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Pre-Algebra • Ratios, Rates & Proportional Reasoning

y=kx & Constant of Proportionality — I can represent proportional relationships with equations of the form y=kx and interpret k as the constant of proportionality.

Discover how one simple equation captures every proportional relationship you'll ever see.

Explore the lesson 9 min read
Middle School Math emblem with sigma notation, geometric shapes, and a ruler.
Inside this lesson 10 sections 5 questions 7 figures

SECTION 1

Historical Context & Motivation

People have been comparing quantities for thousands of years. Ancient builders needed to know how much stone to order for a wall twice as long. Merchants needed to figure out the cost of five bags of grain if they knew the price of one. The idea of proportional relationships — where two quantities grow at the same rate — is one of the oldest ideas in math.

~1800 BCE • Babylonian Ratio Tables

Ancient Babylonians carved clay tablets listing prices. If 1 basket of barley cost 2 silver coins, they wrote tables showing 2 baskets cost 4, 3 baskets cost 6, and so on.

~300 BCE • Euclid's Proportions

The Greek mathematician Euclid wrote rules about proportions in his famous book Elements. He showed that proportional reasoning applies to geometry, not just numbers.

1600s • Variables Enter Math

Mathematicians like René Descartes began using letters like x and y to stand for unknown numbers. This made it possible to write a single equation for an entire relationship.

Today • y = kx Everywhere

Scientists, engineers, and everyday people use y = kx to model unit prices, speed, recipes, currency exchange, and much more.

Here's the big question this lesson answers: when two quantities always change at the same rate, how can we describe that relationship with one short equation? The answer is y = kx, and the letter k tells us exactly how the two quantities are connected.

SECTION 2

Core Principles & Definitions

Before we dive into the equation, let's lock in some key ideas. These four principles are the building blocks for everything that follows.

1

Proportional Relationship

Two quantities are proportional when they always have the same ratio. If you double one, the other doubles too.
2

Constant of Proportionality (k)

The constant of proportionality, written as k, is the number you multiply x by to get y. It stays the same for every pair of values.
3

The Equation y = kx

This equation says: "y equals k times x." It's the shorthand for any proportional relationship. The k tells you the rate.
4

Finding k from a Table

To find k, pick any row of a table and divide: k = y ÷ x. If the answer is the same for every row, the relationship is proportional.
✦

KEY TAKEAWAY

Think of k like a recipe multiplier. If a smoothie recipe uses 2 bananas per serving, then k = 2. For 3 servings you need 2 × 3 = 6 bananas. No matter how many servings you make, the rate (2 bananas per serving) never changes. That steady rate is the constant of proportionality.

SECTION 3

Seeing Proportional Relationships

A proportional relationship has a special look on a graph. It always forms a straight line that passes through the origin (0, 0). The steeper the line, the larger the value of k.

Proportional Relationships on a Graphx (servings)y (bananas)0246801234(1, 2)(2, 4)(3, 6)(4, 8)(3, 4.5)k = 2 (steeper)k = 1.5 (less steep)
Both lines start at the origin (0, 0). The cyan line (k = 2) is steeper because y grows faster. The violet dashed line (k = 1.5) rises more gently. A bigger k always means a steeper line.

Notice that both lines pass through (0, 0). This makes sense: if you buy 0 servings, you need 0 bananas! Every proportional relationship starts at the origin. If a line doesn't go through (0, 0), the relationship is not proportional.

SECTION 4

The Mathematical Framework

Let's look at the equation and the formulas you'll use most often. Don't worry — there are only two main formulas, and they're really the same idea flipped around.

PROPORTIONAL RELATIONSHIP

y=k×xy = k × x

y = the output (what you're finding), k = the constant of proportionality (the rate), x = the input (what you already know).

FINDING k

k=y÷xk = y ÷ x

Divide any y-value by its matching x-value. If the answer is the same for every pair, then k is that number and the relationship is proportional.

EQUIVALENT RATIO TEST

y1÷x1=y2÷x2=y3÷x3=…=ky₁ ÷ x₁ = y₂ ÷ x₂ = y₃ ÷ x₃ = … = k

Every pair in a proportional table gives the same quotient. That shared quotient is k.

💡

Remember!

The constant of proportionality k is also called the unit rate. If bananas cost $0.50 each, then k = 0.50 and the equation is y = 0.50x, where x is the number of bananas and y is the total cost.

SECTION 5

Finding k in Tables

One of the most common ways you'll see proportional relationships is in a table. Let's look at two tables side by side — one that is proportional and one that is not.

✓ Proportional✗ NOT ProportionalHours (x)Pay (y)y ÷ x1$882$1683$2485$408k = 8 → y = 8x ✓Hours (x)Pay (y)y ÷ x1$10102$157.53$206.75$306Ratios differ → NOT y = kx ✗How to Check for Proportionality1. Divide y by x for every row.2. If every quotient is the SAME number, the relationship is proportional and that number is k.
On the left, every row gives y ÷ x = 8, so k = 8 and the equation is y = 8x. On the right, the ratios are all different, so that table does not show a proportional relationship.

The left table might represent a job that pays $8 per hour with no flat starting bonus. The right table might represent a job that pays an hourly rate plus a sign-up bonus. That extra bonus breaks the proportional pattern.

SECTION 6

Worked Example

Let's walk through a full problem from start to finish. Follow each step carefully.

Lemonade Stand Earnings

  1. 1

    Step 1 — Read the Problem

    Maya sells cups of lemonade. She earns $3 for every cup she sells. Write an equation in the form y = kx. Then find how much she earns after selling 15 cups.
  2. 2

    Step 2 — Identify the Variables

    Let x = the number of cups sold. Let y = the total earnings in dollars. The rate (dollars per cup) is the constant of proportionality.
    x = cups, y = dollars, k = dollars per cup
  3. 3

    Step 3 — Find k

    Maya earns $3 per cup. That means for every 1 cup, y goes up by 3. So k = 3.
    k = 3
  4. 4

    Step 4 — Write the Equation

    Substitute k = 3 into y = kx.
    y = 3x
  5. 5

    Step 5 — Solve for 15 Cups

    Plug in x = 15: y = 3 × 15 = 45.
    y = $45
  6. 6

    Step 6 — Check Your Answer

    Does 45 ÷ 15 equal our k? Yes — 45 ÷ 15 = 3. ✓ The answer makes sense.
    Maya earns $45 after selling 15 cups of lemonade.

SECTION 7

Proportional vs. Non-Proportional

Not every relationship between two numbers is proportional. Here's how to tell the difference quickly.

FeatureProportional (y = kx)Non-Proportional
GraphStraight line through (0, 0)Curved line, or straight line that misses (0, 0)
Tabley ÷ x gives the same number every timey ÷ x gives different numbers
Equation formy = kx (no added or subtracted number)y = kx + b, or another form with extra terms
ExampleCost = $5 × (number of tickets)Cost = $5 × (number of tickets) + $2 fee

Comparing proportional and non-proportional relationships

✦

KEY TAKEAWAY

Imagine filling identical water bottles from a faucet. If every bottle takes exactly 10 seconds, that's proportional — k = 10 seconds per bottle. But if you also need 5 seconds to turn on the faucet at the start, those first 5 seconds break the proportion. The "+5" is like the extra number in y = kx + b. Proportional means no extra number — just multiply.

SECTION 8

Connection to Linear Equations

In later math courses you'll meet the equation y = mx + b. This is the general equation for any straight line. The equation y = kx is actually a special case of y = mx + b where b = 0. Knowing y = kx inside and out gives you a head start.

y = kx (This Lesson)y = mx + b (Coming Soon)
What it describesProportional relationships onlyAny straight-line relationship
Slopek is the slopem is the slope
y-interceptAlways 0 (line goes through the origin)b (can be any number)
Exampley = 4xy = 4x + 3

y = kx is a building block for y = mx + b

When you get to slope-intercept form, remember: you already know what the slope means! It's your old friend k, the constant of proportionality. You'll just add one more piece — the y-intercept b — to handle lines that don't start at (0, 0).

SECTION 9

Practice Problems

Try these five problems on your own. They start easy and get harder. Read each answer only after you've given it your best shot!

Question 1: CONCEPTUAL

In the equation y = kx, what does the letter k represent? Explain in your own words why it's called a "constant."
Show answer
The letter k represents the constant of proportionality, which is the rate that connects x and y. It's called a "constant" because its value never changes no matter which (x, y) pair you look at. For example, if k = 5, then y is always 5 times x. Whether x is 1, 10, or 100, you always multiply by 5. That steady multiplier is the "constant" part.

Question 2: BASIC CALCULATION

A car travels at a constant speed. In 2 hours it covers 120 miles. Find the constant of proportionality k (in miles per hour), then write the equation y = kx.
Show answer
Use k = y ÷ x. Here y = 120 miles and x = 2 hours, so k = 120 ÷ 2 = 60 miles per hour. The equation is y = 60x, where x is hours and y is miles. You can verify: when x = 2, y = 60 × 2 = 120 ✓.

Question 3: INTERMEDIATE

A table shows these pairs: (4, 10), (8, 20), (10, 25), (14, 35). Determine whether the relationship is proportional. If it is, state k and write the equation.
Show answer
Check each ratio: 10 ÷ 4 = 2.5, 20 ÷ 8 = 2.5, 25 ÷ 10 = 2.5, and 35 ÷ 14 = 2.5. Every ratio equals 2.5, so the relationship is proportional. The constant of proportionality is k = 2.5 and the equation is y = 2.5x. This means y is always two and a half times x.

Question 4: APPLIED

Sophia is baking cookies. Her recipe says she needs 3 cups of flour for every 24 cookies. She wants to make 60 cookies. Use y = kx to find how many cups of flour she needs. (Hint: let x = number of cookies and y = cups of flour.)
Show answer
First, find k: k = y ÷ x = 3 ÷ 24 = 0.125 cups per cookie. The equation is y = 0.125x. Now plug in x = 60: y = 0.125 × 60 = 7.5 cups of flour. You can double-check by noticing that 60 cookies is 2.5 batches of 24. Since each batch needs 3 cups, 2.5 × 3 = 7.5 cups. ✓

Question 5: CRITICAL THINKING

Two students each claim they have a proportional relationship. Student A says: "y = 7x." Student B says: "y = 7x + 2." Who is correct? Explain why, and describe what Student B's graph would look like compared to Student A's.
Show answer
Student A is correct. The equation y = 7x fits the form y = kx with k = 7, so it is proportional. Student B's equation, y = 7x + 2, has an extra "+2" which means y is not simply k times x. Student B's graph would be a straight line with the same steepness (slope 7), but it would cross the y-axis at (0, 2) instead of at (0, 0). Because it doesn't pass through the origin, it is not a proportional relationship.

SUMMARY

Lesson Summary

Summary

A proportional relationship is one where two quantities always have the same ratio. You can write every proportional relationship as y = kx, where k is the constant of proportionality. To find k, divide any y-value by its matching x-value: k = y ÷ x. If every pair in a table gives the same k, the data is proportional.

On a graph, a proportional relationship always forms a straight line through the origin (0, 0). A larger k makes the line steeper. This concept is the foundation for the slope-intercept form y = mx + b that you'll study next — where y = kx is simply the special case with b = 0.

Varsity Tutors • Pre-Algebra • y=kx & Constant of Proportionality