Compare Ratios

Calculator and formulas to compare two ratios with mathematical background

Ratio Comparison Calculator

What is calculated?

This function compares two ratios using the cross-product method or the decimal comparison. The result shows which ratio is greater, smaller, or equal.

Enter two ratios

:
First ratio

:
Second ratio

Comparison result
Decimal comparison will be shown after calculation
The comparison is performed using the cross-product method

Comparison Info

Comparison methods

Two proven methods:

  • Cross product: a×d vs b×c
  • Decimal comparison: a/b vs c/d
  • Both methods are equivalent
  • Cross product avoids rounding

Tip: Cross product is more accurate with large numbers or many decimal places.

Comparison operators
>

Greater than
First ratio is greater

<

Less than
First ratio is smaller

=

Equal
Both ratios are equal

Cross-product rule

a:b > c:d ⟺ a×d > b×c
Multiply "across" and compare the products


Mathematical formulas for ratio comparisons

Basic comparison
\[\frac{a}{b} \circ \frac{c}{d}\] where \(\circ \in \{<, =, >\}\)
Cross-product method
\[a:b \circ c:d \Leftrightarrow a \cdot d \circ b \cdot c\] Multiply across
Greater relation
\[\frac{a}{b} > \frac{c}{d} \Leftrightarrow a \cdot d > b \cdot c\] If the left cross product is larger
Less relation
\[\frac{a}{b} < \frac{c}{d} \Leftrightarrow a \cdot d < b \cdot c\] If the left cross product is smaller
Equality
\[\frac{a}{b} = \frac{c}{d} \Leftrightarrow a \cdot d = b \cdot c\] Cross-products are equal
Decimal comparison
\[a:b \circ c:d \Leftrightarrow \frac{a}{b} \circ \frac{c}{d}\] Division and decimal comparison

Step-by-step example

Example: compare 3:4 with 1:3

1Cross-product method

3 × 3 = 9 (a × d)
4 × 1 = 4 (b × c)
Compare: 9 ? 4

9 > 4, so 3:4 > 1:3

2Decimal comparison

\[\frac{3}{4} = 0.75\] \[\frac{1}{3} = 0.333...\]

0.75 > 0.333...

3Result

\[3:4 > 1:3\]

Both methods lead to the same result: 3:4 is greater than 1:3

Comparison methods in detail

Cross-product method
Advantages:
  • No rounding errors
  • Works with integers
  • Exact results
  • Faster for large numbers
Example:
\[5:7 \text{ vs } 3:4\] \[5 \times 4 = 20, \quad 7 \times 3 = 21\] \[20 < 21 \Rightarrow 5:7 < 3:4\]
Decimal comparison
Advantages:
  • Intuitive
  • Shows concrete values
  • Good for approximations
  • Visualizes differences
Example:
\[5:7 = \frac{5}{7} ≈ 0.714\] \[3:4 = \frac{3}{4} = 0.750\] \[0.714 < 0.750\]

More comparison examples

Simple comparisons
2:3 vs 3:4 \[2 \times 4 = 8 < 3 \times 3 = 9\] \[2:3 < 3:4\]
1:2 vs 3:6 \[1 \times 6 = 6 = 2 \times 3 = 6\] \[1:2 = 3:6\]
5:4 vs 6:5 \[5 \times 5 = 25 > 4 \times 6 = 24\] \[5:4 > 6:5\]
Practical examples
Price-performance
3€ for 2kg vs 5€ for 4kg
\[3:2 \text{ vs } 5:4\]
3×4 = 12 > 2×5 = 10
First is more expensive
Speed
100km in 2h vs 120km in 3h
\[100:2 \text{ vs } 120:3\]
100×3 = 300 > 2×120 = 240
First trip faster
Edge cases
Very similar ratios
\[7:10 \text{ vs } 71:101\]
7×101 = 707
10×71 = 710
\[7:10 < 71:101\]
Negative numbers
\[-3:4 \text{ vs } 1:-2\]
(-3)×(-2) = 6
4×1 = 4
\[-3:4 > 1:-2\]

Practical applications

Price-performance comparison

Example: supermarket offers

Product A: €3 for 250g
Product B: €5 for 450g

\[3:250 \text{ vs } 5:450\] \[3 \times 450 = 1350 > 250 \times 5 = 1250\]

Product A is more expensive per gram

Efficiency comparison

Example: work performance

Person A: 12 tasks in 3 hours
Person B: 20 tasks in 6 hours

\[12:3 \text{ vs } 20:6\] \[12 \times 6 = 72 > 3 \times 20 = 60\]

Person A is more efficient

Financial analysis

Compare profit-loss ratios:
Company A: €100 profit, €20 cost
Company B: €150 profit, €40 cost

Sports statistics

Compare hit rates:
Player A: 15 hits from 25 attempts
Player B: 22 hits from 40 attempts

Mathematical foundations

Order relations

The comparison of ratios is based on order relations of rational numbers. The cross-product method uses the property that sign is preserved when multiplying by positive numbers.

Equivalence of methods

Cross product and decimal comparison are mathematically equivalent. The cross-product method however avoids rounding errors and is often more practical for large numbers.

Key properties
  • Transitivity: a>b and b>c ⇒ a>c
  • Antisymmetry: a>b ⇒ b
  • Total order: For any a,b either ab
  • Scale invariance: k·a:k·b = a:b for k>0




More Ratio Functions

Aspect Ratio  •  Compare Ratios  •  Decimal to Ratio  •  Fraction to Ratio  •  Ratio-Calculator  •  Ratio Simplifier  •  Ratio to Decimal  •  Ratio to Fraction  •  Scale Ratio  •