Ratio

Simplifying a Ratio or Solving a Proportion

A ratio compares two quantities, and simplifying it means writing it with the smallest whole numbers that keep the same relationship. Enter two numbers to simplify their ratio to lowest terms, or enter three numbers of a proportion (A is to B as C is to what?) to solve for the missing fourth term.

The Formula

Simplifying a ratio: divide both numbers by their greatest common factor (GCF) — the largest whole number that divides evenly into both.

Solving a proportion ($\vA{A}:\vB{B} = \vC{C}:D$): cross-multiplication gives $\vA{A} \times D = \vB{B} \times \vC{C}$, so the missing term is:

$$D = \frac{\vB{B} \times \vC{C}}{\vA{A}}$$

Worked Example

Simplifying 8:12:

1. Greatest common factor of 8 and 12: 4.
2. Divide both parts by 4: $8 \div 4 : 12 \div 4 = 2 : 3$.

Solving 4:6 = 10:?

1. Cross-multiply: $? = \frac{6 \times 10}{4}$.
2. $? = \frac{60}{4} = 15$, so 4:6 = 10:15.

Key Factors to Consider

• A ratio describes a relationship between quantities, not the quantities themselves. A recipe calling for a 2:1 ratio of flour to sugar means exactly the same thing whether it’s scaled to 2 cups flour/1 cup sugar or 200g flour/100g sugar — the RATIO stays identical even though the actual amounts change, which is exactly the property that makes ratios useful for scaling recipes, mixing solutions, or resizing images proportionally.
• A simplified ratio and its unsimplified form represent the exact same relationship. 8:12 and 2:3 describe an identical proportion — simplifying just makes the relationship easier to read and compare at a glance, the same way reducing a fraction to lowest terms doesn’t change its value.
• Proportions (equal ratios) are the basis for scaling recipes, maps, and mixtures consistently. Any time a real-world relationship needs to be scaled up or down while preserving the same proportional balance — like adjusting a recipe for a different serving size, or reading a map’s scale — solving a proportion (the “A is to B as C is to what” pattern) is the underlying math.
• A ratio comparing more than two quantities (like a 1:2:3 mix ratio) extends the same GCF simplification idea, just across every term at once. Simplifying a three-part ratio still means dividing every term by their shared greatest common factor — this calculator’s own two-term scope covers the fundamental case that same idea builds on.

Common Mistakes

• Simplifying only one number in the ratio instead of both. Both parts must be divided by the same greatest common factor, or the relationship between them changes — dividing just one side produces a different ratio, not a simplified version of the original.
• Cross-multiplying incorrectly in a proportion. For A:B = C:D, the cross products are A × D and B × C — mixing up which numbers pair together gives a wrong result even though the arithmetic itself is correct.
• Treating a ratio as if it were a fraction of a whole. A 2:1 ratio of flour to sugar means twice as much flour as sugar, not that flour is two-thirds of the total mixture — that distinction matters when converting a ratio into actual proportions of a whole.

Useful to Know

• Need to simplify a fraction instead of a ratio? Fraction Calculator simplifies fractions and performs fraction arithmetic.
• Scaling a recipe’s ingredient list up or down? Recipe Scaling Calculator adjusts every ingredient amount to a new serving size.
• Working with an image or screen’s width-to-height ratio specifically? Aspect Ratio Calculator handles aspect ratio calculations and comparisons.