Solving Proportions — Solve a proportion using a table or equation. - SHSAT Math

SECTION 1

Where Proportions Came From

Have you ever doubled a recipe? If the original calls for 2 cups of flour to make 12 cookies, you know you need 4 cups to make 24. You just used a proportion — two equal ratios. People have been thinking this way for thousands of years.

~1800 BCE • Ancient Babylon

Babylonian scribes carved clay tablets showing how to scale recipes and trade goods using equal ratios.

~300 BCE • Euclid's Elements

The Greek mathematician Euclid wrote formal rules about proportions. He showed that equal ratios follow predictable patterns.

~600 CE • Indian Mathematicians

Scholars in India developed the "Rule of Three," a shortcut for solving proportions that traders used across Asia.

1700s • Cross-Multiplication Spreads

European textbooks began teaching cross-multiplication as the standard way to solve proportions in schools.

Today • SHSAT & Beyond

Proportions appear on the SHSAT, SAT, and in everyday life — from maps and models to speed and pricing.

The big question has always been the same: if you know three of the four numbers in two equal ratios, how do you find the missing one? That is exactly what this lesson teaches you.

SECTION 2

Core Ideas You Need to Know

Before you solve a proportion, make sure you understand the building blocks. A ratio compares two quantities (like 3 to 5). A proportion is a statement that two ratios are equal (like 3/5 = 6/10). Here are the key ideas.

Ratio

A comparison of two numbers using division. You can write it as 3 : 5, 3/5, or "3 to 5."

Proportion

An equation showing two ratios are equal. Example: 3/5 = 6/10. The cross products are always equal.

Scale Factor

The number you multiply (or divide) both parts of a ratio by to get the other ratio. Here it is × 2.

Cross Products

In a/b = c/d, the cross products are a × d and b × c. If the proportion is true, they are equal.

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KEY TAKEAWAY

Think of a proportion like a balance scale. Both sides must weigh the same. If you change one number, you must change another to keep it balanced. A table method finds the scale factor, while cross-multiplication uses an algebra shortcut. Both get you the same answer!

SECTION 3

Seeing Proportions in Action

The diagram below shows how two equal ratios relate. Notice that you can move from the top ratio to the bottom one by multiplying both parts by the same scale factor. This is the idea behind the table method.

The table shows Ratio 1 (3 and 5) on top and Ratio 2 (12 and ?) on the bottom. Multiply both parts by the scale factor of 4 to find the missing value of 20.

In the diagram, both numbers in the first ratio are multiplied by 4 to get the second ratio. When you spot this pattern, you can find any missing number quickly.

SECTION 4

The Math Behind Proportions

There are two main methods to solve a proportion. The table method works great when the scale factor is easy to spot. The cross-multiplication method works every time, even when the numbers are tricky.

PROPORTION FORMAT

a / b = c / d

This means "a is to b as c is to d." The four values are called the terms of the proportion.

TABLE METHOD

Scale Factor = known term in Ratio 2 ÷ matching term in Ratio 1

Find the scale factor first. Then multiply the other term by the same factor to get the missing value.

CROSS-MULTIPLICATION

a × d = b × c

Multiply across the equals sign diagonally. The two products (called cross products) are always equal in a true proportion. Solve for the unknown.

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When to Use Which?

If one number divides evenly into another (like 3 into 12), the table method is fastest. If the numbers don't divide evenly (like 7 and 15), use cross-multiplication.

SECTION 5

Table vs. Cross-Multiplication — Side by Side

Let's solve the same proportion — 4/7 = 20/x — using both methods so you can see how each one works.

Both the table method (left) and cross-multiplication (right) produce x = 35. Choose whichever feels faster for the problem at hand.

Both methods give x = 35. On the SHSAT, you want to save time. Glance at the numbers first. If you spot a clean scale factor, use the table. Otherwise, cross-multiply.

SECTION 6

Worked Example — Step by Step

A map says 3 centimeters represent 15 miles. If two cities are 8 centimeters apart on the map, how many miles apart are they?

Solving a Map Proportion

1
Step 1 — Write the known ratio

The map tells us 3 cm = 15 miles. Write this as a ratio: 3/15 (cm over miles).
2
Step 2 — Set up the proportion

We need to find how many miles match 8 cm. Call the unknown x. Our proportion is 3/15 = 8/x.
3
Step 3 — Try the table method

Can we find a nice scale factor? 8 ÷ 3 is not a whole number. So the table method is not the easiest here. Let's use cross-multiplication instead.
4
Step 4 — Cross-multiply

3 × x = 15 × 8. That gives us 3x = 120.

3x = 120
5
Step 5 — Solve for x

Divide both sides by 3. x = 120 ÷ 3 = 40.

x = 40 miles
6
Step 6 — Check your answer

Does 3/15 = 8/40? Simplify both: 3/15 = 1/5 and 8/40 = 1/5. Yes! The cities are 40 miles apart.

SECTION 7

Strengths and Limitations of Each Method

Both the table method and cross-multiplication are useful tools. The table below helps you decide which one to grab.

Feature	Table Method	Cross-Multiplication
Speed	Very fast when scale factor is a whole number	Fast for any numbers, whole or decimal
Works with messy numbers?	Not ideal — scale factor may be a fraction or decimal	Yes — always works
Uses algebra?	No — just multiply and divide	Yes — you solve an equation
Good for SHSAT?	Great for quick-pick problems	Great for harder problems
Risk of error	Low — few steps	Medium — watch your multiplication

Table vs. Cross-Multiplication comparison

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KEY TAKEAWAY

Imagine you have two toolbox drawers. The table method is like a screwdriver — simple and quick for the right screw. Cross-multiplication is like a power drill — a little more setup, but it handles every job. Learn both so you always have the right tool.

SECTION 8

Connection to Advanced Topics

Solving proportions is a stepping stone to bigger ideas in math. The table below shows how the skill you just learned connects to topics you will see later.

This Lesson	Future Topic
Solving a/b = c/d for one unknown	Solving linear equations (Algebra 1)
Scale factor between two ratios	Constant of proportionality (y = kx)
Checking equal ratios	Similar triangles and scale drawings (Geometry)
Word problems with proportions	Unit rates, speed, density, and percent problems

How proportions connect to future math

On the SHSAT, proportions appear in many forms — maps, recipes, speed problems, and even probability. Mastering the basics now makes those harder problems feel familiar.

SECTION 9

Practice Problems

Try these five problems. They start easy and get harder. Use a table or cross-multiplication — whichever feels right for the problem.

Question 1: CONCEPTUAL

True or false: The ratios 6/9 and 10/15 form a proportion. Explain how you know.

Show answer

True. Simplify 6/9 by dividing both parts by 3 to get 2/3. Simplify 10/15 by dividing both parts by 5 to get 2/3. Since both ratios equal 2/3, they form a proportion. You can also check with cross products: 6 × 15 = 90 and 9 × 10 = 90. Equal cross products mean it is a true proportion.

Question 2: BASIC CALCULATION

Solve for x: 5/8 = 15/x.

Show answer

Using the table method, notice that 15 ÷ 5 = 3, so the scale factor is 3. Multiply 8 × 3 = 24. So x = 24. You can verify with cross-multiplication: 5 × x = 8 × 15, giving 5x = 120, so x = 120 ÷ 5 = 24. Both methods confirm x = 24.

Question 3: INTERMEDIATE

Solve for n: 7/n = 21/36.

Show answer

Cross-multiply: 7 × 36 = n × 21. That gives 252 = 21n. Divide both sides by 21: n = 252 ÷ 21 = 12. Check: 7/12 simplified stays 7/12, and 21/36 simplified (divide by 3) is 7/12. They match, so n = 12 is correct. Notice the scale factor was 3 (7 × 3 = 21 and 12 × 3 = 36).

Question 4: APPLIED

A train travels 90 miles in 2 hours at a constant speed. How many miles will it travel in 7 hours? Set up and solve a proportion.

Show answer

Set up the proportion with miles over hours: 90/2 = x/7. Cross-multiply: 90 × 7 = 2 × x, giving 630 = 2x. Divide both sides by 2: x = 315. The train will travel 315 miles in 7 hours. You can also use the table method: the scale factor from 2 to 7 is 3.5, and 90 × 3.5 = 315. Always make sure units match on both sides of the proportion.

Question 5: CRITICAL THINKING

A store sells 3 notebooks for $4.50 and 5 erasers for $3.00. Marcus buys 8 notebooks and 10 erasers. He says the total cost is $18.00. Is he right? Use proportions to find each cost and explain your reasoning.

Show answer

First, find the cost of 8 notebooks: 3/4.50 = 8/x. Cross-multiply: 3x = 4.50 × 8 = 36, so x = 36 ÷ 3 = $12.00. Next, find the cost of 10 erasers: 5/3.00 = 10/y. Cross-multiply: 5y = 3.00 × 10 = 30, so y = 30 ÷ 5 = $6.00. The total is $12.00 + $6.00 = $18.00. Marcus is correct! Notice that 8 is not a whole-number multiple of 3, so the table method for the notebooks requires cross-multiplication, while the erasers have a clean scale factor of 2.

SUMMARY

Lesson Summary

Summary

A proportion states that two ratios are equal, like a/b = c/d. When one of the four values is unknown, you have two reliable ways to find it. The table method finds a scale factor (a number you multiply by) between the two ratios. The cross-multiplication method uses the rule that cross products are equal (a × d = b × c) and then solves the resulting equation.

For the SHSAT, check whether the numbers divide evenly — if they do, the table method is fastest. If not, use cross-multiplication. Always check your answer by plugging it back in and simplifying both ratios. Proportions connect to future topics like linear equations, similar figures, and unit rates, so mastering them now gives you a strong foundation.