Proportion Calculator
Proportion calculator — solve a:b = c:d for the missing term with cross-multiplication steps. Free ratios-proportion solver with visuals and practice.
Cross-multiply to clear both denominators, then divide by whatever multiplies the blank. One equation — four possible solutions.
- Grade level
- 6–9
- Curriculum band
- Difficulty
- Beginner
- For this topic
- Time to master
- 1–2 weeks
- With daily practice
- Tools on page
- 10 on page
- Calculator + practice
Calculator
Proportion Terms — blank exactly one
Proportion Summary
Breakdown
| Property | Value |
|---|---|
| Cross a × d | 24 |
| Cross b × c | 24 |
| Decimal a/b | 0.75 |
| Decimal c/d | 0.75 |
Results
Missing term (d)
8
Solved by cross-multiplication
3:4 = 6:8
Both sides share one value
24 = 24
a×d vs b×c — must match
0.75 = 0.75
a/b vs c/d corroboration
Results follow proportion rules for the inputs given. Fill exactly three terms and blank one.
Statistics
Live readouts from your current inputs — the balance, crosses and cards all move together. Blank a different box to solve a different term.
Solved term
d = 8
Cross-solved
Cross a×d
24
Extremes
Cross b×c
24
Means — must match
Shared decimal
0.75
Both sides
Visualizations
Both sides share one value
Balance of the proportion
Scale balance — both pans weigh the same
Formula
A proportion equates two ratios. Cross-multiplying clears both denominators, turning the equality of fractions into one multiplication equation solvable for any single unknown.
Alternative Form
Variables
| Symbol | Name | Unit | Description |
|---|---|---|---|
| a, b | Left ratio | number | First comparison a:b |
| c, d | Right ratio | number | Second comparison c:d |
| a×d = b×c | Cross equation | number | Denominator-free form |
| ? | Missing term | number | Exactly one blank box |
Explanation
Think of a balance scale: whatever multiplies one pan must multiply the other. Cross-multiplication weighs both pans at once, and dividing by the leftover coefficient isolates the blank.
Derivation
From a/b = c/d multiply by b·d: a·d = c·b. Then d = b·c/a (a ≠ 0); c = a·d/b (b ≠ 0); b = a·d/c (c ≠ 0); a = b·c/d (d ≠ 0). One equation, four solutions.
Step-by-Step Calculation
Follow each step to understand how the result is calculated from your inputs.
Write the proportion with the blank marked.
Cross-multiply to clear both denominators.
Isolate the missing term on one side.
Evaluate the arithmetic.
Verify decimals match on both sides.
State the completed proportion.
Worked Examples
See how the calculator works with different scenarios.
Scenario
Solve for the missing fourth term.
Inputs
Calculation
d = 4×6/3 = 24/3 = 8
Result
d = 8
Interpretation
3:4 = 6:8 — the right side is the left doubled.
Understanding Your Result
Your Result: Missing term (d): 8
Typical range: Recipe doubles (×2); map scales (×25000); currency near market rates.
What This Means
The solved term completes the equality — both ratios now share one value, one decimal, one meaning.
Why It Matters
Recipes, dosages, maps, currency, and similar figures all reduce to “three knowns, one blank” proportions.
Professional Interpretation
Report the full equality plus the cross check: “3:4 = 6:8 (3×8 = 4×6 = 24).”
Recommendations
- •Blank exactly one box before solving
- •Cross first, then divide by the leftover
- •Verify with both crosses and decimals
- •Keep units paired per ratio (eggs:flour both sides)
- •Guard zero divisors (a, b, c, d as needed)
- •State the full completed proportion
Learn — The Complete Guide
A proportion a:b = c:d says two ratios match. With three knowns, cross-multiplication finds the fourth.
The core concept
Cross: a×d = b×c. Then divide by whatever multiplies the blank: d = b×c/a, c = a×d/b, b = a×d/c, a = b×c/d.
- A proportion is an equality of two ratios
- Cross once: a×d = b×c clears both denominators
- Divide by the blank’s partner to isolate it
- Exactly one blank gives exactly one answer
Derivations — why the formulas work
Cross — one multiply clears both
- 1a/b = c/d
- 2Multiply both sides by b·d
- 3Left: a·d, right: c·b
- 4a×d = b×c ✓
Solve d — divide by a
- 1a×d = b×c
- 2Divide both sides by a ≠ 0
- 3d = b×c/a
- 4Check: 4×6/3 = 8 ✓
Two blanks — infinite answers
- 1a/b = c/d with c,d unknown
- 2Any scaled pair (c·k, d·k) fits
- 3Infinitely many solutions
- 4Need three knowns ✓
Where you meet this in real life
Learn this first — prerequisites in order
- 1Ratios: read a:b as division
- 2Cross-multiplication mechanics
- 3One-step equations: divide both sides
- 4Decimals: multiply and divide fluently
Careers that use this daily
- •Nurse — weight-based dosing
- •Chef — portion scaling
- •Surveyor — similar-triangle heights
- •FX analyst — rate conversions
- •Architect — scale-model dimensions
Why it helps
- •Solves for ANY one blank term
- •Cross + decimal double verification
- •Balance visual shows equality
- •Steps model isolate-and-divide
- •Handles decimals and fractions directly
Best practices
- •Mark the blank before computing
- •Write the cross equation line first
- •Divide only by the coefficient of the blank
- •Verify crosses AND decimals
- •Restate the full proportion in the answer
Limitations — what this tool does not do
- •Exactly one blank required (not zero, not two)
- •Zero divisors rejected per formula
- •Ratios must pair comparable units
- •Display rounds to 12 significant digits
Curriculum map — where this topic lives in school
| Level | Grades | Standards | Skills |
|---|---|---|---|
| Upper Elementary | 5–6 | 5.NBT, 6.RP | Ratios · Division |
| Middle School | 6–8 | 6.RP, 7.RP | Proportions · Cross-multiply · Unit rates |
| High School | 9+ | HSG-SRT, HSF | Similar figures · Proportional functions |
Timeline — how this idea grew up
~500 AD
Rule of three in India
Aryabhata’s school teaches trairashika — solving “three knowns, one unknown” for trade.
1202
Fibonacci spreads it west
Liber Abaci carries proportion methods to European merchants for currency and cloth.
1500s
Means-extremes language
Textbooks name inner (means) and outer (extremes) terms; product equality becomes standard proof.
Today
Proportions run services
Dosages, maps, mixes, and models all solve missing terms by the same cross routine.
A little history
- • The “rule of three” (three knowns, one unknown) was standard merchant math for a millennium.
- • Means (inner) × means = extremes (outer) × extremes is the classic verification.
- • Proportional reasoning is Piaget’s marker of formal operational thought.
Exam-day checklist
- • Lead with the cross equation for method marks.
- • Name the isolate formula for your blank.
- • Verify with the decimal equality line.
- • Close with the full proportion statement.
Myths vs facts
Practical Tips
- •Blank one box — exactly one
- •Write a×d = b×c first
- •Divide by the blank’s partner
- •Verify crosses match
- •Verify decimals match
- •Pair units per ratio
- •Restate the full equality
Common Mistakes
Assumptions
This calculator makes the following assumptions to provide estimates:
Limitations
Be aware of these limitations when using this calculator:
Frequently Asked Questions
Answers to common questions about this calculator.
What is a proportion?
How do I solve a proportion with one missing term?
What is cross-multiplication in a proportion?
Why must exactly one term be blank?
Can proportions use decimals or fractions?
How do I use this proportion calculator?
Related Calculators
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Related Guides
Deepen your understanding with these educational resources.
Glossary
Technical terms used in mortgage calculations explained in plain language.
- Proportion
- Equality of two ratios: a:b = c:d.
- Cross-multiplication
- Clearing denominators to a×d = b×c.
- Missing term
- The single blank solved for.
- Extremes
- Outer terms a and d.
- Means
- Inner terms b and c.
- Degree of freedom
- One free value per proportion equation.
Authoritative References
Trusted sources and standards referenced by this calculator.
Applicable Standards
Diagrams & Illustrations
Visual intuition for every core idea. Select a card to enlarge it.
Did You Know?
Short intuition builders — great for class warm-ups.
Vitruvius built in proportion
Roman architects scaled temple columns by fixed a:b = c:d ratios — the “orders” are proportion tables in stone.
Nurses cross daily
“250 mg : 5 mL = dose : weight” — drug-dosage math is a missing-term proportion where the cross check saves lives.
The rule of three is ancient
Indian mathematician Aryabhata (~500 AD) and Fibonacci (1202) both taught trairashika — the rule of three — for trade proportions.
Cameras expose by proportion
F-stops halve light per step: aperture ratios form a geometric proportion, which is why settings click in exact ×2 ratios.
Compare Side by Side
One equality, three confirmations — ratio, decimal, and cross views must all agree.
| Check | Left side | Right side | Reading |
|---|---|---|---|
| Ratio form | 3:4 | 6:8 | Same relationship |
| Decimal form | 0.75 | 0.75 | Must be equal |
| Cross form | 24 | 24 | Must be equal |
Recent Calculations
Your last runs on this page — stored only in this browser, never uploaded.
3:4 = 6:8
Current inputs — live
Study Plan
A week-by-week path from prerequisites to mastery. Your checklist progress saves in this browser.
Prerequisites
- ▸Ratios as divisions
- ▸Cross-multiplication basics
- ▸Solving one-step equations (divide both sides)
Mastery progress — 0%
- I can write the cross equation instantly
- I can solve for any single blank
- I verify with crosses and decimals
- I pair units correctly across the equals
- I count blanks before solving
- 1
Week 1 · ~15 min/day
Cross and solve for d
Goals
- Write a×d = b×c from any proportion
- Solve d = b×c/a reliably
- Verify with decimals
Practice
Solve 10 fourth-term proportions (3:4 = 6:d family) — verify each with crosses.
- 2
Week 2 · ~20 min/day
Any blank, any position
Goals
- Solve for a, b, or c with matching formulas
- Handle decimals and fractions
- Pair units correctly
Practice
Solve 5 mixed-blank proportions with decimals; write the isolate line each time.
- 3
Week 3 · ~20 min/day
Proportions in the wild
Goals
- Scale recipes, maps, doses by proportion
- Refuse two-blank problems
- Report full equalities with proof
Practice
Solve one recipe, one map, and one rate problem as proportions with cross checks.
Exam tips
- •Write the a×d = b×c line first for method marks.
- •Circle the blank and name its formula (d = b×c/a…).
- •End with the full “a:b = c:d ✓” statement.
Practice Problems
0/6 solvedTry each problem first, peek at the hint only if stuck, then reveal the solution and mark it solved.
Solve 3:4 = 6:d.
Solve 3:b = 6:8.
Solve a:4 = 6:8.
Solve 0.5:1.5 = 2:d.
Is 3:4 = 6:8 true?
A map scale is 2 cm : 5 km. How far is 7 cm?
Printable Worksheet
Generate a fresh problem set for class or homework. Print it, or copy the answer key separately.
- 1.18 × 47 =
- 2.8 ÷ 11 =
- 3.38 × 26 =
- 4.35 ÷ 4 =
- 5.21 − 16 =
- 6.2 + 26 =
- 7.14 ÷ 12 =
- 8.23 ÷ 3 =
- 9.49 ÷ 7 =
- 10.22 − 17 =
- 11.41 + 18 =
- 12.47 − 35 =
Formula Cheat Sheet
Every formula on this page in one printable table.
| Formula | Meaning | Example |
|---|---|---|
| a×d = b×c | Cross equation | 3×8 = 4×6 = 24 |
| d = b×c/a | Solve fourth | 4×6/3 = 8 |
| c = a×d/b | Solve third | 3×8/4 = 6 |
| b = a×d/c | Solve second | 3×8/6 = 4 |
| a = b×c/d | Solve first | 4×6/8 = 3 |
| a/b = c/d ✓ | Decimal verify | 0.75 = 0.75 ✓ |
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Page Information
2026-09-29
3.1.0-math
Free to use for educational and professional purposes.