Ratios & Proportions

A ratio like $3 : 4$ (also written $\frac{3}{4}$) compares two amounts. A proportion sets two ratios equal:

$\frac{3}{4} = \frac{x}{12}$

This is true exactly when the cross-products match — the two diagonals multiply to the same thing: $3 \times 12 = 4 \times x$. That fact turns any proportion with one unknown into a one-step equation. It works because both sides name the same ratio, just scaled differently — the heart of every "if this, then how much of that" question.

Cross-multiply, then solve. For $\frac{3}{4} = \frac{x}{12}$:

$3 \times 12 = 4 \times x \;\Rightarrow\; 36 = 4x \;\Rightarrow\; x = 9$

So $\frac{3}{4} = \frac{9}{12}$.

Real situations are proportions in disguise. If $3$ apples cost $1.50, what do $8$ cost? Set up price over apples, equal on both sides:

$\frac{1.50}{3} = \frac{x}{8}$

The unit rate is $0.50 per apple, so x = 0.50 \times 8 = \4.00$. Same engine: equal ratios, solve for the missing piece.

Set up the proportion so matching quantities line up, then cross-multiply. Predict before you check.