Ratios, Proportions, and Rule of Three

SAT Questions that focus on Ratios, Proportions, and Rule of Three require knowledge of the following topics.

Ratios and Proportions

Ratio is the relation between two amounts. For example: $\frac{2}{3}$.

Proportion is the equality between two ratios: $\frac{2}{3}=\frac{4}{6}$.

Properties of Proportions

P.1: If $\frac{X}{Y}=\frac{R}{S}$, then $\frac{X+Y}{Y}=\frac{R+S}{S}$
Demonstration:
$\frac{X}{Y}=\frac{R}{S}$
$XS=RY$.
Adding $YS$ to both sides of the equation:
$XS+YS=RY+YS$
$S(X+Y)=Y(R+S)$
$\frac{X+Y}{Y}=\frac{R+S}{S}$


P.2: $\frac{X}{Y}=\frac{R}{S}=\frac{X+R}{Y+S}$.
Demonstration:
$\frac{X}{Y}=\frac{R}{S}=a$, where "a" is a constant.
Thus,
$X=Ya$ and $R=Sa$
Plugging this result into the initial ratio:
$\frac{X+R}{Y+S}=\frac{Ya+Sa}{Y+S}=\frac{(Y+S)a}{Y+S}=a$


P.3: If $\frac{X}{Y}=\frac{R}{S}$, then $\frac{X^2}{Y^2}=\frac{R^2}{S^2}=\frac{XR}{YS}$.
Demonstration:
Let's multiply both sides of the equation by $\frac{X}{Y}$:
$\frac{X}{Y}\frac{X}{Y}=\frac{R}{S}\frac{X}{Y}$
$\frac{X^2}{Y^2}=\frac{XR}{YS}$.

Now let's multiply both sides of the equation by $\frac{R}{S}$:
$\frac{X}{Y}\frac{R}{S}=\frac{R}{S}\frac{R}{S}$
$\frac{XR}{YS}=\frac{R^2}{S^2}$.

Directly and Inversely Proportional Values

Two values are directly proportional, when as one value increases, the other increases at the same rate: $\frac{X_1}{X_2}=\frac{Y_1}{Y_2}$.

Two values are inversely proportional, when as one value increases, the other decreases at the same rate: $\frac{X_1}{X_2}=\frac{Y_2}{Y_1}$.
For example, the speed of a vehicle and the time it takes to go from Los Angeles to San Francisco. If one doubles the speed, the time of travel will be reduced to half.

Rule of Three

For an equation of the form $\frac{X_1}{X_2}=\frac{Y_1}{Y_2}$, the Rule of Three states that:

$X_1=\frac{X_2Y_1}{Y_2}$, or

$X_2=\frac{X_1Y_2}{Y_1}$, or

$Y_1=\frac{X_1Y_2}{X_2}$, or

$Y_2=\frac{X_2Y_1}{X_1}$

The Rule of Three can be used for solving a number of questions. For example: if 3 cans of coke cost 5 dollars, how much will 12 cans cost? Using the Rule of Three:

$\frac{X_1}{X_2}=\frac{Y_1}{Y_2}$

$\frac{3 cans}{12 cans}=\frac{5 dollars}{x dollars}$

$\frac{3}{12}=\frac{5}{x}$

$x=12(5)/3=20$ dollars

Solved SAT Practice Tests

Find Practice Tests in the following link:

SAT Practice Tests - Ratios, Proportions, and Rule of Three and

Additional Practice Tests - Ratios, Proportions, and Rule of Three