12.1 Ratios, Rates, Proportions, and Units
Key Takeaways
- Part-to-part is not part-to-whole: 12 girls and 18 boys is 12:18 = 2:3 as girls to boys, but 12:30 = 2:5 as girls to the class.
- A unit rate divides so the second quantity is 1: 210 miles in 3 hours is 70 miles per hour, so 5 hours is 350 miles.
- Convert time before multiplying by speed: 2 hours 15 minutes is 2.25 hours, not 2.15, so 48 mph covers 108 miles.
- Dimensional analysis cancels labels: 8 m/s times 3600/1000 is 28.8 km/h; multiplying by 60 once stops at meters per minute.
- Unit price and density are rates. $3.60 for 12 oz is $0.30 per ounce; 48 g in 16 cm³ is 3 g/cm³.
Problem-Solving and Data Analysis (PSD) is about 25% of operational PSAT 8/9 Math, 9–11 questions. College Board's Assessment Framework for the Digital SAT Suite lists five PSD skill/knowledge testing points on this test. This independent OpenExamPrep chapter teaches the first three: ratios, rates, proportional relationships, and units; percentages; and one-variable data (distributions, mean, median, and range). Two-variable data, probability, and in-context graphics sit in 13.1 Two-Variable Data: Models and Scatterplots, 13.2 Probability and Conditional Probability, and 13.3 In-Context Math Questions and Graphics. OpenExamPrep is not a College Board partner and does not claim official approval, review, partnership, or exact equivalence with College Board materials.
A calculator is allowed on every Math question, including embedded Desmos. About 30% of Math is in context. About 25% is student-produced response (SPR). Ratios and unit conversions are frequent SPR targets: type the number only—no unit letters, no commas, no dollar signs. SPR box rules are in 15.1 Student-Produced Response Questions. Handheld calculator policy is in 3.1 Devices, What to Bring, and Calculator Policy.
Part-to-part versus part-to-whole
A ratio compares two quantities of the same kind. Write it with a colon (2:3), as a fraction (2/3), or in words (2 to 3). The first job is to decide whether you are comparing two parts or one part to the total.
A classroom has 12 girls and 18 boys. Total students: 12 + 18 = 30.
| Comparison | Numbers | Simplified |
|---|---|---|
| Part-to-part, girls to boys | 12:18 | 2:3 |
| Part-to-part, boys to girls | 18:12 | 3:2 |
| Part-to-whole, girls to students | 12:30 | 2:5 |
| Part-to-whole, boys to students | 18:30 | 3:5 |
The fraction of the class that is girls is 12/30 = 2/5, which is 40%. That 2/5 is the part-to-whole ratio. The 2:3 figure is girls to boys, a different comparison.
Trap. If a stem says the ratio of cats to dogs is 2:5, that is part-to-part. Total parts = 2 + 5 = 7. Cats are 2/7 of the pets, not 2/5. If there are 28 pets in that 2:5 mix, each part is 28/7 = 4 pets, so 8 cats and 20 dogs. Check: 8:20 = 2:5, and 8 + 20 = 28.
Paint uses the same arithmetic. Mix 2 parts blue to 5 parts yellow to make 28 liters. Each part is 4 liters, so 8 L blue and 20 L yellow. If someone treats 2:5 as blue being 2/5 of the mix, they pour 11.2 L of blue and the color is wrong.
Unit rates
A rate compares quantities of different kinds, such as miles per hour or dollars per pound. A unit rate uses 1 in the denominator: 1 hour, 1 pound, 1 minute. College Board's PSD description for the SAT Suite, including PSAT 8/9, names unit rate as a skill you have to understand and apply, not only recite.
A car travels 210 miles in 3 hours at a constant speed.
210 ÷ 3 = 70 miles per hour.
In 5 hours: 70 × 5 = 350 miles. In 12 minutes (12/60 = 0.2 hour): 70 × 0.2 = 14 miles. Check: 12 minutes is 1/5 of an hour, and 70/5 = 14.
A printer produces 45 pages in 3 minutes. Unit rate: 45/3 = 15 pages per minute. In 8 minutes: 15 × 8 = 120 pages.
Proportion check: 45/3 = p/8. Cross-multiply: 45 × 8 = 3p, so 360 = 3p, p = 120.
Proportions
Two ratios form a proportion when they are equal. If four notebooks cost $12, seven notebooks at the same unit price cost:
Unit price = 12/4 = $3 per notebook. 7 × 3 = $21.
Cross-multiply 4/12 = 7/x:
4x = 12 × 7 = 84
x = 21
You can also write 4/7 = 12/x; 4x = 84; x = 21. Same answer. What you must not do is treat three extra notebooks as +$3 total. Each extra notebook is $3, so three extras are $9, and 12 + 9 = 21. That addition works only because the unit price is constant.
Recipe: 3 cups of flour make 24 cookies. Flour f for 40 cookies:
3/24 = f/40
3 × 40 = 24f
120 = 24f
f = 5 cups
Check: 24 cookies use 3 cups, so 8 cookies per cup. 40/8 = 5 cups.
A map scale of 1 cm : 5 km is a proportion. A trail that measures 8.4 cm on the map is 8.4 × 5 = 42 km. In meters: 42 × 1000 = 42,000 m. If this were SPR, type 42000 or 42 depending on the unit the stem asked for—read the unit. Do not type 42,000 with a comma.
Unit conversions and dimensional analysis
Convert so both sides of a rate use matching units. Multiply by conversion fractions that equal 1, and cancel labels. That canceling is dimensional analysis.
| From | To | Multiply by |
|---|---|---|
| Hours → minutes | 2.5 hours | 2.5 × 60 = 150 minutes |
| Minutes → hours | 90 minutes | 90/60 = 1.5 hours |
| Kilometers → meters | 3.5 km | 3.5 × 1000 = 3500 m |
| Meters → kilometers | 450 m | 450/1000 = 0.45 km |
Time trap: 2 hours 20 minutes is not 2.20 hours. Twenty minutes is 20/60 = 1/3 hour, so the time is 2 + 1/3 = 7/3 hours, about 2.333 hours.
Distance at 48 miles per hour for 2 hours 15 minutes: 15 minutes = 0.25 hour, so time = 2.25 hours. Distance = 48 × 2.25. Compute 48 × 2 = 96 and 48 × 0.25 = 12; total 108 miles. Check: 48 × 9/4 = 432/4 = 108.
A runner covers 400 meters in 50 seconds.
400/50 = 8 meters per second.
To kilometers per hour, chain the 1-fractions:
8 m/s × (1 km / 1000 m) × (3600 s / 1 h) = 8 × 3600 / 1000 = 8 × 3.6 = 28.8 km/h
The 1000 and 3600 cancel meters and seconds. If you multiply by 60 once, you get 480 meters per minute, a different unit. Desmos will happily compute 8 × 60 = 480; it will not notice that the stem asked for km/h.
A student walks 1.2 km in 15 minutes.
1.2 km = 1200 m, and 1200/15 = 80 m/min. In m/s: 80/60 = 4/3 m/s (about 1.333 m/s). In km/h: 15 minutes = 0.25 hour, so 1.2 / 0.25 = 4.8 km/h. Check: 80 m/min × 60 min/h = 4800 m/h = 4.8 km/h.
Dollars per unit and density
Five pounds of apples cost $8.75. Unit price: 8.75/5 = $1.75 per pound. Twelve pounds: 12 × 1.75 = $21.00. Check: 8.75 × 12/5 = 8.75 × 2.4 = 21.
Unit-price comparison. Brand A is 12 ounces for $3.60. Brand B is 16 ounces for $4.00.
- A: 3.60/12 = $0.30 per ounce
- B: 4.00/16 = $0.25 per ounce
Brand B is cheaper per ounce. Sixteen ounces of A would cost 16 × 0.30 = $4.80, which is $0.80 more than Brand B's $4.00.
Density is mass per volume, another unit rate. A metal sample has mass 48 grams and volume 16 cubic centimeters.
Density = 48/16 = 3 g/cm³.
A 10 cm³ piece of the same metal has mass 3 × 10 = 30 g. A 5 g piece has volume 5/3 cm³. For SPR, type the improper fraction 5/3 (three characters) rather than a mixed number.
Trail mix: 3 kg costs $18. Cost per 100 g: 3 kg = 3000 g, and 18/3000 = 0.006 dollars per gram, so 100 g costs $0.60. Shortcut: 3000/100 = 30 hundred-gram packs, and 18/30 = 0.6.
A faucet fills 15 liters in 2 minutes. Rate: 15/2 = 7.5 L/min. In 8 minutes: 8 × 7.5 = 60 L. In 40 seconds (2/3 minute): 7.5 × 2/3 = 5 L.
Combining two rates without averaging them blindly
A bus travels 90 km in 1.5 hours, then 60 km in 45 minutes. Overall average speed is total distance over total time, not the mean of the two speeds.
First leg: 90/1.5 = 60 km/h. Second leg: 45 minutes = 0.75 hour, and 60/0.75 = 80 km/h.
Total distance: 90 + 60 = 150 km. Total time: 1.5 + 0.75 = 2.25 hours. Average speed: 150 / 2.25. Write 2.25 as 9/4: 150 × 4/9 = 600/9 = 200/3 km/h, which is 66.6 repeating km/h.
The mean of 60 and 80 is 70, and that is wrong here because the legs did not last the same amount of time. Weighted by time: (60 × 1.5 + 80 × 0.75) / 2.25 = (90 + 60) / 2.25 = 150 / 2.25, the same 200/3. If this is SPR, type 200/3 (five characters). That fraction is exact.
Setting up the proportion on paper
Write the two rates with matching units before you cross-multiply. If one rate is pages per 3 minutes and the unknown is pages in 8 minutes, keep minutes in both denominators. If a stem mixes hours and minutes, convert first.
Desmos can multiply 8.4 × 5 or 48 × 2.25, but it will not notice that you used 2.20 hours instead of 2 hours 20 minutes. Convert on the school scratch paper (write your name; do not tear the packet; do not bring your own paper), then use the scientific keypad as a check.
A science club has 8 sixth graders and 12 seventh graders. What is the part-to-whole ratio of sixth graders to all club members, in simplest form?
A car travels 180 miles in 3 hours at a constant speed. How many miles does it travel in 5 hours at that speed?
A runner moves at a constant 8 meters per second. What is that speed in kilometers per hour?