Live direct ↔ inverse twin for the classic three-knowns problem, with a sentence frame, work peel, sense gate, and what-if rail on c.
Educational proportion literacy only. Not a CAS or exam oracle. Confirm methods with your course when grades matter.
By Ahmet C. Toplutaş·Site owner & editor · Math tools
Last updated May 21, 2026 · Reviewed May 21, 2026 · Planning estimates only - not tax, medical, or investment advice
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Rule of three asks: if a corresponds to b, what corresponds to c? The missing value is x.
This page's framing is a direct ↔ inverse twin, with a sentence frame, a work peel, a sense gate, and a what-if rail on c. It is not the proportion board's a:b = c:d cross-product twin, not ratio simplify/share-split, and not percentage-of jobs.
Decision framing: “Does more of the first amount mean more result (direct) or less result (inverse)?”
Educational proportion literacy only. Not a computer-algebra system or exam oracle. Confirm methods with your course when grades are on the line. See our disclaimers.
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The board restates your inputs as a plain sentence so the story stays attached to the algebra. Swap presets when you want recipe, crew, price, or trip language without rewriting the keys by hand.
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Direct peels x = (b · c) / a and checks a · x against b · c. Inverse peels x = (a · b) / c and checks a · b against c · x. Showing both products keeps the cross step honest.
Tip: load Work crew (inverse). Then flip to Direct on the twin to see how badly the wrong rule mis-tells the story.
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When a, b, and c are positive, the gate checks whether x moves the way the chosen rule usually predicts as c grows or shrinks relative to a. It is a literacy hint, not a proof that the real world must be linear.
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Direct ratio equality and inverse product equality.
Example (direct): 2 cups → 3 servings; 5 cups → x = (3 · 5) / 2 = 7.5 servings. Example (inverse): 4 workers → 6 days; 8 workers → x = (4 · 6) / 8 = 3 days.
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Sibling math boards when you need four-term proportions, ratio rewrite, or percents.
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Scenario A: Recipe. 2 → 3 servings; 5 cups → 7.5 servings on direct. Inverse twin is a different (wrong) story for food scaling.
Load Work crew: 4 workers take 6 days; 8 workers take 3 days on inverse.
Scenario B: Unit price. 3 items cost $12; 7 items cost $28 on direct.
Scenario C: Same trip. 60 mph takes 2 hours; 40 mph takes 3 hours on inverse (distance held constant in the sketch).
Scenario D: Need a:b = c:d. Hand off to proportion when four labeled terms and cross-products are the homework language.
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Stop here for everyday three-knowns direct/inverse literacy. Hand off for four-term proportion checks, ratio simplification, or percent jobs. Nonlinear science models need course-specific formulas.
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It solves the classic three-knowns problem: if a corresponds to b, what corresponds to c. You choose direct or inverse proportion. A direct ↔ inverse twin shows both answers, a work peel shows the products, a sense gate flags direction mismatches, and a what-if rail varies c.
Direct: a/b = c/x so x = (b·c)/a (more of a-side amount → more result). Inverse: a·b = c·x so x = (a·b)/c (more of a-side amount → less result), as in more workers finishing a job sooner.
The same three numbers are closed under both rules. Comparing them is the teaching point when you are unsure which story fits.
When all inputs are positive, it checks whether the solved x moves the way the chosen rule usually predicts as c grows or shrinks relative to a. It is a literacy hint, not a proof of the real-world model.
Proportion solves or checks a:b = c:d with cross-products and means/extremes. Rule of three is the everyday three-knowns sentence with an explicit direct vs inverse choice.
Ratio simplifies and rewrites a single a:b (part:whole, share split). Rule of three finds an unknown fourth quantity from three knowns under direct or inverse proportion.
No. It is educational proportion literacy for homework and quick checks. Confirm methods with your course when grades are on the line.
LIVE RULE-OF-THREE BOARD
Direct ↔ inverse →Scene presets
Proportion kind
First amount
First result
New amount
SENTENCE FRAME
If 2 corresponds to 3, then 5 corresponds to 7.5 (direct).
DIRECT ↔ INVERSE TWIN
Direct x
7.5
Inverse x
1.2
Twin gap 6.3 · selected: direct
WORK PEEL
Direct: a/b = c/x → x = (b · c) / a
WHAT-IF RAIL (vary c)
Updates as you type. Share writes board state; typing does not rewrite the address bar.
Direct x
7.5
Twin other 1.2
Rule of three finds the missing result when three values are known. Direct scales with the new amount; inverse keeps a product constant. Compare both on the twin before you commit to a story.
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