Chapter 10.2 — Exercise 10.2 — Inverse Proportion
Introduction of inverse proportion. This is Lesson 2 of 4 in Chapter 10: Direct and Inverse Proportions.
The Opposite Kind of Partnership
Not every pair of related quantities grows together. Speed up a journey and travel time drops; add more workers to a job and the days needed to finish it shrinks. When one quantity rises exactly as fast as the other falls — so that their product, not their ratio, stays fixed — the two are in inverse proportion, written x ∝ 1/y.
x ∝ 1/y means xy = k (constant)
For any two value-pairs: x₁y₁ = x₂y₂The test is mechanical and reliable: multiply each x-value by its matching y-value across an entire table. If every product comes out the same, the two columns are inversely proportional; if even one product differs, they aren't — no matter how convincing the pattern looks otherwise.
Running the Product Test on Three Tables
Applying that test to three separate data sets makes the difference concrete:
| x | 50 | 40 | 30 | 20 |
|---|---|---|---|---|
| y | 5 | 6 | 7 | 8 |
| xy | 250 | 240 | 210 | 160 |
The products here — 250, 240, 210, 160 — are all different, so x and y in Table A are not in inverse proportion, even though x is clearly decreasing as y increases.
| x | 100 | 200 | 300 | 400 |
|---|---|---|---|---|
| y | 60 | 30 | 20 | 15 |
| xy | 6000 | 6000 | 6000 | 6000 |
Every product in Table B equals 6000, so this pair genuinely is in inverse proportion.
| x | 90 | 60 | 45 | 30 | 20 | 5 |
|---|---|---|---|---|---|---|
| y | 10 | 15 | 20 | 25 | 30 | 25 |
| xy | 900 | 900 | 900 | 750 | 600 | 125 |
Table C starts convincingly — the first three products all equal 900 — but the last three break the pattern, so the columns as a whole are not inversely proportional. This is exactly why the check has to be run across every single pair in a table, not just the first two or three: a relationship that looks inverse for part of a table can stop being inverse partway through.
A Fixed Budget Spreads Across a Variable Price
A school has exactly ₹6000 to spend on books. As the price per book rises, the number of books it can afford falls — a textbook case of xy = k, with k = 6000 throughout:
| Price per book (₹) | Number of books | Check |
|---|---|---|
| 40 | 150 | 40 × 150 = 6000 |
| 50 | 120 | 50 × 120 = 6000 |
| 60 | 100 | 60 × 100 = 6000 |
| 75 | 80 | 75 × 80 = 6000 |
| 80 | 75 | 80 × 75 = 6000 |
The last two rows are worth a second look: at ₹75 a book the school buys 80 books, and at ₹80 a book it buys 75 — price and quantity have literally swapped roles between those two rows, which is only possible because their product stays locked at exactly 6000 either way.
Squares on a Page: Rows Against Columns
Arrange 48 squares from a squared-paper sheet into a grid of R rows and C columns — since the total count is fixed, more rows always means fewer columns:
| Rows (R) | Columns (C) |
|---|---|
| 2 | 24 |
| 3 | 16 |
| 8 | 6 |
Two ratio checks confirm the inverse relationship holds exactly, not just approximately: comparing the R = 2 and R = 3 rows, R₁ : R₂ = 2 : 3, and C₂ : C₁ = 16 : 24 = 2 : 3 — the same ratio, just flipped between rows and columns. Comparing R = 8 (row 3) against a hypothetical R = 12 (row 4) gives the identical pattern: R₃ : R₄ = 4 : 6 = 2 : 3, matching C₄ : C₃ = 8 : 12 = 2 : 3. Whenever one ratio equals the reverse of the other — rather than the same ratio repeated — that is itself the signature of an inverse relationship, the mirror image of the direct-proportion test from Exercise 10.1.
The same idea repeats with any fixed total: 36 squares split into R = 2, C = 18; R = 3, C = 12; R = 4, C = 9; or R = 6, C = 6 — every pair multiplying back to exactly 36, regardless of how the rows and columns are divided between them.
Why the Ratio Flips Instead of Repeating
The R-and-C check above — R₁ : R₂ turning out equal to C₂ : C₁, rather than C₁ : C₂ — isn't a coincidence specific to squared paper; it falls straight out of the inverse-proportion formula itself. Starting from x₁y₁ = x₂y₂ and dividing both sides by x₂y₁ gives x₁/x₂ = y₂/y₁ — the ratio of the x-values equals the reversed ratio of the y-values. Compare that to direct proportion, where x₁/x₂ = y₁/y₂ keeps the same order on both sides. That one flipped pair of subscripts is the entire algebraic difference between the two kinds of proportion, and it's exactly what the R : C check was verifying by hand.
A Partial Match Isn't Proof
Table C is worth returning to for a moment, because it demonstrates a genuine trap: the first three products (900, 900, 900) look like solid evidence of an inverse relationship, and a student checking only the first three columns would confidently — and wrongly — call the whole table inversely proportional. The last two products (750 and 600) show the relationship actually failed partway through. There's no shortcut around checking every column; a rule that holds for 3 out of 6 data points hasn't been shown to hold at all, since the remaining 3 could always break it, exactly as they do here.
Checking the Idea Against a Familiar Situation
A quick sanity check, away from tables of numbers: a car covering 120 km takes 2 hours at 60 km/hr, or 3 hours at 40 km/hr, or 4 hours at 30 km/hr. Multiplying speed by time in every case gives exactly 120 — the fixed distance — confirming that speed and time, for a fixed distance, are inversely proportional in exactly the same xy = k sense as the book prices and the squared-paper grids above. This is worth holding onto, because it's the version of inverse proportion that shows up constantly outside a maths textbook: faster travel always means less time, at a rate that multiplies out to the same fixed total every time. The same pattern governs a shared job split among workers — more workers on the same task always means fewer days needed, with the total "worker-days" required staying fixed — which is precisely the situation the next exercise builds on, using the very same xy = k logic developed here.
Two Sides of One Coin
Direct and inverse proportion answer the same underlying question — "if one quantity changes, what happens to the other?" — but with opposite answers: a constant ratio in Exercise 10.1, a constant product here. Exercise 10.3 takes this same xy = k relationship into real situations — workers and work-days, pipes and fill-times, pumps and tanks — where recognising the inverse pattern correctly is the entire challenge.