How to solve SAT ratios, rates, and unit conversion questions

This skill covers comparing quantities with ratios, working with rates like miles per hour or dollars per pound, scaling recipes and maps with proportions, and converting between units. On the Digital SAT these are usually short word problems where the setup matters more than the arithmetic. Problem-Solving and Data Analysis is about 15% of the Math section, or 5 to 7 of the 44 questions, and is split across seven skills, so expect about one question on this skill, sometimes two. The same reasoning also shows up inside questions from other skills.

What the questions look like

Proportions. A recipe, scale drawing, or rate is given for one amount, and you find the matching value for another amount. For example, if 3 cups of flour make 8 servings, how much flour do 20 servings need?

Unit conversions. You convert a rate from one set of units to another, such as gallons per minute to liters per hour, sometimes through two or three conversion factors. Area and volume conversions, like square yards to square feet, are a favorite.

Ratio parts. Two or three quantities are in a ratio such as $3:5$, and you know the total or one of the parts. You find another part, the total, or a fraction of the whole.

Step-by-step method

1. Write every number with its unit. "42 pages in 1.5 minutes" is $\dfrac{42 \text{ pages}}{1.5 \text{ min}}$. Units tell you what to multiply and divide.
2. Set up proportions with matching units in matching positions. If the left fraction is cups over servings, the right fraction must also be cups over servings. Then cross-multiply.
3. Convert by multiplying by fractions equal to 1. Write each conversion factor so the unit you want to remove appears once on top and once on the bottom, then cancel. To turn minutes into hours in a rate, multiply by $\dfrac{60 \text{ min}}{1 \text{ hr}}$ if minutes are in the denominator.
4. Square or cube the factor for area or volume. Since 1 yard is 3 feet, 1 square yard is $3^2 = 9$ square feet and 1 cubic yard is $3^3 = 27$ cubic feet.
5. Split totals by ratio using parts. For a ratio $3:5$ with a total of 64, there are $3 + 5 = 8$ parts, so each part is $\frac{64}{8} = 8$. The two amounts are $24$ and $40$.
6. Round only at the end. Keep full values through the calculation.

Worked example

Example 1. A pump moves water at 18 gallons per minute. One gallon is approximately 3.785 liters. To the nearest whole number, how many liters does the pump move per hour?

Start with the rate and cancel units one at a time:

$$\frac{18 \text{ gal}}{1 \text{ min}} \times \frac{60 \text{ min}}{1 \text{ hr}} \times \frac{3.785 \text{ L}}{1 \text{ gal}}$$

Gallons cancel, minutes cancel, and liters per hour remain:

$$18 \times 60 \times 3.785 = 4087.8$$

To the nearest whole number, the pump moves about 4,088 liters per hour.

Example 2. A rectangular patio measures 4.5 yards by 6 yards. Paving costs $3.20 per square foot. What is the cost to pave the entire patio? (1 yard = 3 feet)

The price is per square foot, so convert the side lengths to feet before multiplying:

$$4.5 \text{ yd} = 13.5 \text{ ft} \qquad 6 \text{ yd} = 18 \text{ ft}$$

$$\text{Area} = 13.5 \times 18 = 243 \text{ square feet}$$

$$\text{Cost} = 243 \times 3.20 = \$777.60$$

You can also find the area in square yards first, $4.5 \times 6 = 27$, and multiply by 9 to get 243 square feet. The common wrong answer, $86.40, comes from multiplying 27 square yards by $3.20 as if the price were per square yard.

Common traps

• Flipping a conversion factor. If you multiply by $\frac{1 \text{ gal}}{3.785 \text{ L}}$ when you want liters, the units won't cancel and the answer will be far too small. Write the units out and confirm they cancel before computing.
• Converting square units with the linear factor. One square yard is 9 square feet, not 3. One cubic foot is $12^3 = 1{,}728$ cubic inches, not 12.
• Mixing part-to-part and part-to-whole. If boys and girls are in a $3:5$ ratio, boys make up $\frac{3}{8}$ of the group, not $\frac{3}{5}$. Add the parts to get the whole.
• Treating decimal hours as minutes. 1.5 hours is 90 minutes, not 150 minutes and not 1 hour 50 minutes.
• Rounding in the middle. Rounding 3.785 to 3.8 in Example 1 gives 4,104, which may be a listed choice. Use the given values and round once at the end.

Desmos tip

Use Desmos as a calculator and type the whole conversion chain on one line, such as 18*60*3.785. You avoid copying rounded intermediate results, which is where most arithmetic slips happen in multi-step conversions. For a proportion with an unknown, typing an equation like 3/8 = x/20 shows a vertical line at the solution, $x = 7.5$.