Two quantities can rise together with a steady ratio — or trade off with a steady product. Here is how to tell, graph, and use each one.
Point 5 of 5 in this lesson: 4.3.5 Telling them apart, and using them
In Lesson 4.2 you learned that a proportion is two equal ratios, and that "in proportion" means a ratio holds steady as both numbers grow. This lesson takes that idea and lets the numbers actually move. Sometimes two quantities climb together hand in hand — buy twice as much, pay twice as much. Other times they pull against each other — go twice as fast, arrive in half the time. Both are kinds of proportion, but they obey opposite rules, and the whole art is learning to tell which is which.
By the end of this lesson you will be able to do five things: recognize direct proportion and find its constant k from a table; graph a direct relationship as a straight line through the origin; recognize inverse proportion by its constant product; sketch its falling curve; and run a quick test that tells the two apart for any table of numbers. Keep one color habit the whole way: the first quantity (the input, our x) is amber, the second quantity (the output, our y) is blue, and a third term, when one appears, is purple.
Apples cost $3 a pound. Buy 1 pound, pay $3. Buy 2 pounds, pay $6. Buy 5 pounds, pay $15. Notice what stays the same no matter how much you buy: the price per pound. Divide the cost by the weight and you always land back on the same number, 3. When two quantities behave like this — one is always the same fixed multiple of the other — we say they are in direct proportion.
Let the weight be x and the cost be y. The steady fact is that the ratio y / x never changes. Call that fixed value k:
y / x = k (always the same), which rearranges to y = k · x.
| Weight x (lb) | Cost y ($) | Ratio y / x |
|---|---|---|
| 1 | 3 | 3 |
| 2 | 6 | 3 |
| 3 | 9 | 3 |
| 4 | 12 | 3 |
| 5 | 15 | 3 |
The number k has a name worth knowing: the constant of proportionality. It is the value of y when x is exactly 1 — the cost of one pound, the distance in one hour, the amount per single unit. Once you know k, the whole relationship is settled: multiply any x by k to get its y.
Test: divide y by x for every pair. If you always get the same number k, the quantities are in direct proportion.
Rule: y = k · x, where k = y / x is the constant of proportionality. Double x and y doubles too; triple x and y triples.
A car travels at a steady speed. In 2 hours it covers 130 miles. Is distance in direct proportion to time, and if so, what is k?
Divide distance by time: 130 / 2 = 65. Check another point — in 3 hours it would go 195 miles, and 195 / 3 = 65 again. The ratio holds, so yes, direct. Here k = 65 miles per hour, and the rule is y = 65x.
Set the unit price k (the constant) and watch the table fill. The amount x climbs, the cost y = k·x climbs with it — but look at the right-hand y / x column: it stays glued to k.
A picture makes direct proportion unmistakable. Take each x, y pair from the table, treat it as a point, and plot it. For the $3-a-pound apples we plot (1,3), (2,6), (3,9), (4,12). Two things jump out: the points march in a perfectly straight line, and that line, extended back, passes exactly through the origin, the point (0,0).
Why must the line pass through the origin? Because when x = 0 the rule says y = k · 0 = 0. Zero pounds of apples cost zero dollars; zero hours of driving covers zero miles. Buying nothing costs nothing — so the point (0,0) is always on a direct-proportion graph. That is the visual fingerprint: a straight line, and it goes through (0,0).
It is tempting to say "straight line means direct proportion," but that is only half true. The line must also pass through the origin. A line like y = 2x + 5 is perfectly straight, yet it crosses the y-axis at 5, not 0 — so doubling x does not double y, and these quantities are not in direct proportion. Straight and through the origin: both are required.
Slide the slope k and watch the line tilt — but it always pivots through the origin. Step the x-marker along the line to read off the matching point and confirm y / x equals k every time.
Now flip the relationship. Suppose a journey is a fixed 120 miles. The faster you drive, the less time it takes — speed and time pull against each other. Drive 60 mph and it takes 2 hours; drive 40 mph and it takes 3 hours; crawl at 20 mph and it takes 6 hours. As one number goes up, the other goes down. This is inverse proportion.
What stays constant is not the ratio this time — it is the product. Speed times time always rebuilds the same fixed distance: 60 × 2 = 120, 40 × 3 = 120, 20 × 6 = 120. Let x be the speed and y the time:
x · y = k (always the same), which rearranges to y = k / x.
| Speed x (mph) | Time y (h) | Product x · y |
|---|---|---|
| 10 | 12 | 120 |
| 20 | 6 | 120 |
| 30 | 4 | 120 |
| 40 | 3 | 120 |
| 60 | 2 | 120 |
Here the constant k is the whole fixed quantity being shared out — the total distance, the total job, the total amount. And the giveaway of inverse proportion is the trade-off: multiply x by some number and you divide y by that same number. Go three times as fast, take one third as long.
Test: multiply x by y for every pair. If you always get the same number k, the quantities are in inverse proportion.
Rule: x · y = k, or equivalently y = k / x. Double x and y halves; triple x and y drops to a third.
It takes 240 worker-hours to paint a fence. With 4 painters it takes 60 hours; with 6 painters, 40 hours; with 8 painters, 30 hours. Is time in inverse proportion to the number of painters?
Multiply painters by hours: 4 × 60 = 240, 6 × 40 = 240, 8 × 30 = 240. The product holds at 240, so yes, inverse. The constant k = 240 is the total size of the job, and time = 240 / painters. More painters, proportionally less time.
Pick the fixed distance k, then step the speed x through its values. The time y = k / x falls as speed rises — yet the x · y column stays pinned to k.
Plot the inverse pairs the same way — but the picture changes completely. Take the 120-mile table and plot (10,12), (20,6), (30,4), (40,3), (60,2). The points do not line up. They trace a smooth curve that drops steeply at first and then flattens out, gliding closer and closer to the axes without ever touching them. This curve is one branch of a shape called a hyperbola.
The shape makes physical sense. When the speed x is tiny, the time y is enormous, so the curve shoots up near the y-axis. As the speed grows huge, the time shrinks toward zero but can never actually be zero — a finite journey always takes some time — so the curve sinks toward the x-axis without landing on it. And the curve can never touch the y-axis either, because x = 0 would mean dividing by zero, which is undefined: at zero speed you never arrive at all.
Choose the constant k and watch the whole curve redraw. Step the point along it and check that x · y stays equal to k at every spot — even as the point slides toward the flat tail.
Hand someone a table of numbers and ask "direct, inverse, or neither?" Here is the routine that never fails. First glance at the direction: as x rises, does y rise too, or fall? That hints at the answer — but a hint is not a proof. So then you run the matching arithmetic test:
① The numbers rise together → suspect direct → check whether y / x is constant.
② One rises while the other falls → suspect inverse → check whether x · y is constant.
③ If neither the ratio nor the product holds steady → it is neither kind of proportion.
This is the trap that catches everyone. Two quantities can rise together and still not be in direct proportion — what matters is that the ratio stays constant. Look at x = 1, 2, 3 with y = 2, 5, 10: y climbs as x climbs, but y / x = 2, then 2.5, then 3.33… — not constant. And x · y = 2, 10, 30 — not constant either. So this is neither. "Rises together" is a clue, never a conclusion: always run the ratio test.
Table A: x = 2, 4, 6, 8 and y = 5, 10, 15, 20. They rise together — suspect direct. Test the ratio: 5/2, 10/4, 15/6, 20/8 all equal 2.5. Constant! So Table A is direct, with k = 2.5 and rule y = 2.5x.
Table B: x = 1, 2, 4, 5 and y = 20, 10, 5, 4. As x rises, y falls — suspect inverse. Test the product: 1·20, 2·10, 4·5, 5·4 all equal 20. Constant! So Table B is inverse, with k = 20 and rule y = 20 / x.
A small table appears. Decide what kind of relationship it is and tap your answer. The widget then runs both tests on the actual numbers and shows you which one passed — the y / x ratio or the x · y product.
Two quantities can be tied together in two opposite ways. In direct proportion they rise together with a constant ratio: y / x = k, the rule is y = kx, and the graph is a straight line through the origin whose slope is k. In inverse proportion they trade off with a constant product: x · y = k, the rule is y = k / x, and the graph is a falling curve that hugs the axes but never touches them. To classify any table: rise together → test the ratio; trade off → test the product; if neither stays constant, it is neither. Above all, remember that "rising together" is only a clue — the constant ratio is the real proof.
Direct proportion has one especially famous special case: comparing everything to a fixed whole of 100. That is exactly what a percentage is — a ratio "out of 100." In Lesson 4.4 you will see how the constant-ratio thinking you just built turns straight into percentages.
Work each one out first, then open the answer to check your thinking.
Six questions to lock it in. Tap the answer you think is right.
This lesson develops proportional relationships and is aligned to the U.S. Common Core standards for grades 7 and 8. Direct proportion is treated exactly as the standards intend: students decide whether two quantities are in a proportional relationship by testing for a constant ratio, identify the constant of proportionality k (as a unit rate) from tables, equations, and graphs, and recognize that the graph is a straight line through the origin with slope k (7.RP.A.2, including 7.RP.A.2a–d). Comparing the direct line against a non-proportional line, and against the inverse curve, draws on the function ideas of 8.F.A and 8.F.B (interpreting and comparing relationships and their graphs). Inverse variation (xy = k, y = k/x) is included as a deliberate extension that points forward to Algebra I; the emphasis here is conceptual — the constant-product test and the hyperbola's behavior near its asymptotes — rather than formal function notation. The single most common misconception, addressed directly in Section 4.3.5 and the quiz, is the belief that "two quantities rising together must be directly proportional." The antidote, practiced throughout, is to always confirm the relationship with the constant-ratio test (for direct) or the constant-product test (for inverse) rather than relying on direction alone.