Proportional
Relationships
Master how to identify and solve proportional relationships using tables, equations, and graphs: a key skill for the SSAT Middle Level.
What Is a Ratio?
A ratio compares two quantities. If you have 3 apples and 6 oranges, the ratio of apples to oranges is \(\frac{3}{6} = \frac{1}{2}\). This means for every 1 apple, there are 2 oranges.
What Does Proportional Mean?
Two quantities are proportional when they increase or decrease at the same constant rate. If you double one quantity, the other doubles too.
The Constant of Proportionality
In every proportional relationship, the ratio \(\frac{y}{x}\) is always the same number. We call this the constant of proportionality, written as \(k\).
The Equation \(y = kx\)
Every proportional relationship can be written as \(y = kx\). This equation tells you: multiply the input by the constant to get the output.
Reading a Table for Proportions
A table is proportional if every row gives the same value when you divide \(y \div x\).
| Gallons (\(x\)) | Miles (\(y\)) | \(\frac{y}{x}\) |
|---|---|---|
| 2 | 50 | 25 |
| 4 | 100 | 25 |
| 6 | 150 | 25 |
| 8 | 200 | 25 |
Proportional vs. Not Proportional
| Proportional | Not Proportional | |
|---|---|---|
| Equation form | \(y = kx\) | \(y = kx + b\), \(b \neq 0\) |
| Graph | Straight line through (0, 0) | Line misses origin, or curves |
| Table test | \(\frac{y}{x}\) is constant | \(\frac{y}{x}\) varies row to row |
| Real example | \(\$3\) per lb of fruit | \(\$3\) per lb + \(\$2\) bag fee |
How to Solve Proportion Problems
Example 1: Finding \(k\) from a Table
A baker uses the following recipe. Is the relationship between cups of flour and cookies proportional? If so, find \(k\).
| Cups of Flour (\(x\)) | Cookies (\(y\)) |
|---|---|
| 2 | 24 |
| 5 | 60 |
| 8 | 96 |
Example 2: Solving for a Missing Value
A store sells notebooks at a constant price. If 4 notebooks cost \(\$14\), how much do 10 notebooks cost?
Example 3: Graph to Equation
The graph below shows a proportional relationship. Write the equation.
Cross-Multiplication Shortcut
When two ratios form a proportion, \(\frac{a}{b} = \frac{c}{d}\), you can cross-multiply to solve for unknowns.
What Is a Unit Rate?
A unit rate is the value of \(k\) when \(x = 1\). It tells you the amount of \(y\) per one unit of \(x\). The unit rate is the constant of proportionality.
Common Mistakes to Avoid
Summary: Key Concepts
| Concept | What It Means | How to Find It |
|---|---|---|
| Ratio | Comparison of two quantities | \(\frac{a}{b}\) or \(a : b\) |
| Proportion | Two equal ratios | \(\frac{a}{b} = \frac{c}{d}\) |
| Constant \(k\) | Rate that stays the same | \(k = \frac{y}{x}\) for any row |
| Equation | Rule connecting \(x\) and \(y\) | \(y = kx\) |
| Unit Rate | Value of \(y\) when \(x = 1\) | Same as \(k\) |
Question 1 of 5
A table shows that when \(x = 3\), \(y = 15\) and when \(x = 7\), \(y = 35\). What is the constant of proportionality \(k\)?
Question 2 of 5
If \(\frac{4}{6} = \frac{x}{15}\), what is the value of \(x\)?
Question 3 of 5
Which equation represents a proportional relationship?
Question 4 of 5
A car travels 240 miles on 8 gallons of gas. At the same rate, how many miles can it travel on 5 gallons?
Question 5 of 5
A table shows: \(x = 2, y = 10\); \(x = 4, y = 18\); \(x = 6, y = 30\). Is this relationship proportional?