Proportion — Complete SAT Math Guide
Master ratios, proportions, direct & inverse relationships, and SAT-specific strategies
1. Ratio Review
A ratio compares two quantities. It tells you how much of one thing there is relative to another.
Notation 1
$a : b$
"a to b"
Notation 2
$\dfrac{a}{b}$
"a over b"
Notation 3
"a to b"
Written in words
Simplifying Ratios: Ratios simplify like fractions. $\;12:8 = 3:2\;$ (divide both by 4).
Part-to-Part vs Part-to-Whole
Part-to-Part
Ratio of boys to girls = $3:5$
Compares one part to another part.
Total parts = $3 + 5 = 8$
Part-to-Whole
Boys are $\dfrac{3}{8}$ of the total
Girls are $\dfrac{5}{8}$ of the total
Compares one part to the total.
2. Proportion — Definition, Notation & Terms
Definition
A proportion is an equation stating that two ratios are equal:
$$\frac{a}{b} = \frac{c}{d} \quad \text{or} \quad a:b = c:d$$Read: "$a$ is to $b$ as $c$ is to $d$."
Both ratios represent the same relationship. If three of the four values are known, the fourth can be found.
Terms of a Proportion
In $\dfrac{a}{b} = \dfrac{c}{d}$:
| Term | Values | Position |
|---|---|---|
| Extremes | $a$ and $d$ | Outer terms (first & last) |
| Means | $b$ and $c$ | Inner terms (middle) |
$$a \cdot d = b \cdot c$$
Proportion Terms Diagram: Extremes and Means
Red indicates the extremes, blue indicates the means. The products of the extremes and means are equal.
Equivalent Proportions
If $\dfrac{a}{b} = \dfrac{c}{d}$, then all of these are also true: