeds

Runner A runs 12 miles in 2 hours. Runner B runs 15 miles in 3 hours. Who is faster?

Runner A:

Runner A runs 6 miles per hour.

Runner B:

Runner B runs 5 miles per hour.

Answer: Runner A is faster because \(6 > 5\).

8. Worked Example 3: Finding the better price

Store A sells 8 granola bars for $4.80. Store B sells 12 granola bars for $6.60. Which store has the better price?

Store A unit price:

Store A costs $0.60 per bar.

Store B unit price:

Store B costs $0.55 per bar.

Answer: Store B is the better deal because each bar costs less.

9. Worked Example 4: A harder price optimization problem

A small bottle of soap has 18 ounces for $2.70. A large bottle has 30 ounces for $4.20. Which is the better buy?

Small bottle:

The small bottle costs $0.15 per ounce.

Large bottle:

The large bottle costs $0.14 per ounce.

Answer: The large bottle is the better buy because its unit price is lower.

10. Tips for solving unit rate and price questions

• Read carefully to see what is being compared.
• Decide what should be “per 1.”
• Divide in the correct order.
• Always include units in your answer.
• For price optimization, compare the unit prices.
• If you are finding the best deal, the lower cost per unit is usually best.

11. Common mistakes to avoid

• Forgetting units: Write dollars per pound, not just 2.5.
• Dividing the wrong way: Make sure you know whether you want dollars per item or items per dollar.
• Comparing total prices only: A bigger package may cost more total, but still be the better deal.
• Not using the same unit: Compare prices using the same unit, such as per ounce for both items.

12. Quick check

• If 20 pencils cost $5, what is the cost per pencil? $0.25 per pencil
• If a bike travels 36 miles in 3 hours, what is its speed? 12 miles per hour
• Which is the better deal: 10 ounces for $2.50 or 15 ounces for $3.00? 15 ounces for $3.00, because \(3.00 \div 15 = 0.20\) and \(2.50 \div 10 = 0.25\)

Summary

A unit rate compares a quantity to 1 unit. We usually find it by dividing. Unit rates help us compare speeds, prices, and other real-world situations fairly.

When comparing prices, find the unit price for each option. The item with the lower cost per unit is usually the better value. Always keep track of your units so your answer makes sense.

Complex Ratios are ratios where one or both parts are fractions. They may look tricky at first, but they follow the same big idea as all ratios: a ratio compares two quantities.

When we work with complex ratios, we often want to find a unit rate. A unit rate tells us how much there is for 1 of something. For example, miles per 1 hour or cups per 1 serving.

A complex ratio can be written like a fraction divided by another fraction, such as \(\frac{3}{4} \div \frac{1}{2}\). This means, “How many groups of \(\frac{1}{2}\) are in \(\frac{3}{4}\)?”

To solve these, we use a rule for dividing fractions:

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

In words, keep the first fraction, change division to multiplication, and flip the second fraction.

This helps us turn a complex ratio into a simpler value we can understand.

Why complex ratios matter:

• Finding speed, such as miles per hour, when the distance or time is a fraction.
• Finding cost, such as dollars per pound, when the amount bought is a fraction.
• Comparing recipes, measurements, and other real-life situations.

How to solve a complex ratio

1. Write the ratio as a division problem.
2. If needed, rewrite whole numbers as fractions, like \(2 = \frac{2}{1}\).
3. Keep the first fraction.
4. Change division to multiplication.
5. Flip the second fraction.
6. Multiply.
7. Simplify your answer.
8. Check what the unit rate means in the problem.

Important idea: The answer should make sense. If you divide by a fraction smaller than 1, the result often becomes larger, because you are asking how many small parts fit into something.

Worked Example 1: Basic complex ratio

Find \(\frac{3}{4} \div \frac{1}{2}\).

Use the fraction division rule:

$$\frac{3}{4} \div \frac{1}{2} = \frac{3}{4} \times \frac{2}{1}$$

Multiply:

$$\frac{3}{4} \times \frac{2}{1} = \frac{6}{4}$$

Simplify:

$$\frac{6}{4} = \frac{3}{2} = 1\frac{1}{2}$$

So, \(\frac{3}{4} \div \frac{1}{2} = 1\frac{1}{2}\).

This means there are \(1\frac{1}{2}\) groups of \(\frac{1}{2}\) in \(\frac{3}{4}\).

Worked Example 2: Finding a unit rate

A runner goes \(\frac{5}{6}\) mile in \(\frac{1}{3}\) hour. What is the runner’s speed in miles per hour?

We want miles for 1 hour, so divide distance by time:

$$\frac{5}{6} \div \frac{1}{3}$$

Keep, change, flip:

$$\frac{5}{6} \div \frac{1}{3} = \frac{5}{6} \times \frac{3}{1}$$

Multiply:

$$\frac{5}{6} \times \frac{3}{1} = \frac{15}{6}$$

Simplify:

$$\frac{15}{6} = \frac{5}{2} = 2\frac{1}{2}$$

The unit rate is \(2\frac{1}{2}\) miles per hour.

Worked Example 3: Cost with fractions

\(\frac{3}{4}\) pound of grapes costs \(\$2\frac{1}{4}\). What is the cost per pound?

First, change the mixed number to an improper fraction:

$$2\frac{1}{4} = \frac{9}{4}$$

Now divide cost by pounds:

$$\frac{9}{4} \div \frac{3}{4}$$

Keep, change, flip:

$$\frac{9}{4} \times \frac{4}{3}$$

Multiply:

$$\frac{9}{4} \times \frac{4}{3} = \frac{36}{12} = 3$$

The cost is \(\$3\) per pound.

Worked Example 4: One fraction and one whole number

A recipe uses \(\frac{2}{3}\) cup of juice for 2 smoothies. How much juice is used for 1 smoothie?

We divide the total juice by the number of smoothies:

$$\frac{2}{3} \div 2$$

Rewrite 2 as \(\frac{2}{1}\):

$$\frac{2}{3} \div \frac{2}{1}$$

Keep, change, flip:

$$\frac{2}{3} \times \frac{1}{2} = \frac{2}{6} = \frac{1}{3}$$

So each smoothie uses \(\frac{1}{3}\) cup of juice.

Tips for success

• Read the problem carefully to know which quantity should be divided by which.
• “Per” usually means divide.
• Turn mixed numbers into improper fractions before dividing.
• After multiplying, always simplify.
• Include the correct units, like miles per hour or dollars per pound.

Common mistakes to avoid

• Forgetting to flip the second fraction.
• Flipping the first fraction instead of the second.
• Dividing in the wrong order.
• Leaving the answer unsimplified.
• Forgetting the unit in a unit rate problem.

Quick check

Try these on your own:

1. \(\frac{1}{2} \div \frac{1}{4}\)
2. \(\frac{3}{5}\) mile in \(\frac{1}{5}\) hour. What is the speed in miles per hour?
3. \(\$\frac{7}{2}\) for \(\frac{7}{8}\) pound. What is the cost per pound?

Answers

1. $$\frac{1}{2} \div \frac{1}{4} = \frac{1}{2} \times 4 = 2$$
2. $$\frac{3}{5} \div \frac{1}{5} = \frac{3}{5} \times \frac{5}{1} = 3$$ miles per hour
3. $$\frac{7}{2} \div \frac{7}{8} = \frac{7}{2} \times \frac{8}{7} = 4$$ dollars per pound

Summary

Complex ratios compare quantities when one or both numbers are fractions. To find a unit rate, divide the two quantities. When dividing fractions, keep the first fraction, change division to multiplication, and flip the second fraction. Then simplify and write the answer with the correct unit.

Dimensional Analysis and Unit Conversion

Sometimes a math problem gives a measurement in one unit, but the answer is needed in a different unit. For example, you may know a distance in inches but need it in feet, or know a mass in grams but need it in kilograms. Unit conversion helps us change from one unit to another without changing the actual amount.

Dimensional analysis is a method for converting units by using multiplication and division with conversion factors. A conversion factor is a ratio that equals 1 because it shows the same amount written in two different units.

For example, since 12 inches and 1 foot are the same length, the ratios below are both equal to 1:

Because these ratios equal 1, multiplying by them does not change the actual size of the measurement. It only changes the unit.

The big idea: choose a conversion factor so the unit you do not want will cancel out.

For example, if you start with inches and want feet, put inches on the bottom of the fraction so inches cancel:

The inches cancel, leaving feet.

How to use dimensional analysis

1. Write the starting measurement.
2. Write a conversion factor as a fraction.
3. Place the units carefully so unwanted units cancel.
4. Multiply or divide the numbers.
5. Check the final unit to make sure it is the one you wanted.

You can think of units like labels. Just as numbers can be multiplied and divided, units can also cancel when they appear in both the top and bottom of a fraction.

Common conversion facts

• Customary length: 12 in = 1 ft, 3 ft = 1 yd, 5280 ft = 1 mi
• Customary weight: 16 oz = 1 lb
• Customary capacity: 8 fl oz = 1 cup, 2 cups = 1 pint, 4 quarts = 1 gallon
• Metric length: 10 mm = 1 cm, 100 cm = 1 m, 1000 m = 1 km
• Metric mass: 1000 g = 1 kg
• Metric capacity: 1000 mL = 1 L

In the metric system, conversions are often easier because units change by powers of 10. But the same dimensional analysis method still works.

Worked Example 1: One-step customary conversion

Convert 36 inches to feet.

Start with 36 inches. We want inches to cancel and feet to remain.

Now divide:

So,

Why it works: the unit inches appears once on top and once on bottom, so it cancels.

Worked Example 2: One-step metric conversion

Convert 2500 milliliters to liters.

We know that 1000 mL = 1 L. We want milliliters to cancel.

Now compute:

So,

Worked Example 3: Two-step conversion

Convert 72 inches to yards.

There is no direct conversion on our basic list from inches to yards, so we can convert in steps:

• 12 in = 1 ft
• 3 ft = 1 yd

Chain the conversion factors together:

Now cancel units:

• in cancels with in
• ft cancels with ft
• yd is left

Now multiply and divide the numbers:

So,

This is called chaining conversion factors. You use more than one conversion factor in a row until you reach the unit you want.

Worked Example 4: A rate with unit conversion

A student walks 3 miles in 1 hour. How many feet per hour is that?

Start with the rate:

Use the conversion 1 mi = 5280 ft.

Miles cancel, and feet per hour remain:

So the rate is:

This example shows that dimensional analysis works with rates too, not just single measurements.

Tips for choosing the correct conversion factor

• Ask yourself: What unit do I want to get rid of?
• Put that unit in the opposite place of where it started so it cancels.
• If the unit starts on top, put it on the bottom of the conversion factor.
• If the unit starts on the bottom, put it on the top of the conversion factor.

Example: To change 5 kilograms to grams, kilograms must cancel.

Common mistakes to avoid

• Flipping the conversion factor the wrong way. If the wrong unit remains, the factor needs to be turned over.
• Forgetting to write units. Units help you see what cancels.
• Using the wrong conversion fact. Check that the two units in the factor are actually equal.
• Stopping too early. In a multi-step problem, keep going until the final unit is the one asked for.

Quick check

Suppose you want to convert 4 feet to inches. Which conversion factor should you use?

• \(\frac{12\text{ in}}{1\text{ ft}}\)
• \(\frac{1\text{ ft}}{12\text{ in}}\)

Since feet must cancel, use:

Another quick check

Convert 300 centimeters to meters.

The centimeters cancel, leaving meters.

Summary

Dimensional analysis is a smart and reliable way to convert units. You multiply by conversion factors that are equal to 1, and you arrange them so unwanted units cancel out.

Always write the units, choose the conversion factor carefully, and check that the final unit matches the question. If needed, chain together more than one conversion factor. With practice, this method makes unit conversion much easier.

Proportional Relationships in Tables and Graphs

In 7th grade math, a proportional relationship happens when two quantities change together in a very special way. One quantity is always a constant multiple of the other.

This means there is a number you can multiply by each time to go from one quantity to the other. That number is called the constant of proportionality.

For example, if each notebook costs \(\$3\), then the total cost is always \(3\) times the number of notebooks. So the relationship between notebooks and cost is proportional.

In this lesson, you will learn how to recognize proportional relationships in tables and graphs, and how to tell when a relationship is not proportional.

1. What makes a relationship proportional?

A relationship is proportional if:

• the ratio between the two quantities is always the same, and
• the graph is a straight line that passes through the origin.

The origin is the point \((0,0)\) on a graph.

If a relationship is proportional, it can be written like this:

Here:

• \(x\) is one quantity,
• \(y\) is the other quantity,
• \(k\) is the constant of proportionality.

2. How to check a table for proportionality

To decide whether a table shows a proportional relationship, compare the ratio \(\frac{y}{x}\) for each pair of values.

If the ratio stays the same every time, the relationship is proportional. If the ratio changes, it is not proportional.

Here is a table:

Check the ratio:

• \(\frac{2}{1} = 2\)
• \(\frac{4}{2} = 2\)
• \(\frac{6}{3} = 2\)
• \(\frac{10}{5} = 2\)

Since the ratio is always \(2\), this is a proportional relationship. The constant of proportionality is \(2\).

This means each apple costs \(\$2\), and the equation is:

3. What if the table is not proportional?

Look at this table:

Now check the ratios:

• \(\frac{5}{1} = 5\)
• \(\frac{10}{2} = 5\)
• \(\frac{16}{3} \neq 5\)
• \(\frac{20}{4} = 5\)

Most of the ratios are \(5\), but one is different. Because the ratio is not always the same, this relationship is not proportional.

Important: Even if most of the table follows a pattern, all pairs must have the same ratio for the relationship to be proportional.

4. How to recognize proportional relationships on a graph

A graph shows a proportional relationship if:

• the points lie on a straight line, and
• the line passes through the origin, \((0,0)\).

Why does it need to pass through the origin? Because if \(x = 0\), then \(y = 0\) in a proportional relationship. For example, if you buy 0 apples, the cost is \(\$0\).

If the graph is a straight line but does not pass through the origin, then the relationship is not proportional.

5. Connecting tables, graphs, and equations

These three forms all describe the same relationship:

• Table: shows matching values
• Graph: shows how the values change visually
• Equation: shows the rule using \(y = kx\)

If a relationship is proportional, you should be able to move between all three forms.

For example, if the constant of proportionality is \(4\), then:

This means:

• when \(x=1\), \(y=4\)
• when \(x=2\), \(y=8\)
• when \(x=3\), \(y=12\)

These points would make a straight line through the origin.

Worked Example 1: Finding proportionality in a table

Decide whether this table shows a proportional relationship.

Step 1: Find the ratio \(\frac{y}{x}\).

• \(\frac{15}{1}=15\)
• \(\frac{30}{2}=15\)
• \(\frac{60}{4}=15\)
• \(\frac{90}{6}=15\)

Step 2: Compare the ratios.

They are all equal to \(15\).

Answer: Yes, the relationship is proportional.

The constant of proportionality is \(15\), so the equation is:

Worked Example 2: A table that is not proportional

Decide whether this table shows a proportional relationship.

Step 1: Find each ratio.

• \(\frac{12}{2}=6\)
• \(\frac{18}{3}=6\)
• \(\frac{25}{4}=6.25\)
• \(\frac{30}{5}=6\)

Step 2: Compare the ratios.

One ratio is different, so the relationship is not proportional.

Answer: No, this table does not show a proportional relationship.

Worked Example 3: Using a graph description

A graph shows points that lie on a straight line. The line passes through \((0,0)\), \((1,3)\), \((2,6)\), and \((4,12)\).

Is this relationship proportional?

Step 1: Check whether the graph is a straight line.

Yes, it is.

Step 2: Check whether it passes through the origin.

Yes, it passes through \((0,0)\).

Step 3: Check the ratio if needed.

• \(\frac{3}{1}=3\)
• \(\frac{6}{2}=3\)
• \(\frac{12}{4}=3\)

Answer: Yes, the relationship is proportional, and the constant of proportionality is \(3\).

The equation is:

Worked Example 4: Straight line but not proportional

A graph is a straight line through the points \((0,2)\), \((1,5)\), \((2,8)\), and \((3,11)\).

Is this relationship proportional?

Step 1: Is it a straight line?

Yes.

Step 2: Does it pass through the origin?

No. It passes through \((0,2)\), not \((0,0)\).

Answer: No, the relationship is not proportional.

Even though the graph is a straight line, it does not go through the origin, so it is not proportional.

6. A simple step-by-step method

When you are given a table:

1. Take each pair of values.
2. Find \(\frac{y}{x}\).
3. If all the ratios are equal, the relationship is proportional.
4. The common ratio is the constant of proportionality.

When you are given a graph:

1. Check whether the points make a straight line.
2. Check whether the line passes through \((0,0)\).
3. If both are true, the relationship is proportional.

7. Common mistakes to avoid

• Looking only at addition: In proportional relationships, you focus on multiplying, not just adding the same amount.
• Forgetting the origin: A straight line alone is not enough. It must pass through \((0,0)\).
• Checking only one ratio: You must check all the pairs in the table.
• Mixing up \(x\) and \(y\): Be consistent when finding \(\frac{y}{x}\).

8. Quick practice questions

Try these on your own:

1. Table: \((1,7), (2,14), (5,35)\). Is it proportional?
2. Table: \((2,8), (4,16), (6,25)\). Is it proportional?
3. A graph is a straight line through \((0,0)\), \((2,10)\), and \((4,20)\). Is it proportional?
4. A graph is a straight line through \((0,4)\), \((1,6)\), and \((2,8)\). Is it proportional?

Answers:

• 1. Yes, because \(\frac{7}{1}=\frac{14}{2}=\frac{35}{5}=7\)
• 2. No, because \(\frac{25}{6}\) is not equal to \(4\)
• 3. Yes, straight line and passes through the origin
• 4. No, it does not pass through the origin

Summary

A proportional relationship has a constant multiplier between two quantities. In a table, the ratio \(\frac{y}{x}\) must stay the same for every pair of values.

On a graph, a proportional relationship appears as a straight line that passes through the origin. If either of these conditions is missing, the relationship is not proportional.

Always ask yourself: Is there a constant ratio? and Does the graph go through \((0,0)\)? If yes, the relationship is proportional.

Constant of Proportionality: Understanding \(y = kx\)

Sometimes two quantities grow together in a very special way. When one quantity is always a constant multiple of the other, we call the relationship proportional.

The rule for a proportional relationship is:

$$y = kx$$

In this equation:

• \(x\) is one quantity
• \(y\) is the other quantity
• \(k\) is the constant of proportionality

The constant of proportionality tells us how much \(y\) there is for each 1 unit of \(x\).

For example, if each notebook costs $3, then the total cost is proportional to the number of notebooks. The equation is:

$$y = 3x$$

Here, \(x\) could be the number of notebooks, \(y\) could be the total cost, and the constant of proportionality is \(k = 3\).

What does the constant of proportionality mean?

The constant of proportionality is the number you multiply by. It is the same in every pair of values in a proportional relationship.

You can find \(k\) by dividing \(y\) by \(x\):

$$k = \frac{y}{x}$$

If the relationship is proportional, this value stays the same for every pair in the table, graph, or situation.

How to tell if a relationship is proportional

• The relationship can be written as \(y = kx\).
• The ratio \(\frac{y}{x}\) is always the same.
• On a graph, the points form a straight line that goes through the origin, \((0,0)\).
• In a table, the output is always the input multiplied by the same number.

Finding the constant of proportionality from a table

Look at each pair of values and divide \(y\) by \(x\). If you keep getting the same answer, that answer is \(k\).

Worked Example 1: From a table

A table shows the number of bags of apples and the total cost.

\[ \begin{array}{c|c} x & y \\ \hline 1 & 4 \\ 2 & 8 \\ 3 & 12 \\ 5 & 20 \end{array} \]

Find the constant of proportionality.

Divide \(y\) by \(x\):

• \(\frac{4}{1} = 4\)
• \(\frac{8}{2} = 4\)
• \(\frac{12}{3} = 4\)
• \(\frac{20}{5} = 4\)

The ratio is always 4, so the constant of proportionality is:

$$k = 4$$

The equation is:

$$y = 4x$$

This means the cost is $4 for each bag of apples.

Finding the constant of proportionality from a word problem

In real-life situations, \(k\) often represents a rate. It might be cost per item, miles per hour, or cups of water per batch.

Worked Example 2: From a context

A machine fills 6 bottles in 2 minutes. Assume the number of bottles filled is proportional to time. Find the constant of proportionality.

Let:

• \(x\) = number of minutes
• \(y\) = number of bottles

Use the formula:

$$k = \frac{y}{x} = \frac{6}{2} = 3$$

So the constant of proportionality is 3.

The equation is:

$$y = 3x$$

This means the machine fills 3 bottles every 1 minute.

Finding the constant of proportionality from a graph

On a graph of a proportional relationship, the line goes through the origin. You can pick any point on the line, then divide \(y\) by \(x\) to find \(k\).

Worked Example 3: From a graph point

A graph shows a line passing through \((0,0)\) and the point \((4,10)\). Find the constant of proportionality.

Use the point \((x,y) = (4,10)\):

$$k = \frac{y}{x} = \frac{10}{4} = \frac{5}{2} = 2.5$$

So the constant of proportionality is:

$$k = 2.5$$

The equation is:

$$y = 2.5x$$

This means \(y\) increases by 2.5 for every increase of 1 in \(x\).

Using the equation \(y = kx\)

Once you know \(k\), you can find missing values.

Worked Example 4: Finding a missing value

A proportional relationship has constant of proportionality \(k = 7\). If \(x = 5\), find \(y\).

Substitute into \(y = kx\):

$$y = 7(5) = 35$$

So, \(y = 35\).

Now suppose \(y = 56\) and you want to find \(x\).

Use the equation:

$$56 = 7x$$

Divide both sides by 7:

$$x = 8$$

Important idea: the relationship must start at 0

In a proportional relationship, when \(x = 0\), then \(y = 0\). That is because:

$$y = k(0) = 0$$

If a graph does not go through the origin, then it is not proportional.

For example, the equation \(y = 3x + 2\) is not proportional because it does not match the form \(y = kx\).

Common mistakes to avoid

• Mixing up \(x\) and \(y\): Be careful to divide \(y \div x\), not \(x \div y\).
• Forgetting to check all values: In a table, one pair is not enough. Make sure the ratio stays the same.
• Thinking every straight line is proportional: A proportional graph must be a straight line and pass through \((0,0)\).
• Using addition instead of multiplication: Proportional relationships are about multiplying by the same number.

Quick step-by-step guide

1. Identify \(x\) and \(y\).
2. Compute \(\frac{y}{x}\).
3. If the value stays the same, call it \(k\).
4. Write the equation as \(y = kx\).
5. Use the equation to find missing values if needed.

Mini practice to think about

• If 3 pencils cost $6, what is \(k\)?
  \(k = \frac{6}{3} = 2\), so \(y = 2x\).
• If a point on a proportional graph is \((6,18)\), what is \(k\)?
  \(k = \frac{18}{6} = 3\), so \(y = 3x\).
• If \(y = 5x\), what is \(y\) when \(x = 9\)?
  \(y = 45\).

Summary

The constant of proportionality is the fixed number that connects \(x\) and \(y\) in a proportional relationship. It appears in the equation \(y = kx\), and you can find it by dividing \(y\) by \(x\).

You can find \(k\) from a table, a graph, or a real-world situation. If the ratio \(\frac{y}{x}\) is always the same and the graph goes through the origin, then the relationship is proportional.

When you understand \(k\), you can write equations, solve problems, and explain what the relationship means in real life.

Solving Cross-Proportions helps us find a missing number when two ratios are equal.

A proportion is an equation that says two ratios or fractions are equivalent. For example, \(\frac{2}{3} = \frac{4}{6}\) is a proportion because both ratios represent the same comparison.

When one number is missing, we can solve the proportion by using cross multiplication. This is a quick way to find the unknown value.

In this lesson, you will learn what cross-proportions are, when to use them, and how to solve them step by step.

1. What is a proportion?

A proportion compares two equal ratios. It often looks like this:

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

If the two fractions are equal, then the proportion is true.

Here is an example:

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

Both fractions have the same value, so this is a true proportion.

2. What does cross multiplication mean?

In a proportion, you multiply diagonally across the equal sign.

For

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

you multiply:

• \(a \times d\)
• \(b \times c\)

If the proportion is true, these products are equal:

$$a \times d = b \times c$$

This is called cross multiplication.

3. Steps for solving a cross-proportion

1. Write the proportion clearly as two fractions.
2. Cross multiply the numerator of one fraction with the denominator of the other.
3. Write an equation.
4. Solve for the missing variable.
5. Check your answer to make sure the ratios are equal.

4. Worked Example 1

Solve:

$$\frac{3}{5} = \frac{x}{10}$$

Step 1: Cross multiply.

Multiply diagonally:

$$3 \times 10 = 5 \times x$$

$$30 = 5x$$

Step 2: Solve for \(x\).

Divide both sides by 5:

$$x = 6$$

Step 3: Check.

Substitute \(x=6\):

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

Since \(\frac{6}{10}\) simplifies to \(\frac{3}{5}\), the answer is correct.

5. Worked Example 2

Solve:

$$\frac{7}{x} = \frac{14}{8}$$

Step 1: Cross multiply.

$$7 \times 8 = x \times 14$$

$$56 = 14x$$

Step 2: Solve for \(x\).

Divide both sides by 14:

$$x = 4$$

Step 3: Check.

$$\frac{7}{4} = \frac{14}{8}$$

Both sides are equal, so \(x=4\).

6. Worked Example 3

A recipe uses 2 cups of flour for 3 batches of muffins. How many cups of flour are needed for 9 batches?

Set up a proportion:

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

Step 1: Cross multiply.

$$2 \times 9 = 3 \times x$$

$$18 = 3x$$

Step 2: Solve.

$$x = 6$$

Answer: 6 cups of flour are needed.

7. Worked Example 4

A map uses the scale 1 inch = 5 miles. If two towns are 15 miles apart, how many inches apart should they be on the map?

Write the ratios so the units match:

$$\frac{1}{5} = \frac{x}{15}$$

Step 1: Cross multiply.

$$1 \times 15 = 5 \times x$$

$$15 = 5x$$

Step 2: Solve.

$$x = 3$$

Answer: The towns should be 3 inches apart on the map.

8. Tips for setting up proportions correctly

• Make sure the ratios compare the same kinds of things.
• Keep the order the same in both fractions.
• If one fraction has items over total, the other should also have items over total.
• Always check that your units match, such as cups to cups or miles to miles.

For example, if you compare inches to miles in one fraction, do the same in the other fraction.

9. Common mistakes to avoid

• Mixing up the order: If you start with apples/oranges, keep apples/oranges in the second ratio too.
• Forgetting to divide: After cross multiplying, you may still need to divide to find the variable.
• Not checking the answer: A quick check can catch mistakes.
• Using cross multiplication when the ratios are not set up as a proportion: The fractions must be equal.

10. Another way to think about proportions

Sometimes you can solve proportions by finding how one ratio scales to the other.

For example:

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

Since \(6\) becomes \(12\) by multiplying by 2, multiply the top by 2 also:

$$x = 8$$

This works because equivalent fractions are made by multiplying or dividing the numerator and denominator by the same number.

Cross multiplication is especially useful when the scale factor is not easy to see.

11. Practice thinking steps

When you see a proportion with a missing value, ask yourself:

• Are these two ratios set up in the same order?
• Can I solve it by scaling?
• If not, can I cross multiply?
• After I solve, does my answer make sense?

12. Summary

A proportion is an equation showing two equal ratios.

To solve a cross-proportion, multiply across the diagonals and set the products equal. Then solve the equation for the missing value.

Always make sure the ratios are written in the same order, and check your answer at the end.

Direct and Inverse Variation are two ways that two quantities can be related.

In this lesson, you will learn how to tell the difference between them, how to write equations for them, and how to solve simple problems.

This is an important skill in ratios, rates, and proportions because variation shows how one quantity changes when another quantity changes.

1. What is direct variation?

In a direct variation, two quantities change together in the same direction.

• If one quantity increases, the other also increases.
• If one quantity decreases, the other also decreases.

This happens at a constant rate, which means the ratio between the two quantities stays the same.

The equation for direct variation is:

$$y = kx$$

Here:

• y and x are the two variables.
• k is the constant of variation.

The constant of variation tells how many units of y there are for each 1 unit of x.

Another way to think about direct variation is this:

$$\frac{y}{x} = k$$

If the value of \(\frac{y}{x}\) stays the same, the relationship is a direct variation.

Real-life examples of direct variation:

• Total cost and number of identical items
• Distance traveled and time, if speed stays constant
• Pay earned and hours worked, if pay rate stays constant

2. What is inverse variation?

In an inverse variation, two quantities change in opposite directions.

• If one quantity increases, the other decreases.
• If one quantity decreases, the other increases.

In inverse variation, the product of the two quantities stays constant.

The equation for inverse variation is:

$$y = \frac{k}{x}$$

Here, k is still the constant of variation.

Another way to write it is:

$$xy = k$$

If the value of \(xy\) stays the same, the relationship is an inverse variation.

Real-life examples of inverse variation:

• The number of workers and the time needed to finish the same job
• Speed and travel time for the same distance
• The number of people sharing something and the amount each person gets

3. How to tell the difference

To decide whether a relationship is direct or inverse, ask:

• Do the quantities move together? That suggests direct variation.
• Do the quantities move in opposite directions? That suggests inverse variation.

You can also test the numbers:

• For direct variation, check whether \(\frac{y}{x}\) is constant.
• For inverse variation, check whether \(xy\) is constant.

4. Worked Example 1: Direct variation from a table

A table shows the number of notebooks and the total cost.

$$ \begin{array}{c|c} \text{Notebooks } (x) & \text{Cost } (y) \\ \hline 1 & 3 \\ 2 & 6 \\ 4 & 12 \end{array} $$

Step 1: Check the ratio \(\frac{y}{x}\).

$$\frac{3}{1} = 3, \quad \frac{6}{2} = 3, \quad \frac{12}{4} = 3$$

The ratio is always 3, so this is a direct variation.

Step 2: Find the equation.

Since \(k = 3\), the equation is:

$$y = 3x$$

Step 3: Use the equation.

If you buy 5 notebooks, the cost is:

$$y = 3(5) = 15$$

So, 5 notebooks cost 15.

5. Worked Example 2: Finding the constant in direct variation

Suppose \(y\) varies directly with \(x\), and \(y = 18\) when \(x = 6\).

Because direct variation has the form \(y = kx\), substitute the known values:

$$18 = 6k$$

Now solve for \(k\):

$$k = 3$$

So the equation is:

$$y = 3x$$

Now find \(y\) when \(x = 10\):

$$y = 3(10) = 30$$

So, when \(x = 10\), \(y = 30\).

6. Worked Example 3: Inverse variation from a table

A table shows the number of workers and the number of hours needed to finish one job.

$$ \begin{array}{c|c} \text{Workers } (x) & \text{Hours } (y) \\ \hline 2 & 12 \\ 3 & 8 \\ 4 & 6 \end{array} $$

Step 1: Check the product \(xy\).

$$2 \cdot 12 = 24, \quad 3 \cdot 8 = 24, \quad 4 \cdot 6 = 24$$

The product is always 24, so this is an inverse variation.

Step 2: Find the equation.

Since \(k = 24\), the equation is:

$$y = \frac{24}{x}$$

Step 3: Use the equation.

If there are 6 workers, the number of hours is:

$$y = \frac{24}{6} = 4$$

So, 6 workers need 4 hours.

7. Worked Example 4: Finding the constant in inverse variation

Suppose \(y\) varies inversely with \(x\), and \(y = 5\) when \(x = 8\).

Use the form:

$$xy = k$$

Substitute the values:

$$8 \cdot 5 = k$$

$$k = 40$$

So the equation is:

$$y = \frac{40}{x}$$

Now find \(y\) when \(x = 10\):

$$y = \frac{40}{10} = 4$$

So, when \(x = 10\), \(y = 4\).

8. Key differences to remember

• Direct variation: values go up or down together.
• Inverse variation: one goes up while the other goes down.
• Direct variation equation: \(y = kx\)
• Inverse variation equation: \(y = \frac{k}{x}\)
• For direct variation, \(\frac{y}{x}\) stays constant.
• For inverse variation, \(xy\) stays constant.

9. Quick tips

• If twice as much of one quantity gives twice as much of the other, think direct variation.
• If twice as much of one quantity gives half as much of the other, think inverse variation.
• Always check whether you should use a ratio or a product.

10. Summary

Direct and inverse variation both describe relationships between two variables.

In direct variation, the variables change together, and the equation is \(y = kx\).

In inverse variation, the variables change in opposite directions, and the equation is \(y = \frac{k}{x}\).

To tell them apart, check whether the ratio \(\frac{y}{x}\) stays the same or whether the product \(xy\) stays the same.

Percent as a Proportional Rate

Have you seen prices marked 25% off, test scores like 90%, or sports stats such as 60% wins? In each case, the percent tells how much out of 100.

A percent is a special kind of ratio. The word percent means per hundred. So when we write 35%, we mean 35 out of 100.

This makes percent a proportional rate: it compares one amount to a total of 100. Understanding this helps you move between fractions, decimals, and percents.

1. What percent means

Percent compares a part to a whole using 100 as the whole. You can write a percent in three connected ways:

• As a percent: 45%
• As a fraction: \(\frac{45}{100}\)
• As a decimal: \(0.45\)

These all mean the same amount. They are just different forms of the same proportional relationship.

For example:

This means 35 parts out of every 100 parts.

2. Why percent is a proportional rate

A rate compares two quantities with different units or meanings. A percent compares:

• the part
• the whole

but it rewrites that comparison so the whole is always 100.

For example, if 12 out of 20 students prefer pizza, the ratio is:

To write this as a percent, we find an equivalent ratio with denominator 100.

So, 60% of the students prefer pizza.

This works because equivalent ratios are proportional. We multiplied both 12 and 20 by 5.

3. Converting between fractions, decimals, and percents

Because percent means “out of 100,” you can switch between forms using simple steps.

Percent to fraction: Put the percent number over 100, then simplify if possible.

Percent to decimal: Divide by 100, or move the decimal point 2 places left.

Decimal to percent: Multiply by 100, or move the decimal point 2 places right, then add the percent sign.

Fraction to percent: Rewrite the fraction as an equivalent fraction out of 100, or divide the numerator by the denominator and turn the decimal into a percent.

4. Finding the percent, part, or whole

Percent problems often involve three ideas:

• Part: the piece you are talking about
• Whole: the total amount
• Percent: the part compared to 100

You can organize the relationship like this:

This formula shows that percent is a proportional comparison of part to whole.

Worked Example 1: Simple percent as a fraction and decimal

Write 65% as a fraction and as a decimal.

Step 1: Write the percent as a fraction out of 100.

Step 2: Simplify the fraction.

Step 3: Write it as a decimal.

Answer: \(65\% = \frac{13}{20} = 0.65\)

Worked Example 2: Fraction to percent

Convert \(\frac{2}{5}\) to a percent.

Method 1: Make the denominator 100.

So:

Method 2: Divide numerator by denominator.

Then convert the decimal to a percent:

Answer: \(\frac{2}{5} = 40\%\)

Worked Example 3: Finding a percent from a part and whole

Out of 25 questions, Maya got 20 correct. What percent did she get correct?

Step 1: Write the part over the whole.

Step 2: Rewrite as a decimal or as a fraction out of 100.

Step 3: Write as a percent.

Answer: Maya got 80% correct.

Worked Example 4: Finding the part using a percent

A store gives a 30% discount on a \(\$50\) backpack. How much is the discount?

30% means:

To find 30% of 50, multiply:

Answer: The discount is \(\$15\).

If you wanted the sale price, you would subtract the discount from the original price:

So the sale price would be \(\$35\).

5. Using proportion to solve percent problems

Because percent is a proportional rate, you can also solve problems with equivalent ratios.

Suppose 18 out of 30 students wore sneakers. What percent is that?

Set up a proportion:

Now solve. First simplify \(\frac{18}{30}\):

Then write it as a denominator of 100:

So:

Answer: 60% of the students wore sneakers.

6. Helpful patterns to remember

• \(50\% = \frac{1}{2} = 0.5\)
• \(25\% = \frac{1}{4} = 0.25\)
• \(75\% = \frac{3}{4} = 0.75\)
• \(10\% = \frac{1}{10} = 0.1\)
• \(1\% = \frac{1}{100} = 0.01\)

These common percents can help you estimate quickly.

7. Common mistakes to avoid

• Forgetting that percent means out of 100. For example, 8% means \(\frac{8}{100}\), not \(\frac{8}{10}\).
• Moving the decimal the wrong way. Percent to decimal: move left 2 places. Decimal to percent: move right 2 places.
• Mixing up part and whole. The part goes on top, and the whole goes on the bottom.
• Forgetting the percent sign. \(0.6\) and \(60\%\) mean the same amount, but they are written differently.

8. Quick check for understanding

1. Write 42% as a decimal and a fraction.
2. Convert \(\frac{7}{10}\) to a percent.
3. What percent of 40 is 10?
4. Find 20% of 60.

Answers:

1. \(0.42\), \(\frac{42}{100} = \frac{21}{50}\)
2. \(70\%\)
3. \(\frac{10}{40} = 0.25 = 25\%\)
4. \(0.20 \times 60 = 12\)

Summary

A percent is a ratio out of 100. That is why percent is called a proportional rate.

You can convert between fractions, decimals, and percents because they are different ways to show the same amount. When solving percent problems, always think about the part, the whole, and how they relate to 100.

Percent Equations: Part, Whole, and Percent

Percent problems show up in many real-life situations, like discounts, test scores, tips, and sales tax. To solve these problems, it helps to understand the relationship between the part, the whole, and the percent.

In this lesson, you will learn how to write and solve percent equations when one of these three parts is missing.

1. What do part, whole, and percent mean?

• Whole: the total amount
• Part: a piece of the whole
• Percent: how much out of 100

For example, if 15 out of 20 students finished their homework, then:

• The part is 15
• The whole is 20
• The percent tells what portion of 20 that 15 is

2. The main percent equation

The most important equation for percent problems is:

There is one important rule: before multiplying, write the percent as a decimal.

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

• \(25\% = 0.25\)
• \(6\% = 0.06\)
• \(125\% = 1.25\)

3. Solving for the part

If you know the percent and the whole, multiply to find the part.

Example 1: Find the part

What is \(30\%\) of 50?

Use the equation:

Substitute the values:

Answer: \(30\%\) of 50 is 15.

This makes sense because 15 is less than 50, and \(30\%\) is less than half.

4. Solving for the whole

If you know the part and the percent, divide by the decimal form of the percent.

Start with:

To solve for the whole:

Example 2: Find the whole

12 is \(40\%\) of what number?

Let the whole be \(w\).

Divide both sides by \(0.40\):

Answer: 12 is \(40\%\) of 30.

Check:

5. Solving for the percent

If you know the part and the whole, divide the part by the whole. Then change the decimal to a percent.

Start with:

To solve for the percent:

Then multiply by 100 to write the answer as a percent.

Example 3: Find the percent

18 is what percent of 24?

Let the percent be \(p\).

Divide both sides by 24:

Convert to a percent:

Answer: 18 is 75% of 24.

6. A word problem with a percent equation

Example 4: Real-life problem

A jacket costs \(\$80\). It is on sale for \(25\%\) off. How much is the discount?

The discount is the part, because it is part of the original price.

Use:

Substitute the values:

Answer: The discount is \(\$20\).

If you also wanted the sale price, subtract the discount from the original price:

So the sale price is \(\$60\).

7. How to recognize what is missing

When solving a percent problem, ask yourself these questions:

1. What is the whole?
2. What is the part?
3. What is the percent?
4. Which one is missing?

Then use the percent equation to solve.

• If the part is missing, multiply percent and whole.
• If the whole is missing, divide part by percent.
• If the percent is missing, divide part by whole.

8. Common mistakes to avoid

• Forgetting to change the percent to a decimal
  Do not use 25 in the equation for \(25\%\). Use \(0.25\).
• Mixing up part and whole
  The whole is the total amount. The part is only a piece of it.
• Forgetting to convert back to percent
  If you find \(0.6\), the percent is \(60\%\), not just 0.6.

9. Quick guide

• To find the part: $$\text{Part} = \text{Percent} \times \text{Whole}$$
• To find the whole: $$\text{Whole} = \frac{\text{Part}}{\text{Percent}}$$
• To find the percent: $$\text{Percent} = \frac{\text{Part}}{\text{Whole}}$$ then change to percent

10. Summary

Percent equations connect the part, whole, and percent in one simple relationship:

If you know any two of the three values, you can solve for the missing one. Always be careful to change percents to decimals before solving, and change decimals back to percents when needed.

With practice, percent equations become a useful tool for solving everyday math problems.

Percent means “out of 100.” When we talk about percent of change or percent error, we are comparing how much a value changed to the original or correct value.

These ideas are useful in everyday life. Stores use percent increase and decrease for prices. Scientists and students use percent error to see how close a measured answer is to the actual answer.

In this lesson, you will learn:

• what percent of change means,
• how to tell the difference between percent increase and percent decrease,
• what percent error means,
• and how to solve problems step by step.

1. Percent of Change

Percent of change tells how much a number goes up or down compared to its original value.

There are two main types:

• Percent increase: when the new value is greater than the original value.
• Percent decrease: when the new value is less than the original value.

To find percent of change, follow these steps:

1. Find the amount of change.
2. Divide by the original value.
3. Multiply by 100 to turn it into a percent.

The formula is:

$$\text{Percent of Change} = \frac{\text{Amount of Change}}{\text{Original Value}} \times 100\%$$

And the amount of change is:

$$\text{Amount of Change} = |\text{New Value} - \text{Original Value}|$$

After you find the percent, decide whether it is an increase or a decrease by comparing the new value to the original value.

2. Percent Increase

If the new value is larger, the quantity increased.

Example: A video game cost \(\$40\) last month and now costs \(\$50\).

• Original value: \(40\)
• New value: \(50\)
• Amount of change: \(50 - 40 = 10\)

Now use the formula:

$$\frac{10}{40} \times 100\% = 25\%$$

The price went up by 25% increase.

3. Percent Decrease

If the new value is smaller, the quantity decreased.

Example: A jacket cost \(\$60\) and is now on sale for \(\$45\).

• Original value: \(60\)
• New value: \(45\)
• Amount of change: \(60 - 45 = 15\)

Now use the formula:

$$\frac{15}{60} \times 100\% = 25\%$$

The price went down by 25% decrease.

Important reminder: Always divide by the original value, not the new value. This is a very common mistake.

4. Percent Error

Percent error tells how far a measured or estimated value is from the actual or correct value.

This is helpful when a result is close, but not exactly right.

To find percent error:

1. Find the difference between the estimated or measured value and the actual value.
2. Use the absolute value so the difference is positive.
3. Divide by the actual value.
4. Multiply by 100 to make a percent.

The formula is:

$$\text{Percent Error} = \frac{|\text{Estimated Value} - \text{Actual Value}|}{\text{Actual Value}} \times 100\%$$

Notice that percent error uses the actual value in the denominator because we are comparing the mistake to the correct amount.

Worked Example 1: Percent Increase

A school had \(200\) students in the art fair last year. This year, \(250\) students joined. What is the percent increase?

Step 1: Find the amount of change.

$$250 - 200 = 50$$

Step 2: Divide by the original value.

$$\frac{50}{200} = 0.25$$

Step 3: Convert to a percent.

$$0.25 \times 100\% = 25\%$$

Answer: The number of students increased by 25%.

Worked Example 2: Percent Decrease

A water bottle held \(32\) ounces when full. After some water was used, it held \(24\) ounces. What is the percent decrease?

Step 1: Find the amount of change.

$$32 - 24 = 8$$

Step 2: Divide by the original value.

$$\frac{8}{32} = 0.25$$

Step 3: Convert to a percent.

$$0.25 \times 100\% = 25\%$$

Answer: The amount of water decreased by 25%.

Worked Example 3: Percent Error

A student measures the length of a desk as \(118\) cm. The actual length is \(120\) cm. What is the percent error?

Step 1: Find the difference.

$$|118 - 120| = 2$$

Step 2: Divide by the actual value.

$$\frac{2}{120} = 0.0166\ldots$$

Step 3: Convert to a percent.

$$0.0166\ldots \times 100\% \approx 1.67\%$$

Answer: The percent error is about 1.67%.

Worked Example 4: Choosing Between Percent Change and Percent Error

A toy was priced at \(\$80\), but now it costs \(\$68\). A shopper guessed the new price was \(\$70\).

This situation has two different comparisons.

Part A: Find the percent decrease in price.

• Original price: \(80\)
• New price: \(68\)
• Change: \(80 - 68 = 12\)

$$\frac{12}{80} \times 100\% = 15\%$$

The price decreased by 15%.

Part B: Find the percent error in the shopper’s guess.

• Estimated value: \(70\)
• Actual value: \(68\)
• Error: \(|70 - 68| = 2\)

$$\frac{2}{68} \times 100\% \approx 2.94\%$$

The shopper’s guess had a percent error of about 2.94%.

5. How to Know Which Formula to Use

• Use percent of change when something changes from an original value to a new value.
• Use percent error when you compare an estimate or measurement to the actual correct value.

Ask yourself:

• Is this about a value going up or down? Use percent increase or decrease.
• Is this about how close a guess or measurement is to the correct answer? Use percent error.

6. Common Mistakes to Avoid

• Using the wrong denominator. For percent change, divide by the original value. For percent error, divide by the actual value.
• Forgetting to multiply by 100. A decimal like \(0.2\) becomes \(20\%\).
• Mixing up increase and decrease. Check whether the new value is bigger or smaller than the original.
• Not using absolute value for percent error. Error should be positive.

7. Quick Practice Thinking

If a value changes from \(50\) to \(60\), the change is \(10\). Then:

$$\frac{10}{50} \times 100\% = 20\%$$

Since \(60\) is greater than \(50\), it is a 20% increase.

If an estimate is \(52\) but the actual value is \(50\), the error is \(2\). Then:

$$\frac{2}{50} \times 100\% = 4\%$$

The percent error is 4%.

Summary

Percent of change compares how much a value increases or decreases to the original value. Percent error compares how far an estimate or measurement is from the actual value.

Remember these formulas:

$$\text{Percent of Change} = \frac{|\text{New} - \text{Original}|}{\text{Original}} \times 100\%$$

$$\text{Percent Error} = \frac{|\text{Estimated} - \text{Actual}|}{\text{Actual}} \times 100\%$$

If the new value is bigger, it is a percent increase. If the new value is smaller, it is a percent decrease. If you are comparing a guess or measurement to the correct amount, use percent error.

Scale Drawings and Models help us show very large or very small objects at a size we can work with.

A scale drawing is a picture of a real object that is bigger or smaller than the real object, but keeps the same shape and proportions.

A model is a physical version of an object, such as a toy car, a globe, or a building model. Models also use a scale.

In this lesson, you will learn how to read a scale, find actual lengths, find drawing lengths, and use a scale factor to enlarge or reduce figures.

Why do we use scale drawings?

• Maps show large places on small paper.
• Blueprints show buildings before they are built.
• Models show small versions of real objects.
• Diagrams help us measure objects that are too big or too small to draw at actual size.

Main Idea: In a scale drawing or model, all lengths change by the same ratio.

That means if one side is doubled, every side is doubled. If one side is cut in half, every side is cut in half.

This keeps the figure proportional. The shape stays the same, even though the size changes.

1. Understanding Scale

A scale compares a length in the drawing or model to the actual length in real life.

For example, a map scale might say:

\(1\text{ cm} = 5\text{ km}\)

This means every 1 centimeter on the map stands for 5 kilometers in real life.

A model scale might say:

\(1:20\)

This means 1 unit on the model matches 20 of the same units in real life.

If the units are not written, the units on both sides must be the same kind. For example, 1 inch to 20 inches, or 1 centimeter to 20 centimeters.

2. Scale Factor

The scale factor tells how much a figure is enlarged or reduced.

It can be written as a ratio:

$$\text{scale factor} = \frac{\text{new length}}{\text{original length}}$$

If the scale factor is greater than 1, the figure is enlarged.

If the scale factor is between 0 and 1, the figure is reduced.

Examples:

• Scale factor \(2\): the new figure is twice as large.
• Scale factor \(\frac{1}{2}\): the new figure is half as large.
• Scale factor \(3\): each side is multiplied by 3.

3. Finding Missing Lengths

To solve scale problems, use a proportion or multiply/divide by the scale factor.

There are two common situations:

• Find the actual length from a drawing or model.
• Find the drawing length from a real object.

Helpful strategy:

1. Write the scale clearly.
2. Make sure the units match.
3. Set up a proportion or multiply/divide.
4. Check whether your answer makes sense.

Worked Example 1: Finding an Actual Distance on a Map

A map uses the scale \(1\text{ cm} = 4\text{ km}\). Two towns are \(6\text{ cm}\) apart on the map. What is the actual distance?

Step 1: Write the scale.

\(1\text{ cm} \to 4\text{ km}\)

Step 2: Multiply by 6.

If \(1\text{ cm}\) represents \(4\text{ km}\), then \(6\text{ cm}\) represents:

$$6 \times 4 = 24$$

Answer: The actual distance is 24 km.

Worked Example 2: Finding a Length on a Drawing

A blueprint uses the scale \(1\text{ in} = 3\text{ ft}\). A real wall is \(15\text{ ft}\) long. How long should the wall be on the blueprint?

Step 1: Compare the real length to the scale.

\(3\text{ ft}\) in real life matches \(1\text{ in}\) on the blueprint.

Step 2: Divide the real length by 3.

$$15 \div 3 = 5$$

Answer: The wall should be 5 inches long on the blueprint.

Worked Example 3: Using a Scale Factor to Enlarge a Figure

A rectangle has length \(4\text{ cm}\) and width \(3\text{ cm}\). It is enlarged by a scale factor of \(2\).

Step 1: Multiply each side by 2.

New length:

$$4 \times 2 = 8\text{ cm}$$

New width:

$$3 \times 2 = 6\text{ cm}$$

Answer: The enlarged rectangle is 8 cm by 6 cm.

Notice that both side lengths were multiplied by the same number. That is why the new figure keeps the same shape.

Worked Example 4: Using a Scale Factor to Reduce a Figure

A triangle has side lengths \(12\text{ cm}\), \(15\text{ cm}\), and \(18\text{ cm}\). It is reduced by a scale factor of \(\frac{1}{3}\).

Step 1: Multiply each side by \(\frac{1}{3}\).

$$12 \times \frac{1}{3} = 4\text{ cm}$$

$$15 \times \frac{1}{3} = 5\text{ cm}$$

$$18 \times \frac{1}{3} = 6\text{ cm}$$

Answer: The reduced triangle has side lengths 4 cm, 5 cm, and 6 cm.

4. Solving with a Proportion

Sometimes a proportion helps organize the information.

Example scale:

\(2\text{ cm} = 10\text{ m}\)

If a drawing length is \(7\text{ cm}\), let the actual length be \(x\text{ m}\).

Set up the proportion:

$$\frac{2}{10} = \frac{7}{x}$$

Now solve by noticing that \(10\) is \(5\) times \(2\), so \(x\) is \(5\) times \(7\):

$$x = 35$$

So the actual length is \(35\text{ m}\).

5. Important Unit Reminders

Units matter a lot in scale problems.

If the scale says \(1\text{ in} = 4\text{ ft}\), do not treat inches and feet as the same unit. Read the scale carefully and match each side correctly.

If a scale is written like \(1:50\), it usually means the same unit is used for both parts. For example:

• \(1\text{ cm}\) on the drawing = \(50\text{ cm}\) in real life
• or \(1\text{ in}\) on the model = \(50\text{ in}\) in real life

6. What Stays the Same and What Changes?

In scale drawings and models:

• The shape stays the same.
• The side lengths change by the scale factor.
• The figures remain proportional.

For example, if every side of a square is multiplied by 3, the new figure is still a square.

If only one side changes and the others do not, then it is not a correct scale drawing.

7. Common Mistakes to Avoid

• Mixing up the order of the scale. Know which number represents the drawing and which represents the real object.
• Forgetting units. Always include units in your answer.
• Using different scale factors for different sides. Every side must change by the same factor.
• Multiplying when you should divide. Ask: am I going from drawing to real life, or from real life to drawing?

Quick Check

• If a map scale is \(1\text{ cm} = 8\text{ km}\), then \(3\text{ cm}\) represents \(24\text{ km}\).
• If a model car uses scale \(1:10\), then a \(5\text{ cm}\) model length represents \(50\text{ cm}\) in real life.
• If a figure is reduced by scale factor \(\frac{1}{4}\), each side becomes one-fourth of the original length.

Summary

Scale drawings and models use ratios to show real objects at a different size while keeping the same shape.

The scale tells how drawing lengths and actual lengths compare. The scale factor tells how much a figure is enlarged or reduced.

To solve problems, read the scale carefully, keep track of units, and multiply or divide so that all side lengths change proportionally.