Read the notes, then try the practice. It adapts as you go.When you're ready.
Session Length
~17 min
Adaptive Checks
15 questions
Transfer Probes
15
Ratios, rates, and proportions form the backbone of quantitative reasoning on the SAT Math section. These concepts appear in contexts ranging from recipe scaling and unit conversion to speed calculations and population density problems. A ratio compares two quantities, a rate compares quantities with different units, and a proportion states that two ratios are equal. The SAT tests your ability to move fluidly between these representations and apply them to multi-step word problems grounded in real-world scenarios.
Beyond simple cross-multiplication, the SAT expects students to handle unit rates, dimensional analysis, direct and inverse variation, and proportional reasoning embedded in tables, graphs, and verbal descriptions. You may need to convert between units (miles per hour to feet per second), identify whether a relationship is proportional by checking for a constant ratio, or recognize that inverse variation means the product of two quantities remains constant. These skills connect algebra, geometry, and data analysis, making ratios and proportions a high-leverage topic for score improvement.
Mastering this domain requires comfort with setting up equations from word problems, canceling units systematically, and distinguishing between additive and multiplicative relationships. Students who build strong proportional reasoning can tackle a wide range of SAT questions more efficiently, often avoiding complex algebra entirely by recognizing scaling patterns. This topic bridges arithmetic fluency and algebraic problem-solving, making it essential preparation for both the no-calculator and calculator sections of the SAT.
One step at a time.
A comparison of two quantities expressed as $a:b$ or $\frac{a}{b}$. Ratios can describe part-to-part or part-to-whole relationships and remain equivalent when both terms are multiplied or divided by the same nonzero number.
Example: If a class has 12 boys and 18 girls, the ratio of boys to girls is $12:18 = 2:3$. The ratio of boys to total students is $12:30 = 2:5$.
A rate in which the denominator is 1 unit. Unit rates make it easy to compare different rates and to scale quantities up or down proportionally.
Example: A car travels 180 miles in 3 hours. The unit rate is $\frac{180}{3} = 60$ miles per hour.
An equation stating that two ratios are equal: $\frac{a}{b} = \frac{c}{d}$. Proportions can be solved by cross-multiplication, yielding $ad = bc$.
Example: If $\frac{3}{5} = \frac{x}{20}$, then $3 \times 20 = 5 \times x$, so $60 = 5x$ and $x = 12$.
A relationship where $y = kx$ for some constant $k$. As one variable increases, the other increases proportionally. The graph passes through the origin and is a straight line with slope $k$.
Example: If $y$ varies directly with $x$ and $y = 15$ when $x = 3$, then $k = 5$ and the equation is $y = 5x$. When $x = 7$, $y = 35$.
A relationship where $xy = k$ or equivalently $y = \frac{k}{x}$. As one variable increases, the other decreases so that their product remains constant.
Example: If 4 workers can complete a job in 6 hours, the constant is $k = 24$ worker-hours. With 8 workers, it takes $\frac{24}{8} = 3$ hours.
A method for converting units by multiplying by conversion factors written as fractions equal to 1. Units cancel algebraically, leaving the desired unit in the answer.
Example: Convert 5 miles to feet: $5 \text{ mi} \times \frac{5280 \text{ ft}}{1 \text{ mi}} = 26{,}400$ feet.
The constant multiplier that relates corresponding measurements in two similar figures or proportional situations. If the scale factor is $k$, then every length in the original is multiplied by $k$ in the scaled version.
Example: A map uses a scale of 1 inch = 50 miles. If two cities are 3.5 inches apart on the map, the actual distance is $3.5 \times 50 = 175$ miles.
The fixed ratio $k$ in a proportional relationship $y = kx$. It represents the unit rate and can be found by dividing any $y$-value by its corresponding $x$-value.
Example: A table shows $(2, 6)$, $(5, 15)$, $(8, 24)$. Since $\frac{6}{2} = \frac{15}{5} = \frac{24}{8} = 3$, the constant of proportionality is $k = 3$.
More terms are available in the glossary.
Choose a different way to engage with this topic — no grading, just richer thinking.
Explore your way — choose one:
See how the key ideas connect. Nodes color in as you practice.
Walk through a solved problem step-by-step. Try predicting each step before revealing it.
This is guided practice, not just a quiz. Hints and pacing adjust in real time.
Small steps add up.
What you get while practicing:
The best way to know if you understand something: explain it in your own words.
More ways to strengthen what you just learned.