SAT Ratios, Rates, and Units: The Complete Guide

A proportional graph passes through the origin. A service with a fixed initial fee plus a charge per hour has a constant rate of change but is not proportional to hours: zero hours still costs the initial fee. Do not force every straight-line situation into a proportion. Check whether the context includes a starting amount.

A useful table test is to divide each output by its corresponding nonzero input. Equal quotients support a direct proportion when the context assumes one consistent rule. Equal differences alone do not: outputs can increase steadily while retaining a nonzero starting value. For a map scale or a constant material requirement, a proportional model is natural. For a rental bill with a delivery charge, account for the starting charge separately before using an hourly rate.

For density, rearrange according to the unknown: $m=\rho V$ and $V=m/\rho$. A density of $1.6\text{ g/cm}^3$ means each cubic centimeter has mass 1.6 grams under the stated model. Multiply by volume in cubic centimeters to get grams. If volume is given in milliliters, use $1\text{ mL}=1\text{ cm}^3$.

Area and volume conversions require powers. If a length unit contains 100 smaller length units, its square contains 10,000 smaller square units and its cube contains 1,000,000 smaller cubic units. Convert the whole squared or cubed unit, rather than applying the length conversion once. Units in a denominator need the same care as units in a numerator.

For a ratio with a known total, add the ratio parts before dividing. For a ratio with a known difference, subtract the ratio parts. These lead to different scale factors. If the amounts are in a ratio of 7 to 3, the total represents ten parts, while the difference represents four parts. Neither seven nor three alone represents the whole.

Distinguish a fixed task from a fixed duration. For a fixed task, doubling a constant production rate halves the time required. For a fixed duration, doubling that rate doubles output. The first relationship is inverse, while the second is direct. If two independent scanners process different pages of the same archive simultaneously, add their pages-per-minute rates to find the combined rate. Then divide the archive's page count by that sum to find minutes. This assumes they can split the work and maintain their stated rates; it would not describe two required stages that every page must pass through in sequence.

A rate can also describe a concentration, such as milligrams of solute per liter of solution. Multiplying concentration by solution volume gives solute mass when the volume units match. Dividing the mass by concentration gives volume. This is the same unit-cancellation structure as density, even though the numerator measures only the dissolved material rather than the whole solution.

Use the calculator for unwieldy arithmetic after the setup is stable. Parentheses matter when a rate has a product in its denominator. Keep an exact fraction until the final line if a conversion produces a repeating decimal. Early rounding can change the last digit of an otherwise correct answer, especially when you later multiply by a large total.

Average speed is total distance divided by total time. Taking the arithmetic mean of two speeds works when the time spent at each speed is equal. It generally fails for equal distances, because the slower leg takes longer and receives more time weight. Write each travel time first if the legs have different speeds.

Check whether the answer should be larger or smaller before converting. A numerical length in centimeters is larger than the same length in meters. A rate in items per second is smaller than the same rate in items per minute. This size check complements unit cancellation; neither requires a second full solution.

When a group changes, use a scale variable for the original counts. If two inventories are $5k$ and $8k$ and the first gains 18 objects, a new ratio of $7:8$ gives $(5k+18)/(8k)=7/8$. Cross multiplication gives $40k+144=56k$, so $k=9$. The original counts were 45 and 72; the new counts 63 and 72 reduce to $7:8$.

That check is essential because a correct scale factor is often only an intermediate answer. A question may request the original total, the final total, or one category. Label the stage of each quantity. If only one group changes, do not add the change to both sides merely because both appear in the new ratio.

Compound-unit conversions can be handled one factor at a time. To convert a density in grams per cubic centimeter to kilograms per cubic meter, multiply by $\frac{1\text{ kg}}{1000\text{ g}}$ and $\frac{1000000\text{ cm}^3}{1\text{ m}^3}$. The net numerical factor is 1,000. The large factor comes from the cubed volume conversion, not from a change in the material.

Spend about 20 minutes on a first practice session. Use the opening five minutes to write conversion chains without calculating them. Include a time unit, a rate with two changing units, and a squared or cubed measurement. The goal is to see the requested unit survive after cancellation, rather than to memorize a direction for every conversion.

Next, solve a small Solar set in ratios and rates. For each problem, mark whether the relationship uses a total, a difference, a rate, or density. Work accurately before adding time pressure. In the last five minutes, redo two missed setups with the numbers covered and explain aloud why your fraction order is correct.

On a later day, mix this skill with percentages and linear models. If a problem is taking too long, preserve your labeled setup before moving on. A correct expression is useful progress and makes a later return faster. Repeated speed should come from recognizing the structure, not from skipping the unit check.