Ratio Comparison Calculator
Ratio comparison calculator — compare a1:b1 vs a2:b2 by cross-multiplication and decimals. Free online tool with visuals, practice and study plan.
Draw an X between the ratios: each diagonal multiplies, and the bigger product sits under the bigger ratio. Decimals corroborate; crosses rule.
- Grade level
- 6–9
- Curriculum band
- Difficulty
- Beginner
- For this topic
- Time to master
- 1 week
- With daily practice
- Tools on page
- 10 on page
- Calculator + practice
Calculator
Two Ratios — enter all four terms
Comparison Summary
Breakdown
| Property | Value |
|---|---|
| Cross L = a1 × b2 | 10 |
| Cross R = a2 × b1 | 9 |
| Decimal gap |d1−d2| | 0.066666666667 |
| As unit rates | 0.666666666667 : 0.6 per 1 |
Results
Winner
2:3 is larger
By cross-multiplication
10 vs 9
a1×b2 vs a2×b1
0.666666666667 vs 0.6
a1/b1 vs a2/b2 — must agree
0.066666666667
Absolute difference
Results follow cross-multiplication rules for the inputs given. All four terms must be positive and comparable.
Statistics
Live readouts from your current inputs — the bars, crosses and cards all move together. Change any term to watch the verdict flip.
Verdict
First wins
2:3 is larger
Cross L (a1×b2)
10
2×5
Cross R (a2×b1)
9
3×3
Decimals
0.666666666667 / 0.6
Must agree with crosses
Visualizations
Decimal values head to head
First ratio share of combined decimals
Cross-multiplication — the X that decides
Formula
Cross-multiplication clears both denominators at once: multiplying the inequality a1/b1 ? a2/b2 by the positive product b1×b2 leaves exactly the cross products to compare.
Alternative Form
Variables
| Symbol | Name | Unit | Description |
|---|---|---|---|
| a1:b1 | First ratio | ratio | Left contender |
| a2:b2 | Second ratio | ratio | Right contender |
| L, R | Cross products | number | L = a1×b2, R = a2×b1 |
| d1, d2 | Decimals | number | a1/b1 and a2/b2 corroboration |
Explanation
Draw an X between the ratios: each diagonal multiplies, and the bigger product sits under the bigger ratio. Decimals tell the same story with rounding; crosses tell it exactly.
Derivation
For b1,b2 > 0: a1/b1 ? a2/b2 ⟺ a1·b2 ? a2·b1 (multiply by b1·b2 > 0, direction preserved). Equal crosses ⟺ equal ratios ⟺ equivalent fractions.
Step-by-Step Calculation
Follow each step to understand how the result is calculated from your inputs.
Write both ratios as divisions.
Cross-multiply upward-left: a1 × b2.
Cross-multiply upward-right: a2 × b1.
Compare the cross products — larger side wins.
Corroborate with decimals.
State the verdict in plain words.
Worked Examples
See how the calculator works with different scenarios.
Scenario
Which ratio is larger?
Inputs
Calculation
2×5 = 10 vs 3×3 = 9 → 10 > 9
Result
2:3 is larger
Interpretation
Decimals agree: 0.6667 > 0.6.
Understanding Your Result
Your Result: Winner: 2:3 is larger
Typical range: Grocery unit prices within 10%; sports ratios near 0.5; efficiency ratios near 1.0.
What This Means
The verdict names the ratio with the bigger value — bigger cross product, bigger decimal, bigger share per unit.
Why It Matters
Best-buy, fuel-efficiency, speed, and odds decisions are all ratio comparisons — crosses pick winners without rounding traps.
Professional Interpretation
Report crosses (exact) plus decimals (readable): “10 vs 9 (0.667 vs 0.600) — first wins.”
Recommendations
- •Cross first (exact), decimals second (readable)
- •Check units are comparable before comparing
- •Treat equal crosses as proven equivalent
- •Mind close calls — decimals may round them away
- •State the winner in plain words
- •Convert to unit rates for “per one” intuition
Learn — The Complete Guide
Comparing ratios means deciding which division is bigger — cross-multiplication answers exactly, decimals answer readably.
The core concept
For positive terms, a1/b1 vs a2/b2 ⟺ a1×b2 vs a2×b1. The X-diagonals multiply; the bigger product marks the bigger ratio.
- Cross products decide exactly; decimals corroborate
- Bigger cross sits under the bigger ratio
- Equal crosses prove equivalence
- Only comparable ratios may be compared
Derivations — why the formulas work
Cross rule — clear denominators
- 1Compare a1/b1 ? a2/b2
- 2Multiply by b1·b2 > 0
- 3Get a1·b2 ? a2·b1
- 4Order preserved — crosses rule ✓
Tie — equivalence proof
- 1L = R means a1·b2 = a2·b1
- 2Divide by b1·b2: a1/b1 = a2/b2
- 3Equal values = equivalent ratios
- 4Simplified forms match ✓
Decimals — same verdict, rounded
- 1d1 = a1/b1, d2 = a2/b2
- 2d1 ? d2 has same direction as L ? R
- 3Rounding may blur hairlines
- 4Trust crosses for close calls ✓
Where you meet this in real life
Learn this first — prerequisites in order
- 1Ratios: read a:b as the division a/b
- 2Multiplication: reliable two-digit products
- 3Inequalities: >,<,= meanings
- 4Decimals: divide to 3–4 places
Careers that use this daily
- •Procurement officer — supplier rate comparisons
- •Sports analyst — performance ratios
- •Financial analyst — return comparisons
- •Logistics planner — efficiency verdicts
- •Consumer advocate — best-buy testing
Why it helps
- •Exact verdict via cross products
- •Decimal corroboration side by side
- •Tie detection proves equivalence
- •X-diagram makes the method stick
- •Unit-rate intuition for shoppers
Best practices
- •Verify comparability (units, context) first
- •Run crosses before trusting decimals
- •Confirm ties by simplifying both ratios
- •Quote both crosses and decimals in answers
- •Reduce to unit rates for final intuition
Limitations — what this tool does not do
- •Needs four positive terms
- •Ratios must be meaningfully comparable
- •Huge products can lose float precision
- •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 | Division · Inequalities |
| Middle School | 6–8 | 6.RP, 7.RP | Cross-multiply · Unit rates · Best-buy math |
| High School | 9+ | HSN, HSS | Rate comparisons · Data reasoning · Decision math |
Timeline — how this idea grew up
~300 BC
Euclid compares magnitudes
Elements Book V defines when one ratio exceeds another — cross-style reasoning for line segments.
1202
Rule of three
Fibonacci’s Liber Abaci teaches merchants cross-multiplication for currency and price comparisons.
1500s
Cross becomes standard
Arithmetic textbooks drill the X method for proportions, interest, and mixture verdicts.
Today
Comparison engines
From supermarket labels to sports analytics, ratio comparison picks winners at machine speed.
A little history
- • Euclid’s Elements Book V formalized comparing ratios of magnitudes.
- • The medieval “rule of three” is cross-multiplication applied to proportions.
- • Unit pricing laws (1970s) made ratio comparison a consumer right.
Exam-day checklist
- • Show both cross products — never just the verdict.
- • Add a decimal corroboration line.
- • Write “equivalent” for ties, with the 12 = 12 style proof.
- • Check units match before comparing rates.
Myths vs facts
Practical Tips
- •Draw the X — diagonals multiply
- •Bigger cross = bigger ratio
- •Equal crosses = proven tie
- •Back it up with decimals
- •Check comparability first
- •Unit-rate both sides for intuition
- •Watch rounding on close calls
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.
How do you compare two ratios?
Why does cross-multiplication work?
Can I compare ratios with decimals instead?
How do I tell if two ratios are equal?
Do the ratios need the same units?
How do I use this ratio comparison 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.
- Cross product
- Diagonal product: a1×b2 or a2×b1.
- Equivalent ratios
- Equal-value ratios; crosses tie.
- Unit rate
- Ratio scaled to denominator 1.
- Inequality direction
- Preserved when multiplying by positives.
- Corroboration
- Decimals confirming the cross verdict.
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.
Supermarkets run on crosses
Comparing 420g-for-$3 vs 550g-for-$4 is a ratio comparison — cross products pick the cheaper gram without a single decimal.
Medieval merchants crossed too
The “rule of three” taught to 15th-century traders is cross-multiplication — comparing exchange rates across currencies.
Olympic ties need exactness
When photo-finishes round to equal thousandths, judges fall back to exact counts — the same instinct as preferring crosses over rounded decimals.
Slope is a ratio contest
Asking which hill is steeper compares rise:run ratios — cross-multiplying gradients decides without dividing.
Compare Side by Side
Three lenses on the same verdict — crosses decide exactly, decimals display readably, unit rates build intuition.
| Method | Computation | Result | Strength |
|---|---|---|---|
| Cross products | 10 vs 9 | First wins | Exact |
| Decimals | 0.666666666667 vs 0.6 | First wins | Readable |
| Unit rates | per 1 unit each | 0.666666666667 / 0.6 | Intuitive |
Recent Calculations
Your last runs on this page — stored only in this browser, never uploaded.
2:3 vs 3:5 → 2:3 is larger
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
- ▸Multiplying multi-digit numbers
- ▸Inequalities: reading > and <
Mastery progress — 0%
- I cross-multiply without pairing errors
- I verify every verdict with decimals
- I prove ties by equal crosses
- I check comparability before comparing
- I report winner + evidence fluently
- 1
Week 1 · ~15 min/day
Cross mechanics
Goals
- Pair a1×b2 and a2×b1 reliably
- Read verdicts from crosses
- Verify with decimals
Practice
Compare 10 pairs (2:3 vs 3:5, 7:11 vs 5:8…) by crosses — check each with decimals.
- 2
Week 2 · ~20 min/day
Ties and equivalence
Goals
- Prove equivalence with equal crosses
- Simplify tied ratios to confirm
- Convert verdicts to unit rates
Practice
Find 5 equivalent pairs and 5 winners; write each verdict two ways (cross + decimal).
- 3
Week 3 · ~20 min/day
Comparisons in the wild
Goals
- Judge best-buys by unit rates
- Compare speeds and efficiencies
- Refuse incomparable comparisons
Practice
Compare two real labels (price/weight) and two real rates (fuel, speed); report crosses + decimals.
Exam tips
- •Write both cross products explicitly — method marks live there.
- •Add the decimal check line even when crosses suffice.
- •State “equivalent” (not “equal”) for tied ratios.
Practice Problems
0/6 solvedTry each problem first, peek at the hint only if stuck, then reveal the solution and mark it solved.
Which is larger: 2:3 or 3:5?
Are 2:3 and 4:6 equivalent?
Which is larger: 7:11 or 5:8?
400g for $3 vs 550g for $4 — better grams-per-dollar?
Car A: 420km per 30L. Car B: 550km per 40L. More efficient?
Team X wins 17:25, Team Y wins 23:34. Better win ratio?
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 |
|---|---|---|
| L = a1×b2, R = a2×b1 | Cross products | 2:3 vs 3:5 → 10 vs 9 |
| L > R → first wins | Verdict rule | 10 > 9 → 2:3 wins |
| L = R → equivalent | Tie rule | 12 = 12 → 2:3 = 4:6 |
| d1 = a1/b1, d2 = a2/b2 | Decimal check | 0.667 vs 0.600 ✓ |
| a1/b1 ? a2/b2 ⟺ L ? R | Why it works | ×b1b2 > 0 preserves order |
| unit rate = a/b per 1 | Intuition finish | 14.0 vs 13.75 km/L |
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Page Information
2026-09-29
3.1.0-math
Free to use for educational and professional purposes.