Unitary method — direct and inverse proportion: Worked examples

This page is a case gym. The Unitary method — direct and inverse proportion parent taught the two laws; here we run them through every kind of situation an exam or the real world can throw at you — including the weird "edge" cases that trip people up: zeros, a mix of both proportions in one problem, and limiting behaviour.

The scenario matrix

Here is the full map of cases. Every example below is tagged with the cell it fills.

Cell	Case class	What makes it tricky
A	Direct, scale up (more)	plain "value of one, then multiply"
B	Direct, scale down (fewer)	answer must come out smaller
C	Inverse, plain	product stays fixed, not ratio
D	Inverse, non-integer unit	the "value of one" is a fraction
E	Zero / degenerate input	0 workers, 0 items — what breaks?
F	Limiting behaviour	as workers $\to \infty$ , days $\to$ ?
G	Real-world word problem	must extract which quantities pair
H	Exam twist: chain of both	direct and inverse in one problem

Cells A, B → direct. C, D → inverse. E, F → edge/limit. G, H → applied.

Figure — Unitary method — direct and inverse proportion

Example 1 — Cell A (Direct, scale up)

Example 2 — Cell B (Direct, scale down)

Example 3 — Cell C (Inverse, plain)

Example 4 — Cell D (Inverse, non-integer "value of one")

Example 5 — Cell E (Zero / degenerate input)

Worked example 5 workers dig a trench in 12 days. What happens with

0 workers? And what does "0 days" require? Forecast: With nobody digging, the trench is never finished. So "days" should blow up — not a normal number.

Step 1 — Direction & product. Inverse. Fixed job $= 5 \times 12 = 60$ worker-days.

Step 2 — Set workers to 0. $0 \times t = 60$ . Why this breaks: no value of $t$ makes $0 \times t = 60$ , because $0 \times t = 0$ always. Zero workers → the equation has NO solution → the task can never be done. The unitary method flags this honestly as "impossible", not as a number.

Step 3 — Now ask: 0 days. Set $t = 0$ : $x \times 0 = 60$ again has no solution. Finishing in zero time is equally impossible.

Verify: Both degenerate cases give the contradiction $0 = 60$ , so both are ruled out. A correct model should refuse to produce a finite answer here. ✓

Example 6 — Cell F (Limiting behaviour)

Worked example Same 60-worker-day trench. What happens to the number of days as workers grow

without bound ( $x \to \infty$ )? And as $x \to 0^{+}$ (a tiny team)? Forecast: Loads of workers → time crashes toward almost nothing. A single sluggish helper → time balloons.

Step 1 — Write the rule. $t = \frac{60}{x}$ (days as a function of workers $x$ ). Why this form? Inverse proportion is $x y = k$ , so $y = k / x$ — this is the same $y = k / x$ curve you meet in Linear Equations as a non-linear (reciprocal) relation.

Step 2 — Push $x$ large. As $x \to \infty$ , $\frac{60}{x} \to 0$ . Time shrinks toward zero but never reaches zero (Example 5 showed 0 days is impossible).

Step 3 — Push $x$ small. As $x \to 0^{+}$ , $\frac{60}{x} \to \infty$ : with a whisker of a workforce, days explode toward infinity — the smooth version of Example 5's "never finishes".

Verify (spot values): $x = 60 \Rightarrow t = 1$ day; $x = 120 \Rightarrow t = 0.5$ day; $x = 600 \Rightarrow t = 0.1$ day. Falling toward 0. ✓

Example 7 — Cell G (Real-world word problem)

Worked example A recipe for 8 people needs 600 g rice and takes 45 min of cooking. You are cooking for

20 people. How much rice, and does cooking time change? Forecast: Rice scales with people → more rice. Cooking time of a pot is set by the pot, not the head-count → probably unchanged.

Step 1 — Sort the quantities.

Rice vs people: more people → more rice → direct.
Cooking time vs people: boiling rice takes the same 45 min whether it feeds 8 or 20 (bigger pot, same simmer) → not proportional at all. This is the trap: not every pair is a proportion. Why this step? The hardest part of word problems is deciding which quantities pair up. Extract that before touching arithmetic — see Ratio and Proportion.

Step 2 — Rice, value of one person. $\frac{600}{8} = 75$ g per person. Why divide? Direct proportion → go to one first.

Step 3 — Scale to 20. $75 \times 20 = 1500$ g $= 1.5$ kg.

Step 4 — Time. Stays $45$ min (independent variable, not proportional).

Verify: $\frac{600}{8} = 75 = \frac{1500}{20}$ ✓. Time unchanged, matching forecast. ✓

Example 8 — Cell H (Exam twist: chain of direct AND inverse)

Worked example 12 workers, each working 8 hours a day, build

3 houses in 20 days. How many days will 16 workers, working 6 hours a day, need to build 5 houses? Forecast: More workers → faster (down). Fewer hours/day → slower (up). More houses → more work (up). Several forces pull in different directions, so we combine them carefully.

Step 1 — List each factor and its direction (relative to DAYS).

Workers $12 \to 16$ : more workers → fewer days → inverse, multiply by $\frac{12}{16}$ .
Hours/day $8 \to 6$ : fewer hours → more days → inverse on hours, multiply by $\frac{8}{6}$ .
Houses $3 \to 5$ : more houses → more days → direct, multiply by $\frac{5}{3}$ . Why this step? The "chain rule" of proportion: start from the known days and multiply one correcting ratio per changing quantity. Each ratio is written so its effect matches the up/down forecast.

Step 2 — Assemble. $t = 20 \times \frac{12}{16} \times \frac{8}{6} \times \frac{5}{3}$ Why written this way? Inverse factors put the new value on the bottom (so more workers → smaller factor); direct factors put the new value on top (more houses → bigger factor).

Step 3 — Compute. $t = 20 \times 0.75 \times 1.3333 \dots \times 1.6667 \dots = \frac{20 \times 12 \times 8 \times 5}{16 \times 6 \times 3} = \frac{9600}{288} = 33. \overline{3} days .$ So $t = 33 \frac{1}{3}$ days.

Verify (total work must balance). Work $=$ workers $\times$ hours/day $\times$ days, per house. Old: $\frac{12 \times 8 \times 20}{3} = 640$ worker-hours per house. New: $\frac{16 \times 6 \times 33. 3}{5} = \frac{3200}{5} = 640$ worker-hours per house. ✓ Same effort per house — the model is consistent. Connect this to Time and Work.

Coverage check

Recall Did we hit every cell? (cover the answers)

A Direct up ::: Ex 1 — ₹225
B Direct down ::: Ex 2 — 12 kg
C Inverse plain ::: Ex 3 — 15 h
D Inverse fractional ::: Ex 4 — 12.6 h
E Zero input ::: Ex 5 — impossible (no solution)
F Limit ::: Ex 6 — $t \to 0$ as $x \to \infty$
G Word problem ::: Ex 7 — 1.5 kg rice, time unchanged
H Chain of both ::: Ex 8 — $33 \frac{1}{3}$ days

Recall One-line "why" for each edge case

Why does 0 workers give no answer? ::: $0 \times t = k$ can never equal a positive $k$ .
Why does the reciprocal curve never touch zero? ::: $60/ x$ shrinks forever but stays positive for all finite $x$ .
Why isn't cooking time proportional to diners? ::: The pot simmers the same 45 min regardless of head-count.

Connections

Ratio and Proportion — extracting which quantities pair (Ex 7) is proportion-spotting.
Percentages — "per person" (75 g) is a unitary value, exactly like "per 100".
Speed Distance Time — Ex 3 & 6 are speed–time inverse cousins.
Time and Work — Ex 8's worker-hours balance is the master identity there.
Linear Equations — $t = 60/ x$ is the reciprocal curve; contrast with the straight line $y = k x$ .