Solve proportional relationships using a table or equation....

SECTION 1

Where Do Proportions Come From?

People have used proportions (statements that two ratios are equal) for thousands of years. Ancient builders needed them to scale up small plans into huge temples. Traders needed them to figure out fair prices when buying different amounts of goods. Let's look at a few key moments in this story.

~1800 BCE

Babylonian Tablets

Babylonian scribes carved ratio tables into clay tablets to help merchants convert between weights of silver and barley. These are some of the earliest proportion tables ever found.

~300 BCE

Euclid's Elements

The Greek mathematician Euclid wrote Book V of his famous textbook, laying out the formal rules of ratios and proportions that students still learn today.

~600 CE

Indian 'Rule of Three'

Indian mathematicians developed the Rule of Three — a shortcut for solving proportions — and it spread along trade routes to the Arab world and then to Europe.

Today

Proportions Everywhere

From cooking recipes to map scales to video-game design, proportional reasoning is one of the most commonly used math skills in daily life and on standardized tests like the SSAT.

The big question behind all of these moments is the same: If I know one ratio, how can I find a missing value in a related ratio? That's exactly what this lesson will teach you.

SECTION 2

Core Principles of Proportional Relationships

Before we solve anything, let's nail down four ideas that every proportion problem depends on. Once you understand these, the rest of the lesson will feel much easier.

Ratio

A ratio compares two quantities by division. You can write it as 3 to 5, 3 : 5, or 3/5. All three mean the same thing.

Proportion

A proportion is an equation that says two ratios are equal, like 3/5 = 6/10. If one number is missing, you can find it.

Constant of Proportionality

The constant of proportionality (often called k) is the fixed number you multiply one quantity by to get the other. In a table, every y ÷ x gives the same k.

Cross-Multiplication

When you have a/b = c/d, you can cross-multiply to get a × d = b × c. This is the fastest way to solve for a missing number.

✦ KEY TAKEAWAY

Think of a proportion like a recipe. If 2 cups of flour make 12 cookies, you can double everything to get 4 cups → 24 cookies. The ratio stays the same no matter how much you scale up or down. That fixed ratio is what makes the relationship proportional.

SECTION 3

Seeing Proportions in a Table and on a Graph

One of the best ways to understand proportions is to see the numbers laid out in a table and then plotted on a graph. The diagram below shows a proportional relationship between the number of hours worked and the amount of money earned at $8 per hour.

On the left, notice how every row in the table gives the same result when you divide y by x: the constant k = 8. On the right, the graph shows a straight line through the origin (0, 0). A proportional relationship always makes a straight line through the origin.

Look at the table first. Each time the hours go up by 1, the dollars go up by 8. That steady jump is a big clue. Now look at the graph. The dots all sit on a line that starts at (0, 0). If you ever see a straight line that passes through the origin, you know the relationship is proportional.

SECTION 4

The Math Behind Proportions

There are two main formulas you need. The first one is the equation form of a proportional relationship. The second is the cross-multiplication method for solving a proportion with a missing value.

PROPORTIONAL EQUATION

y = k × x

Here, y is the output, x is the input, and k is the constant of proportionality (the rate that stays the same). You find k by dividing any y by its matching x.

FINDING THE CONSTANT

k = y ÷ x

Pick any row of your table and divide the y-value by the x-value. If the relationship is truly proportional, every row will give the same k.

CROSS-MULTIPLICATION

a/b = c/d → a × d = b × c

When two fractions are equal, you can multiply diagonally. This lets you solve for a missing number without needing to find k first. Just multiply the known diagonal, then divide by the remaining known number.

💡 When to Use Which Method

Use y = k × x when a table gives you several pairs and you need to find k or fill in missing values. Use cross-multiplication when you have a single proportion like 3/5 = x/20 and need to solve for x quickly.

SECTION 5

Table Method vs. Equation Method

You'll usually see two setups on the SSAT: a problem that gives you a table of values, or a problem that gives you a word problem you turn into an equation. The diagram below walks through both methods side by side for the same problem.

Both methods reach $8.00. The table method is great when you already have a table and need to find k. The equation method is faster when you can set up the proportion directly from a word problem.

On the SSAT, the table method is most useful when the problem already shows you a table with one missing entry. The equation method shines when the problem gives you a word problem like "If 5 pens cost $3, how much do 20 pens cost?" Either way, the answer is the same, so pick whichever feels easier for that problem.

SECTION 6

Worked Example: Solving Step by Step

Let's solve a complete problem from start to finish. Read the problem, then follow each step carefully.

📝 PROBLEM

A car travels at a constant speed. In 2 hours, it covers 90 miles. How many miles will it cover in 7 hours?

Full Solution

Step 1 — Identify the Given InformationWe know that 2 hours → 90 miles. We want to find how many miles correspond to 7 hours. Because the car travels at a constant speed, this is a proportional relationship.

Step 2 — Find the Constant kDivide miles by hours: k = 90 ÷ 2 = 45. The car goes 45 miles per hour.

k = 45 miles per hour

Step 3 — Write the EquationUsing y = k × x, we get y = 45 × x, where x is the number of hours and y is the number of miles.

y = 45 × x

Step 4 — Substitute and SolvePlug in x = 7: y = 45 × 7 = 315.

y = 315 miles

Step 5 — Check with Cross-MultiplicationSet up the proportion 90/2 = 315/7. Cross-multiply: 90 × 7 = 630 and 2 × 315 = 630. Both sides match, so our answer is correct.

630 = 630 ✓

SECTION 7

Strengths and Limitations of Each Method

Both the table method and the equation method will get you the right answer. But depending on the problem, one method might be faster or easier. Here's a quick comparison.

Comparison of the two main methods for solving proportional relationships.

Feature	Table Method	Equation Method (Cross-Multiply)
Best when…	A table of values is given and one entry is missing.	You have a word problem with two known values and one unknown.
Speed	Slightly slower — you find k first, then multiply.	Fast — you set up and cross-multiply in two quick steps.
Strengths	Helps you see all the pairs at once. Great for checking whether a relationship is proportional.	Works directly. No need to build a full table.
Limitations	Takes extra time if you have to create the table yourself.	You must set the fractions up correctly — a flipped fraction gives a wrong answer.
Common mistake	Mixing up which column is x and which is y.	Putting hours in one fraction's numerator but dollars in the other's numerator.

✦ KEY TAKEAWAY

Think of it like choosing between stairs and an elevator. Both take you to the same floor. The table method is like the stairs — you see every step clearly. The equation method is like the elevator — faster but you need to push the right button (set up the fractions correctly).

SECTION 8

From Proportions to Linear Equations

Proportional relationships are actually a special type of something bigger called a linear equation. In later math classes, you'll learn about equations like y = mx + b, where m is the slope and b is the y-intercept (the starting value when x = 0). Proportional relationships are the version where b = 0.

Proportional relationships are a special case of linear equations where b = 0.

Feature	Proportional (y = kx)	Linear (y = mx + b)
Graph shape	Straight line through (0, 0)	Straight line, but may not pass through (0, 0)
Starting value	Always 0 (no starting amount)	Can be any number (b)
Example	Earning $10/hour: y = 10x	Earning $10/hour + $20 bonus: y = 10x + 20
y ÷ x always the same?	Yes — always equals k	No — only the rate of change (slope) stays constant

For now, you don't need to worry about y = mx + b on the SSAT. Just know that mastering proportions gives you a head start on linear equations in algebra class. The skills you're building here — setting up ratios, finding k, and cross-multiplying — will be useful for years.

SECTION 9

Practice Problems

Try these five problems. They start easy and get harder. For each one, pick the best answer from the five choices. After you choose, read the explanation to make sure you understand.

PROBLEM 1 — CONCEPTUAL

A table shows these pairs: (2, 6), (4, 12), (6, 18), (8, 24). What is the constant of proportionality k? (A) 2 (B) 3 (C) 4 (D) 6 (E) 8

PROBLEM 2 — BASIC CALCULATION

If 4 notebooks cost $10, how much do 10 notebooks cost? (A) $14 (B) $20 (C) $25 (D) $40 (E) $50

PROBLEM 3 — INTERMEDIATE

A recipe uses 3 cups of flour for every 2 cups of sugar. If you use 9 cups of flour, how many cups of sugar do you need? (A) 4 (B) 5 (C) 6 (D) 7 (E) 8

PROBLEM 4 — APPLIED

On a map, 2 centimeters represent 50 kilometers. Two cities are 7 centimeters apart on the map. What is the actual distance between the cities? (A) 100 km (B) 150 km (C) 175 km (D) 200 km (E) 350 km

PROBLEM 5 — CRITICAL THINKING

A table shows the proportional relationship below: x: 5, 10, 15, 25 y: 8, 16, ?, 40 What is the missing value of y when x = 15? (A) 20 (B) 22 (C) 24 (D) 26 (E) 28

SUMMARY

Proportional Relationships — Quick Review

A proportion is an equation stating that two ratios are equal. To solve one, you can use the table method — find the constant of proportionality (k = y ÷ x) and then use y = k × x — or the equation method — set up two equal fractions and cross-multiply (a × d = b × c).

Remember: a proportional relationship always passes through the origin (0, 0) on a graph and every y ÷ x gives the same constant k. Whether you use a table or an equation, always check your answer by plugging it back in. Proportions show up everywhere — in recipes, maps, money, and speed problems — so this is a skill you will use again and again on the SSAT and beyond.