Two ways quantities relate. Direct: the value pair always sits on a straight line through the origin, because the ratio stays fixed. Inverse: the pair rides a curve that swoops down and flattens, because the product stays fixed.
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.
4.3.1 Quantities in direct proportion
Apples cost $3 a pound. Buy 1 pound, pay $3. Buy 2 pounds, pay $6. Buy 5 pounds, pay $15. Notice what stays the same no matter how much you buy: the price per pound. Divide the cost by the weight and you always land back on the same number, 3. When two quantities behave like this — one is always the same fixed multiple of the other — we say they are in direct proportion.
Let the weight be x and the cost be y. The steady fact is that the ratioy / x never changes. Call that fixed value k:
y / x = k (always the same), which rearranges to y = k · x.
Weight x (lb)
Cost y ($)
Ratio y / x
1
3
3
2
6
3
3
9
3
4
12
3
5
15
3
The cost grows, the weight grows — but the highlighted ratio column stays locked at 3. That unchanging 3 is the constant of proportionality k: here it is just the price per pound.
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.
Direct proportion — the test and the rule
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.
Worked example — finding k
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.
🎮 Try itA direct-proportion table
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.
Unit price k3
Rows up to x =4
4.3.2 The graph of direct proportion
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).
Every pair lands on one straight line, and the line runs through the origin. The steepness of that line — how far y climbs each time x moves one step — is exactly the constant k = 3. In graphing language, k is the slope.
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).
A straight line is not enough
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.
🎮 Try itGraph y = k·x
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.
Slope k2
Point at x =3
4.3.3 Quantities in inverse proportion
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
As the speed rises, the time falls — they move in opposite directions. Yet the highlighted product column never budges from 120, the fixed distance. That fixed product is k.
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 dividey by that same number. Go three times as fast, take one third as long.
Inverse proportion — the test and the rule
Test:multiplyx 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.
Worked example — sharing a fixed job
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.
🎮 Try itAn inverse-proportion table
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.
Fixed distance k
Rows up to step5
4.3.4 The graph of inverse proportion
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 inverse graph is a curve, not a line. It plunges where x is small (a slow speed means a long time), then levels off where x is large. The dashed axes are asymptotes — lines the curve forever approaches but never quite reaches.
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.
🎮 Try itGraph y = k / x
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.
Constant k
Point at x =3
4.3.5 Telling them apart, and using them
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.
The decision routine in one picture. The direction is your first clue; the constant-ratio or constant-product test is the proof.
"They both go up" does NOT prove direct
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.
Worked example — classify two tables
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.
🎮 Try itDirect, inverse, or neither?
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.
★ The big ideas, side by side
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.
Coming up next — 4.4
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.
✎ Exercises 4.3
Work each one out first, then open the answer to check your thinking.
A direct-proportion table has x = 4 paired with y = 28. Find the constant of proportionality k, and write the rule connecting x and y.
Show answer
k = y / x = 28 / 4 = 7. The rule is y = 7x. (So when x = 1, y = 7 — that is what k means.)
Pencils cost a steady price. The table shows x (pencils) and y (cost in ¢): (2, 30), (5, ?), (8, 120). Complete the missing value.
Show answer
Find k from a known pair: 30 / 2 = 15¢ each (check: 120 / 8 = 15 ✓). So for 5 pencils, y = 15 × 5 = 75¢.
A graph is a perfectly straight line, but it crosses the y-axis at 4 (not at 0). Are the two quantities in direct proportion? Explain.
Show answer
No. Direct proportion requires the line to pass through the origin (0,0). This line hits the y-axis at 4, so when x = 0, y = 4 — not 0. Doubling x would not double y, so it fails the test. Straight is not enough; it must also go through the origin.
In an inverse relationship, x · y = 48. If x = 6, find y. If instead x = 16, find y.
Show answer
Use y = 48 / x. When x = 6: y = 48 / 6 = 8. When x = 16: y = 48 / 16 = 3. (Check: 6 × 8 = 48 ✓ and 16 × 3 = 48 ✓.)
For each table, say whether it is direct, inverse, or neither, and give k when it exists. (a) x: 1,2,3,4 — y: 6,12,18,24 (b) x: 2,3,4,6 — y: 12,8,6,4 (c) x: 1,2,3,4 — y: 1,4,9,16
Show answer
(a) Direct. y/x = 6 every row, so k = 6, rule y = 6x. (b) Inverse. x·y = 24 every row (2·12, 3·8, 4·6, 6·4), so k = 24, rule y = 24/x. (c) Neither. y/x = 1, 2, 3, 4 (not constant) and x·y = 1, 8, 27, 64 (not constant). These are the squares — a different rule entirely.
A job needs 72 worker-hours. Build the inverse table for 2, 3, 4, 6, and 8 workers (hours each), and state the rule.
Show answer
k = 72, so hours = 72 / workers:
2 → 36, 3 → 24, 4 → 18, 6 → 12, 8 → 9. (Each product is 72.) Rule: y = 72 / x. More workers, proportionally fewer hours.
A car covers a fixed 150 miles. At 50 mph it takes 3 hours. How long at 75 mph? Is this direct or inverse — and explain how you knew.
Show answer
Inverse — distance is fixed, so speed × time = 150 stays constant; faster speed must mean less time. At 75 mph: time = 150 / 75 = 2 hours. (Speed went up by a factor of 1.5, so time dropped to 3 ÷ 1.5 = 2.)
Sketch (or describe) the graph of y = 5x and the graph of y = 5 / x. Name two features that tell them apart.
Show answer
y = 5x is a straight line through the origin, climbing steadily (slope 5). y = 5/x is a curve (a hyperbola branch) that falls steeply then flattens, approaching the axes but never touching them and never passing through the origin. Two distinguishing features: line vs curve, and passes through (0,0) vs hugs but never reaches the axes.
Challenge. A recipe for 4 people uses 6 eggs. (a) Eggs needed for 10 people? Is this direct or inverse? (b) That same recipe takes one cook 40 minutes; with 2 cooks sharing the work it takes 20 minutes. Is that direct or inverse?
Show answer
(a) Direct: eggs rise with people at a fixed ratio, 6/4 = 1.5 eggs per person. For 10 people: 1.5 × 10 = 15 eggs. (b) Inverse: the job is fixed, so cooks × time stays constant (1 × 40 = 40 = 2 × 20). More cooks → proportionally less time. The same situation contains both kinds of proportion — one for ingredients, one for labor.
Challenge. Is the table direct, inverse, or neither? x: 1, 2, 5, 10 y: 60, 30, 12, 6. If it is one of them, give the rule and predict y when x = 4.
Show answer
As x rises, y falls — suspect inverse. Test the product: 1·60, 2·30, 5·12, 10·6 all equal 60. Constant, so inverse, k = 60, rule y = 60 / x. When x = 4: y = 60 / 4 = 15.
🎯 Quick check
Six questions to lock it in. Tap the answer you think is right.
§ For teachers and parents
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.
eastmath.com · Stage 4 · 4.3 Direct & Inverse Proportion · Intuition before notation
eastmath.com · 4.3 Direct and Inverse Proportion · 4.3.1 Quantities in direct proportion