By Contributor • Updated Aug 30, 2022

Ratios compare two quantities by division. While they often resemble fractions, ratios are read as “X to Y” (e.g., 3/4 is “3 to 4”). Some authors write them with a colon, such as 3:4. This article walks you through two reliable methods for solving algebraic ratio problems: equivalent ratios and cross‑multiplication.

Using Equivalent Ratios

Step 1 – Identify the Variable

Begin by locating the term that contains the unknown. In the example 5/12 = 20/n, the second set of numbers (12 and n) includes the variable. Remember, the numbers in a ratio are not denominators, though the logic mirrors that of fractions.

Step 2 – Understand the Relationship of the Known Set

Next, examine how the two known numbers in the first set relate. Here, 5 is multiplied by 4 to give 20. Recognizing this multiplier (4) is essential.

Step 3 – Apply the Same Multiplier to the Unknown Set

To maintain equality, multiply the other known number (12) by the same factor. 12 × 4 = 48, so n = 48.

Result

Thus, 5/12 = 20/48, confirming that the ratio holds.

Using Cross‑Multiplication

Step 1 – Recognize a Proportion

When the ratio’s numbers do not share a clear multiplier, treat the equation as a proportion: 7/m = 2/4. Here, cross‑multiplication is the most efficient path.

Step 2 – Identify Cross Products

Place an “X” over the proportion to pair the diagonally opposite terms: 7 and 4, and m and 2.

Step 3 – Set Up the Equation

Equate the cross products: 7 × 4 = 2 × m.

Step 4 – Simplify

Compute the known side: 7 × 4 = 28, giving 28 = 2 × m.

Step 5 – Solve for the Variable

Isolate m by dividing both sides by 2: m = 28 ÷ 2 = 14.

Result

Therefore, 7/14 = 2/4, confirming the proportion.

TL;DR (Too Long; Didn’t Read)

After solving a ratio problem, always substitute your solution back into the original equation to verify its correctness. This quick check can catch any procedural or calculation errors.