ratios, rates, and proportional relationships

Ratios and rates are the backbone of SAT word problems. You’ll see them in recipes, conversions, speed, density, and “how many per how much” setups. This walkthrough covers all the SAT’s favorite ratio tricks, unit conversions, direct and inverse variation, and proportional reasoning—with step-by-step practice and toggle explanations for every question.

what are ratios and rates?

A ratio compares two quantities—how much of one thing to another, often written as $a:b$ or $\\frac{a}{b}$ . A rate is a special kind of ratio that compares different units (like miles per hour or dollars per pound). The SAT uses both to model real-world relationships.

For example:

\text{ratio: boys to girls} = 2:3

\text{rate: speed} = \frac{60\ \text{miles}}{1\ \text{hour}} = 60\ \text{mph}

In a group with a 2:3 ratio of boys to girls, for every 2 boys, there are 3 girls. In rates, $60\ \text{miles per hour}$ means traveling 60 miles in one hour.

proportional relationships & cross-multiplying

If two ratios are equal, they form a proportion. The SAT loves questions like “If 5 apples cost $6, how much for 8 apples?” Set up a proportion and cross-multiply to solve.

Practice Question:

If $\frac{5}{6} = \frac{8}{x}$ , what is the value of $x$ ?

5 \cdot x = 6 \cdot 8

5x = 48

x = \frac{48}{5} = 9.6

Always set up $\\frac{a}{b} = \\frac{c}{d}$ , cross-multiply, and solve for the unknown.

rates, unit conversions, & SAT traps

Rates can be per second, per dollar, per liter—you name it. Always write the units and make sure they “cancel” correctly when converting. SAT tip: Write out the units for every step. If the units don’t work out, the answer is wrong.

Example (unit conversion):

If a car travels 90 kilometers in 1.5 hours, what is the speed in meters per second?

90\ \text{km} = 90,000\ \text{meters}

1.5\ \text{hours} = 5400\ \text{seconds}

\text{speed} = \frac{90,000\ \text{m}}{5400\ \text{s}} = 16.67\ \text{m/s}

Always check units—convert everything before dividing!

direct & inverse variation

In direct variation, two variables increase or decrease together: $y = kx$ (k is a constant). In inverse variation, one goes up as the other goes down: $y = \frac{k}{x}$ . The SAT may ask for the constant or how one variable changes if the other is doubled or halved.

Direct variation example:

y = 4x

If $x = 3$ , $y = 12$ . If $x$ doubles, $y$ doubles.

Inverse variation example:

y = \frac{24}{x}

If $x = 4$ , $y = 6$ . If $x$ doubles to 8, $y$ halves to 3.

word problems & setup

SAT ratio/rate word problems require careful setup: decide what’s being compared, assign variables, and always write out the relationship (with units!). Use tables, “for every” statements, or the “part/whole” method as needed.

Example:

A recipe uses flour and sugar in a ratio of 5:2. If 30 cups of flour are used, how many cups of sugar are needed?

\frac{\text{flour}}{\text{sugar}} = \frac{5}{2} = \frac{30}{x}

5x = 60 \Rightarrow x = 12

So, 12 cups of sugar are needed.

Always label your variables and keep units clear!

Ready to test your skills? Try these!

Question 1

The International Space Station orbits Earth at an average speed of 4.76 miles per second. What is the space station’s average speed in miles per hour?

Question 2

For a person $m$ miles from a flash of lightning, the time interval to hear the thunder is $k$ seconds. The ratio of $m$ to $k$ is estimated at 1 to 5. If the interval is 25 seconds, how many miles from the lightning?

Question 3

The population density of Iceland, in people per square kilometer, increased from 2.5 in 1990 to 3.3 in 2014. During this period, the land area was 100,250 sq km. By how many people did Iceland’s population increase from 1990 to 2014?

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