# Ratios, rates, proportions and units on the SAT Canonical: https://duckyhelper.com/learn/sat-math/ratios-rates-and-units/ Updated: 2026-10-01 A ratio compares two amounts, a rate compares amounts with different units (like miles per hour), and a proportion says two ratios are equal. On the SAT, write the units next to every number and multiply by conversion factors so the units you do not want cancel. The classic trap is converting area or volume like length: 1 square yard is 9 square feet, not 3. ## The key idea A conversion factor is a fraction equal to 1, like \(\frac{60 \text{ min}}{1 \text{ h}}\). Multiplying by it changes the units without changing the amount. Line them up so each unwanted unit appears once on top and once on the bottom: $$ 66\,\frac{\text{ft}}{\text{s}} \times \frac{3600\,\text{s}}{1\,\text{h}} \times \frac{1\,\text{mi}}{5280\,\text{ft}} = 45\,\frac{\text{mi}}{\text{h}} $$ Squared and cubed units need the factor squared or cubed. 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. For a ratio like \(4 : 7\), think in parts: there are \(4 + 7 = 11\) parts in the whole, so first find how much one part is worth. ## Worked examples **Example 1: a proportion** Problem: A recipe uses 3 cups of flour for 24 cookies. How many cups of flour are needed for 60 cookies? 1. Set flour over cookies equal in both cases. Call the unknown amount of flour f. $$ \frac{3}{24} = \frac{f}{60} $$ 2. Cross-multiply. $$ 24f = 180 $$ 3. Divide by 24. $$ f = 7.5 $$ Answer: 7.5 cups **Example 2: a chain of unit conversions** Problem: A car travels 66 feet per second. What is its speed in miles per hour? (1 mile = 5,280 feet) 1. Seconds to hours: there are 3,600 seconds in an hour, so multiply by \(\frac{3600 \text{ s}}{1 \text{ h}}\). $$ 66 \times 3600 = 237600 $$ 2. That is 237,600 feet per hour. Feet to miles: divide by 5,280. $$ \frac{237600}{5280} = 45 $$ Answer: 45 miles per hour **Example 3: ratio parts** Problem: A jar holds red and blue marbles in the ratio \(4 : 7\). There are 132 marbles. How many are blue? 1. Count the parts. $$ 4 + 7 = 11 $$ 2. Find one part. $$ \frac{132}{11} = 12 $$ 3. Blue is 7 parts. $$ 7 \times 12 = 84 $$ 4. Check: red is \(4 \times 12 = 48\), and \(48 + 84 = 132\). Answer: 84 blue marbles **Example 4 (SAT-hard): area units** Problem: A rectangular room is 4 yards by 5 yards. Carpet costs $3.50 per square foot. How much does it cost to carpet the room? 1. Area in square yards. $$ 4 \times 5 = 20 $$ 2. Convert to square feet. One square yard is \(3 \times 3 = 9\) square feet. $$ 20 \times 9 = 180 $$ 3. Multiply by the price per square foot. $$ 180 \times 3.5 = 630 $$ 4. The trap answer multiplies by 3 instead of 9: \(60 \times 3.50 = 210\). Answer: $630 ## Common mistakes - **Flipping the conversion factor.** Multiplying 66 by \(\frac{1 \text{ h}}{3600 \text{ s}}\) leaves "ft s per h" units, which is a sign you flipped it. Fix: write the units and check they cancel. - **Using 3 for square yards to square feet.** Area conversions square the factor and volume conversions cube it. Fix: picture a 1 by 1 yard square as a 3 by 3 grid of square feet. - **Using the ratio numbers as the amounts.** \(4 : 7\) does not mean 7 blue marbles. Fix: find the value of one part first. - **Setting up a proportion upside down on one side.** \(\frac{3}{24} = \frac{60}{f}\) mixes flour over cookies with cookies over flour. Fix: say the units out loud for both fractions. - **Forgetting to convert minutes and hours.** A rate in miles per hour cannot be used with a time in minutes. Fix: convert every quantity to matching units before you compute. ## Quick method > **Tip: Units are your answer check** > > If the units of your final number are the units the question asks for, the setup is almost always right. If you get "square feet per hour" when the question wants "hours", a fraction is upside down. This check is fast and catches most rate mistakes. ## Practice **5 SAT-style questions** 1. A printer prints 18 pages in 45 seconds. At this rate, how many pages does it print in 4 minutes? A. 72 B. 96 C. 108 D. 160 Answer: 96. The rate is \(\frac{18}{45} = 0.4\) pages per second. 4 minutes is 240 seconds, and \(0.4 \times 240 = 96\). 72 multiplies 18 by 4 as if 45 seconds were a minute. 2. The ratio of boys to girls in a club is \(5 : 3\). The club has 40 members. How many are girls? A. 8 B. 15 C. 24 D. 25 Answer: 15. There are 8 parts, so one part is \(40 \div 8 = 5\). Girls are 3 parts: 15. 25 is the number of boys, and 24 uses 3/5 instead of 3/8. 3. How many square centimeters are in 2.5 square meters? A. 250 B. 2,500 C. 25,000 D. 250,000 Answer: 25,000. 1 meter is 100 centimeters, so 1 square meter is \(100^2 = 10{,}000\) square centimeters. \(2.5 \times 10{,}000 = 25{,}000\). 250 uses the length factor instead of the area factor. 4. A cyclist rides 18 miles in 1.5 hours. At the same rate, how many minutes does it take to ride 30 miles? A. 120 B. 135 C. 150 D. 180 Answer: 150. The speed is \(18 \div 1.5 = 12\) miles per hour. 30 miles takes \(30 \div 12 = 2.5\) hours, which is 150 minutes. 5. Student-produced response: a pool holds 12,000 gallons. A hose fills it at 8 gallons per minute. How many hours does it take to fill the empty pool? Answer: 25. \(12{,}000 \div 8 = 1{,}500\) minutes, and \(1{,}500 \div 60 = 25\) hours. Entering 1500 answers in the wrong unit. ## Frequently asked questions ### What is the difference between a ratio and a rate? A ratio compares two amounts, often of the same kind, like 4 red marbles to 7 blue marbles. A rate compares amounts with different units, like 12 miles per hour or $3.50 per square foot. A unit rate is a rate per one unit. ### Do I need to memorize unit conversions for the SAT? Know the everyday ones: 60 seconds in a minute, 60 minutes in an hour, 12 inches in a foot, 3 feet in a yard, 100 centimeters in a meter. Less common ones, like feet in a mile, are given in the question. ### How do I convert square or cubic units? Square the length factor for area and cube it for volume. Since 1 foot is 12 inches, 1 square foot is 144 square inches and 1 cubic foot is 1,728 cubic inches. ## Sources - [College Board: SAT Math, Problem-Solving and Data Analysis skills](https://satsuite.collegeboard.org/sat/whats-on-the-test/math/types/problem-solving), accessed 2026-10-01 ## Related - [Percentages on the SAT](https://duckyhelper.com/learn/sat-math/percentages/) - [Area and volume on the SAT](https://duckyhelper.com/learn/sat-math/area-and-volume/) - [Linear functions and slope on the SAT](https://duckyhelper.com/learn/sat-math/linear-functions/) - [SAT Math study guides](https://duckyhelper.com/learn/sat-math/) ## Try asking Ducky - "How do I know which way to flip the conversion fraction?" - "Why is a square yard 9 square feet and not 3?" - "Check my unit chain on this problem." ## Get DuckyHelper Free to start. The web app works in any browser, Chromebooks included; the Mac app can also draw on your real screen. 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