Ratios and Proportional Relationships

What a Ratio Actually Compares

A ratio compares two quantities by division. If a trail mix uses $3$ cups of almonds for every $2$ cups of raisins, the almond-to-raisin ratio is $3:2$, which you can also write as the fraction $\tfrac{3}{2}$. The single most important property is that a ratio is unchanged when you multiply or divide both quantities by the same number. Scaling $3:2$ up by a factor of $4$ gives $12:8$ — a different amount of food, but the same ratio, so the mix tastes identical.

Two quantities $x$ and $y$ are proportional when $y = kx$ for a fixed number $k$ called the constant of proportionality. In a table of proportional data, dividing any output by its input always returns the same $k$. That gives you a one-move test for proportionality: compute $\tfrac{y}{x}$ for each row and check that the values match. If even one row disagrees, the relationship is not proportional.

Setting Up and Solving a Proportion

A proportion is an equation stating that two ratios are equal, such as $\tfrac{a}{b} = \tfrac{c}{d}$. The reliable solving move is cross-multiplication: $\tfrac{a}{b} = \tfrac{c}{d}$ becomes $a \cdot d = b \cdot c$. The single discipline that prevents most mistakes is keeping the same units in the same position in both fractions. Miles over hours must equal miles over hours — never miles over hours set equal to hours over miles.

For example, a map is drawn so that $4$ centimeters represents $18$ real kilometers, and two towns are $10$ centimeters apart on the map. Put map centimeters on top and real kilometers on the bottom in both ratios:

$\frac{4 \text{ cm}}{18 \text{ km}} = \frac{10 \text{ cm}}{d \text{ km}}$

Cross-multiplying gives $4d = 180$, so $d = 45$ kilometers. Notice the constant of proportionality is $\tfrac{18}{4} = 4.5$ kilometers per centimeter, so multiplying the $10$ cm directly by $4.5$ produces the same $45$ km — a shortcut worth trusting once you see why it works.

Part-to-Part vs. Part-to-Whole

SAT ratio questions frequently hide a total. If marbles are red and blue in a $3:5$ ratio, the $3$ and the $5$ are parts, and the whole is $3 + 5 = 8$ parts. So red marbles make up $\tfrac{3}{8}$ of the total, not $\tfrac{3}{5}$. When a problem says there are $56$ marbles, each part equals $\tfrac{56}{8} = 7$ marbles, giving $3 \cdot 7 = 21$ red and $5 \cdot 7 = 35$ blue. The "divide the total by the sum of the parts" move is the backbone of every ratio-splitting question.

Recognizing a Proportional Relationship

A proportional relationship always passes through the origin: when $x = 0$, $y = 0$, because $y = kx$. A line such as $y = 2x + 3$ is linear but not proportional — the $+3$ means doubling $x$ does not double $y$. This distinction matters on the SAT, where "directly proportional" is a precise claim: the graph is a straight line through $(0, 0)$ whose slope is the constant $k$. If a graph or table has a nonzero starting value, reject "proportional," even though it may still be linear.

Choosing an Approach Under Time Pressure

• A missing value inside equal ratios → set up a proportion and cross-multiply.
• A total split by a ratio → add the parts, divide the total by that sum, then multiply each part.
• "Is this proportional?" → check that $\tfrac{y}{x}$ is constant and that the graph hits the origin.
• Scaling a recipe, map, or model → multiply every quantity by the same factor.

Every one of these is the same underlying idea — equal ratios — wearing a different costume. Master the setup and the units, and proportional-relationship questions become some of the fastest points on the section.