Ratio Calculator — Simplify, Compare & Find Missing Values

Q: What does it mean to simplify a ratio?

Simplifying a ratio means dividing both parts by their greatest common divisor so that the two numbers have no common factor other than 1. For example, 12 : 18 simplifies to 2 : 3 because gcd(12, 18) = 6.

Q: How do I compare two ratios?

Convert each ratio to its decimal value by dividing the first number by the second, then compare the decimals. For example, 3 : 4 = 0.75 and 4 : 5 = 0.8, so 4 : 5 is the larger ratio. Equivalently, cross-multiply: a : b vs c : d, compare a times d with c times b.

Q: How do I find a missing value in a ratio?

If you have a : b = c : x and need x, cross-multiply to get a times x = b times c, so x = (b times c) / a. This is the same as solving a proportion. For example, 3 : 4 = 9 : x gives x = (4 times 9) / 3 = 12.

Q: Can a ratio have a zero in it?

The first number can be zero, giving a ratio like 0 : 5, which simplifies to 0 : 1. But the second number cannot be zero because a ratio is fundamentally a division, and division by zero is undefined. This calculator will show an error if either denominator position is zero where it matters.

Q: What is the difference between a ratio and a fraction?

A ratio a : b compares two quantities of possibly different units and does not necessarily sum to a whole. A fraction a/b represents a part of a single whole. In practice they use the same arithmetic, but the interpretation differs. Ratios are often written with a colon, fractions with a horizontal bar.

Q: Can ratios have decimal or fractional terms?

Yes, but the usual convention is to write ratios with whole numbers. To convert 1.5 : 2.5 to whole numbers, multiply both sides by 10 to get 15 : 25, then simplify to 3 : 5.

What this ratio calculator does

A ratio expresses how two quantities compare. This calculator works in three modes. In Simplify mode, it reduces a ratio to its lowest whole-number terms by dividing both parts by their greatest common divisor. In Compare mode, it takes two ratios and tells you which is greater, along with each ratio's decimal value and cross-multiplication check. In Missing Value mode, it solves a proportion of the form a : b = c : x for the unknown x using cross-multiplication. Every mode shows the working steps so you can see exactly how the answer was produced.

Formulas used in each mode

Simplify a ratio a : b

            Let g = gcd(a, b). Then a : b simplifies to (a/g) : (b/g).
          

Compare two ratios a : b and c : d

            Compute a/b and c/d as decimals and compare. Equivalently, compare a·d with c·b (cross-multiplication).

            If a·d > c·b, then a : b > c : d.

Find missing value in a : b = c : x

            Cross-multiply: a · x = b · c. Therefore x = (b · c) / a.
          

Variable definitions

a, b, c, d — the four numbers in the two ratios you supply.
x — the unknown value in the missing-value mode.
gcd(a, b) — greatest common divisor of a and b.

Worked examples

Example 1: Simplify 12 : 18

Find gcd(12, 18). Euclid: gcd(18, 12) → gcd(12, 6) → gcd(6, 0) = 6.
Divide both parts by 6: 12 ÷ 6 = 2 and 18 ÷ 6 = 3.
Result: 2 : 3

Example 2: Simplify 100 : 250

gcd(100, 250) = 50.
Divide both by 50: 100 ÷ 50 = 2 and 250 ÷ 50 = 5.
Result: 2 : 5

Example 3: Compare 3 : 4 with 4 : 5

Decimal values: 3/4 = 0.75 and 4/5 = 0.80.
Since 0.80 > 0.75, the second ratio is larger.
Cross-multiplication check: 3 × 5 = 15, 4 × 4 = 16. Since 15 < 16, 3 : 4 < 4 : 5. ✓
Result: 4 : 5 is the greater ratio.

Example 4: Solve 3 : 4 = 9 : x

Cross-multiply: 3 × x = 4 × 9.
Simplify: 3x = 36, so x = 36 ÷ 3.
Result: x = 12. The second ratio is 9 : 12, which simplifies to 3 : 4. ✓

Example 5: Solve 5 : 8 = 15 : x

Cross-multiply: 5 × x = 8 × 15 = 120.
Divide: x = 120 ÷ 5 = 24.
Result: x = 24. Check: 15 : 24 simplifies to 5 : 8. ✓

Where ratios appear in Class 9 & 10

Ratios are one of the most widely used mathematical ideas in the Punjab board syllabus, often without being called out by that name. You meet them in:

Trigonometry — sin, cos, and tan are ratios of triangle sides.
Similar triangles — corresponding sides are in the same ratio.
Chemical equations — mole ratios of reactants and products.
Physics numericals — speed, density, and pressure are all ratios.
Scale drawings — maps and diagrams use ratios like 1 : 50,000.

Common mistakes to avoid

Simplifying only one side. A ratio is simplified by dividing both parts by the same number. Dividing only the first term changes the ratio's value entirely.
Adding the same number instead of multiplying. To go from 2 : 3 to 4 : 6, multiply both sides by 2 — do not add 2 to each side, which would give 4 : 5 and a different ratio.
Comparing ratios by comparing their parts. 5 : 8 and 7 : 10 — the first term is smaller in the first ratio but the second term is also smaller, so you cannot tell which is larger without computing decimals or cross-multiplying.
Cross-multiplying the wrong pairs. For a : b = c : x, cross-multiplication gives a·x = b·c. Multiply the outer pair, then the inner pair — do not multiply a·b or c·x.
Leaving decimal terms in a ratio. A finished ratio should have whole-number terms. Multiply both sides by 10 or 100 as needed to clear decimals before simplifying.
Using zero as a denominator. A ratio like 5 : 0 is undefined because it corresponds to division by zero. This calculator will show an error.

FAQ

What does it mean to simplify a ratio?

Simplifying a ratio means dividing both parts by their greatest common divisor so that the two numbers have no common factor other than 1. For example, 12 : 18 simplifies to 2 : 3 because gcd(12, 18) = 6.

How do I compare two ratios?

Convert each ratio to its decimal value by dividing the first number by the second, then compare the decimals. For example, 3 : 4 = 0.75 and 4 : 5 = 0.8, so 4 : 5 is the larger ratio. Equivalently, cross-multiply: a : b vs c : d, compare a times d with c times b.

How do I find a missing value in a ratio?

If you have a : b = c : x and need x, cross-multiply to get a times x = b times c, so x = (b times c) / a. This is the same as solving a proportion. For example, 3 : 4 = 9 : x gives x = (4 times 9) / 3 = 12.

Can a ratio have a zero in it?

The first number can be zero, giving a ratio like 0 : 5, which simplifies to 0 : 1. But the second number cannot be zero because a ratio is fundamentally a division, and division by zero is undefined. This calculator will show an error if either denominator position is zero where it matters.

What is the difference between a ratio and a fraction?

A ratio a : b compares two quantities of possibly different units and does not necessarily sum to a whole. A fraction a/b represents a part of a single whole. In practice they use the same arithmetic, but the interpretation differs. Ratios are often written with a colon, fractions with a horizontal bar.

Can ratios have decimal or fractional terms?

Yes, but the usual convention is to write ratios with whole numbers. To convert 1.5 : 2.5 to whole numbers, multiply both sides by 10 to get 15 : 25, then simplify to 3 : 5.

Related tools

Related Hira Academy resources

Ratios and proportions are used throughout the BISE Punjab mathematics and physics syllabi, from similar triangles to trigonometric ratios. Practise with our free Class 9 notes and Class 10 notes.