Chapter 10.3 — Exercise 10.3 — Inverse Applications

Four Word Problems, Same Underlying Equation

Notice the direction of the change is different in every row — price rises, population rises, workers fall, pipes rise — yet the arithmetic is identical throughout: multiply the known pair, divide by the new x to get the new y. Recognising "fixed total, more/fewer shares" is the actual skill; the calculation that follows is routine once that recognition is made.

When the Units Need Sorting First

A ship covers a fixed distance in 10 hours at 16 nautical miles per hour; how much faster must it go to cover the same distance in 8 hours? Speed and time are inversely proportional for a fixed distance, so x₁y₁ = x₂y₂ gives 16 × 10 = x₂ × 8, so x₂ = 20 nautical miles per hour — an increase of 4 nmph over the original 16, not 20 itself, since the question asks how much the speed should go up by, not what the new speed is. (A nautical mile — used specifically for distances at sea — equals 1852 metres, slightly longer than the 1000-metre land kilometre; the proportion itself doesn't care about that difference, since both speeds are measured in the same unit throughout, but it's worth knowing why "nautical miles" shows up instead of the more familiar kilometres in a problem about a ship.)

A related pair of problems needs a time-unit conversion before the proportion can even be set up. 5 pumps fill a tank in 1½ hours (= 90 minutes); how many pumps are needed to fill it in just half an hour (= 30 minutes)?

15 workers build a wall in 48 hours; how many are needed to finish in 30 hours?

And a school day divided into 8 periods of 45 minutes each is reorganised into just 6 periods, with the total number of school hours staying the same:

Every one of these four is the same x₁y₁ = x₂y₂ relationship as before — the only new demand is converting hours to minutes (or vice versa) before plugging numbers in, since 1½ hours and 30 minutes can't be compared meaningfully until they're expressed in the same unit.

A Percentage Change Instead of a Number

A trickier version of inverse proportion asks about percentage change rather than concrete values: if z is directly proportional to x and inversely proportional to y, what's the percentage change in z when x rises 12% and y falls 20%? Rather than picking arbitrary numbers for x and y, the problem is solved entirely through ratios. A 12% rise in x means the new-to-old ratio is x/x₁ = 100/112. A 20% fall in y means y/y₁ = 100/80. Since z ∝ x gives z/z₁ = x/x₁, and z ∝ 1/y gives z/z₁ = y₁/y:

So z₁ is 140% of z — in other words, z increases by 40%. The two percentage changes don't simply add or subtract (12% + 20% ≠ 40%); they combine multiplicatively through their ratios, which is exactly what makes this a compound calculation rather than two separate ones.

Letting the Unknown Stay a Letter

Not every proportion problem resolves to a plain number. If (x + 1) men complete a job in (x + 1) days, how many days will (x + 2) men need for the same job? The inverse relationship x₁y₁ = x₂y₂ still applies, even though every quantity here is an algebraic expression rather than a specific value:

The answer stays in terms of x because the problem never fixes a specific number of men to begin with — the proportion still holds exactly, it simply describes a whole family of possible situations at once instead of a single one. Substituting any specific value confirms the formula behaves sensibly: at x = 3, the original 4 men (x+1) take exactly 4 days (x+1), while the enlarged group of 5 men (x+2) needs only (3+1)²÷(3+2) = 16÷5 = 3.2 days — one extra worker on the same job reduces the time needed, exactly as inverse proportion predicts, even though the whole calculation was carried out without ever fixing a single concrete number until this very last check.

A Rectangle That Isn't Simply Inverse

The last question in this exercise is a deliberate contrast to everything before it. A rectangle has a fixed perimeter of 24 m, so length + breadth = 12 m always. As the length increases from 3 m to 9 m in steps of 1 m, the breadth decreases to match, and the area does something less predictable:

Length and breadth here genuinely move in opposite directions, just as in every inverse-proportion problem above — but their product isn't constant at all; it climbs from 27 up to a peak of 36 exactly where length and breadth are equal (a square), then falls straight back down the same values in reverse. This is precisely why length and breadth in this problem are not in inverse proportion despite one rising as the other falls: inverse proportion requires the product to stay fixed, and here it visibly doesn't. The table also quietly proves a broader fact — among all rectangles sharing one fixed perimeter, the square always encloses the most area.

Telling the Two Patterns Apart Going Forward

Every genuine inverse-proportion problem in this exercise passed the same test from Exercise 10.2: multiply x and y, and the product stays fixed. The rectangle problem is a reminder that not every "one goes up, one goes down" situation clears that bar — sometimes, as here, neither direct nor inverse proportion actually applies. Exercise 10.4 returns to situations that genuinely are proportional, but now with three quantities linked together instead of two.