These questions ask you to classify tables, graphs, equations and everyday relationships. Do not decide from words such as increasing or from a graph merely looking straight; apply a numerical or structural test.
Choose the test that fits the representation
For direct proportion, choose x as the input and y as the dependent amount. In a table, divide every y by its matching x in the same order. In an equation, look for y = kx with no added starting amount. On a graph, require both a straight line and passage through the origin. If direct proportion fails, test inverse proportion by multiplying each matched x and y; only a constant product establishes the inverse case.
| Pattern | Test | Tiny example |
|---|---|---|
| Direct proportion | y divided by x is constant; equivalently y = kx. | The pairs (3, 7.5), (8, 20) and (12, 30) all give y divided by x equal to 2.5. |
| Inverse proportion | x times y is constant. | The pairs (5, 18) and (10, 9) both have product 90, so doubling x halves y. |
| Neither | Neither the quotient nor the product stays constant. | For y = 4x + 6, the added 6 gives a nonzero starting amount, so the relationship is not direct proportion. |
A constant rate of change is weaker than a constant ratio. A relationship can add the same amount for each new unit yet begin above zero; then y divided by x changes. Likewise, two quantities can both rise without any fixed multiplier connecting them. Use the zero-input check and at least two ratios rather than relying on the story's direction.
Reject the tempting wrong judgments
| Tempting judgment | How to reject it |
|---|---|
| Any straight or increasing graph must show direct proportion. | A direct graph must also pass through the origin. A curve or a straight line with a nonzero intercept fails. |
| Adding the same amount per unit guarantees direct proportion. | Substitute zero for the input. A remaining starting amount makes the quotient vary even though the rate of change is constant. |
| The input values must rise by equal gaps for a table to be proportional. | The gaps may differ. Calculate y divided by x for every row; equality of those quotients is what matters. |
| Values that are close, positive or increasing together must have a constant ratio. | Appearance does not establish a multiplier. Compute and compare at least two quotients. |
| A relationship that is not direct must therefore be inverse. | Test the products. If they also change, classify the relationship as neither. |
Try it
Which tables show y directly proportional to x? Select all that apply.
- Pairs (1, 4), (3, 12), and (5, 20)
- Pairs (1, 5), (2, 9), and (3, 13)
- Pairs (2, 7), (4, 14), and (9, 31.5)
- Pairs (3, 8), (6, 15), and (12, 29)
A fixed total of 120 counters is packed into an increasing number of equal bags. How should the relationship between bag count and counters per bag be classified?
- Inverse, because their product remains 120
- Direct, because both quantities describe the counters
- Not related, because the total never changes
- Direct, because adding bags changes the amount per bag
Which situations are not directly proportional? Select all that apply.
- Parking time and cost when there is a fixed entry fee plus an hourly charge.
- Travel time and distance from the start at one constant speed.
- A temperature in degrees Celsius and the same temperature in degrees Fahrenheit.
- Servings and each ingredient amount in one consistently scaled recipe.
- A square's side length and its area.