Determining Proportions — I can determine whether two...

SECTION 1

Where Did Proportions Come From?

People have been comparing amounts for thousands of years. Ancient builders needed to know if a small drawing of a temple would look the same when built full-size. Traders had to figure out fair prices when buying different amounts of grain. The idea of a proportion — two ratios that are equal — was the tool that solved these problems.

~1800 BCE

Egyptian Builders

Ancient Egyptians used proportional ratios to build the pyramids. They scaled small plans up to massive structures while keeping shapes the same.

~500 BCE

Greek Mathematicians

Euclid wrote formal rules about proportions in his book Elements. He showed how to prove two ratios are equal.

~200 CE

Chinese & Indian Traders

Merchants along the Silk Road used proportions to convert currencies and measure goods across different systems.

1500s–1600s

Modern Notation

Mathematicians began writing ratios with colons (3:4) and proportions with equals signs (3:4 = 6:8), giving us the notation we use today.

The big question people kept asking was simple: "Are these two ratios really the same?" That is exactly what you will learn to answer in this lesson.

SECTION 2

Core Principles & Definitions

Before we can determine whether two ratios form a proportion, we need to be clear about a few key ideas. Let's build up from the basics.

Ratio

A ratio compares two quantities. You can write it as 3:4, 3 to 4, or ³⁄₄. It tells you how much of one thing there is for every amount of another.

Equivalent Ratios

Equivalent ratios are different ratios that represent the same comparison. For example, 1:2 and 3:6 are equivalent because you can multiply both parts of 1:2 by 3 to get 3:6.

Proportion

A proportion is an equation that says two ratios are equal. For example, 2/3 = 4/6 is a proportion. If the ratios are NOT equal, they do NOT form a proportion.

Simplify to Check

One way to check is to simplify both ratios to their lowest terms. If they simplify to the same ratio, they form a proportion.

Cross Multiply to Check

Another way is cross multiplication. Multiply diagonally across the equals sign. If both products are the same number, the ratios form a proportion.

✦ KEY TAKEAWAY

Think of a proportion like a recipe. If a cookie recipe calls for 2 cups of flour and 1 cup of sugar (2:1), and you double it to 4 cups of flour and 2 cups of sugar (4:2), the taste stays the same. The ratios 2:1 and 4:2 form a proportion because they are equivalent. But if you used 4 cups of flour and 3 cups of sugar (4:3), the cookies would taste different — those ratios do NOT form a proportion.

SECTION 3

Seeing Proportions

A picture can make proportions much easier to understand. The diagram below shows two pairs of ratios. One pair forms a proportion, and the other does not. Look at how the bar models compare.

The top pair of bar models shows that 2:3 and 4:6 have the same shape — they form a proportion. The bottom pair shows that 2:3 and 3:5 have different shapes — they do not form a proportion.

Notice how the top pair of bars has the exact same shape. When you split the 4:6 bar in half, each piece matches the 2:3 bar perfectly. That tells you the ratios are equivalent. In the bottom pair, no amount of splitting or combining will make 2:3 match 3:5.

SECTION 4

Two Methods to Check for a Proportion

There are two main methods to determine whether two ratios form a proportion. Both are reliable, and you can choose whichever feels easier for a given problem.

Method 1: Simplify Both Ratios

SIMPLIFY TO LOWEST TERMS

a/b → divide both a and b by their GCF

The GCF (Greatest Common Factor) is the largest number that divides evenly into both parts of the ratio. If both ratios simplify to the same fraction, they form a proportion.

For example, consider 6/9 and 10/15. The GCF of 6 and 9 is 3, so 6/9 simplifies to 2/3. The GCF of 10 and 15 is 5, so 10/15 simplifies to 2/3. Since both simplify to 2/3, these ratios form a proportion.

Method 2: Cross Multiplication

CROSS MULTIPLICATION

a/b = c/d → a × d = b × c

Multiply the numerator of the first ratio by the denominator of the second. Then multiply the denominator of the first by the numerator of the second. If the two products are equal, the ratios form a proportion.

Using the same example: Does 6/9 = 10/15? Cross multiply: 6 × 15 = 90 and 9 × 10 = 90. The cross products are equal, so yes, these ratios form a proportion.

💡 Which Method Should I Use?

Both methods always give the correct answer. Simplifying is great when the numbers have obvious common factors. Cross multiplication is often faster when the numbers are larger or you can't easily spot the GCF. Try both and see which one clicks for you!

SECTION 5

Side-by-Side: Simplify vs. Cross Multiply

Let's see both methods in action on the same problem. We want to know: Do 8/12 and 14/21 form a proportion? The diagram below walks through each method step by step.

Both the simplifying method (left, in amber) and the cross-multiplication method (right, in violet) confirm that 8/12 and 14/21 form a proportion.

As you can see, both methods arrive at the same answer. The simplify method shows why the ratios are equal (they are both 2/3). The cross-multiplication method just tells you whether they are equal. On a test, either method is a valid justification.

SECTION 6

Worked Example

Let's work through a complete problem from start to finish. We'll use both methods so you can see the full process.

Do 15/25 and 9/16 form a proportion?

Step 1 — Write Both Ratios as FractionsThe two ratios are already in fraction form: 15/25 and 9/16. We need to determine if they are equal.

15/25 and 9/16

Step 2 — Try SimplifyingThe GCF of 15 and 25 is 5. Divide both by 5: 15 ÷ 5 = 3 and 25 ÷ 5 = 5, so 15/25 simplifies to 3/5. The GCF of 9 and 16 is 1 (they share no factors), so 9/16 is already in simplest form.

15/25 → 3/5 and 9/16 → 9/16

Step 3 — Compare the Simplified RatiosWe compare 3/5 and 9/16. These are clearly different fractions, so the ratios are NOT equivalent.

3/5 ≠ 9/16

Step 4 — Confirm with Cross MultiplicationLet's double-check. Cross multiply: 15 × 16 = 240 and 25 × 9 = 225. Since 240 ≠ 225, the cross products are NOT equal.

240 ≠ 225 — Not a proportion

Step 5 — State Your ConclusionBoth methods agree: 15/25 and 9/16 do not form a proportion. A complete answer would say: "15/25 and 9/16 do not form a proportion because 15/25 simplifies to 3/5, which is not equal to 9/16."

15/25 and 9/16 do NOT form a proportion.

SECTION 7

Strengths & Limitations of Each Method

Both methods work every time, but each one has situations where it shines. Here's a comparison to help you decide which to reach for first.

Comparison of the two methods for determining proportions

Feature	Simplifying	Cross Multiplication
When it's fastest	When you can easily spot common factors (like 10/15 → 2/3)	When the numbers are large or don't simplify nicely
What it shows	Shows the actual simplified ratio — helps you understand WHY they are equal	Gives a yes/no answer — tells you WHETHER they are equal
Possible challenge	Finding the GCF can be tricky with large numbers	Large multiplications can lead to arithmetic mistakes
Best for justifying	"These form a proportion because both simplify to 2/3."	"These form a proportion because 6 × 21 = 9 × 14 = 126."

✦ KEY TAKEAWAY

Think of it like checking if two photos are the same picture. Simplifying is like zooming both photos to the same size and comparing them directly. Cross multiplying is like measuring specific features (nose width, eye spacing) and seeing if the measurements match. Both work — pick the one that feels more natural to you!

SECTION 8

Connection to Solving Proportions

So far, you've learned to check whether two ratios form a proportion. The next step in your math journey is to solve proportions — finding a missing value when you know the other three numbers.

From checking proportions to solving them

What You Do Now	What Comes Next
Given two complete ratios, decide if they are equal.	Given a proportion with one missing number, find that number.
Example: Does 3/4 = 9/12? Yes!	Example: 3/4 = x/12. Solve for x. (x = 9)
Method: Simplify or cross multiply to check.	Method: Cross multiply and divide to solve.
Skill: Checking equivalence	Skill: Using proportional reasoning to find unknowns

Everything you are learning now — simplifying, cross multiplying, and justifying — is the foundation for solving proportions. You'll use these same tools in algebra, science, cooking, map reading, and even video game design. Mastering today's skill means you're building a tool you'll use for years to come.

SECTION 9

Practice Problems

Try these five problems on your own. They start easy and get harder. For each one, decide whether the ratios form a proportion and explain how you know.

PROBLEM 1 — CONCEPTUAL

In your own words, what does it mean for two ratios to form a proportion? Give one example of a proportion and one example of two ratios that are NOT a proportion.

PROBLEM 2 — BASIC CALCULATION

Do 6/8 and 15/20 form a proportion? Show your work using either method.

PROBLEM 3 — INTERMEDIATE

A store sells 3 notebooks for $4.50. Another store sells 7 notebooks for $10.00. Do these prices form a proportion? If not, which store has the better deal?

PROBLEM 4 — APPLIED

A recipe for 12 muffins calls for 2 cups of flour and 3 eggs. You want to make 30 muffins, so you plan to use 5 cups of flour and 8 eggs. Does each ingredient keep the same proportion as the original recipe? Justify your answers.

PROBLEM 5 — CRITICAL THINKING

Marcus says that 4/6 and 6/9 form a proportion because "both have a difference of 2 between the numerator and denominator." Is Marcus correct? Explain why his reasoning does or does not work, and give a counterexample if needed.

SUMMARY

Lesson Summary

A proportion is an equation stating that two ratios are equal. To check whether two ratios form a proportion, you can use simplifying (reduce both ratios to lowest terms and compare) or cross multiplication (multiply diagonally and compare the products). If the simplified forms match or the cross products are equal, the ratios form a proportion.

Always justify your answer by showing your work. Saying "Both ratios simplify to 2/3" or "The cross products are both 60" is a complete justification. Remember that proportions are about multiplicative relationships, not additive differences. This skill is the foundation for solving proportions, which you will learn next.