25 hard SAT Ratios, rates, proportions, units questions
Real questions from the SAT Climb bank, all at the hard difficulty tier. Pick an answer before you open the explanation. Every question tells you why the answer is right and why each wrong choice is tempting.
Math · Problem-Solving and Data Analysis~2 per testHard tier
Flips the conversion factor (multiplies when you should divide).
Drops a unit step in a multi-step conversion.
Mismatched units (feet vs. inches) inside one calculation.
Question 1Hard
A car travels at a constant speed of 72 kilometers per hour. To the nearest whole number, how many feet does the car travel in 5 seconds? 1 kilometer = 3281 feet
Show the answer and explanation
Why D is right
The car travels 72 kilometers per hour. Converting to feet per hour: 72 × 3281 = 236232 feet per hour. Converting to feet per second: 236232/3600 = 65.62 feet per second. In 5 seconds, the car travels 65.62 × 5 = 328.1 feet, which rounds to 328 feet.
Why the others are wrong
ADistance cannot be negative. This results from a sign error in the calculation.
BThis is the speed in meters per second without properly applying the conversion to feet.
CThis results from inverting part of the calculation, multiplying by 3600 instead of dividing when converting hours to seconds.
Question 2Hard
A chemical solution has a concentration of 45 milligrams of active ingredient per liter. A laboratory needs to prepare 3.6 liters of a diluted solution with a concentration of 18 milligrams per liter. How many milliliters of the original solution are needed?
Show the answer and explanation
Why C is right
The diluted solution needs 18 milligrams per liter × 3.6 liters = 64.8 milligrams total of active ingredient. The original solution has 45 milligrams per liter, so the volume needed is 64.8 ÷ 45 = 1.44 liters. Converting to milliliters: 1.44 liters × 1000 = 1440 milliliters.
Why the others are wrong
AThis results from inverting the concentration ratio and calculating (18 ÷ 45) × 3.6 × 1000 incorrectly.
BThis results from an arithmetic error in calculating the volume needed from the original solution.
DThis results from failing to account for the concentration difference and simply converting 1.8 liters to milliliters.
Question 3Hard
A solution contains 0.045 kilograms of salt per liter. A container holds 2400 milliliters of this solution. What is the mass, in grams, of salt in the container? (1 kilogram = 1000 grams, 1 liter = 1000 milliliters)
Show the answer and explanation
Why C is right
The container holds 2400 milliliters, which equals 2.4 liters. The solution contains 0.045 kilograms of salt per liter, so the total mass of salt is 2.4 × 0.045 = 0.108 kilograms. Converting to grams: 0.108 × 1000 = 108 grams.
Why the others are wrong
AThis results from converting 2400 milliliters to 2.4 liters but failing to convert the final answer from kilograms to grams, leaving it as 0.108 instead of 108.
BThis results from inverting the conversion ratio, computing 2400 ÷ 0.045 ÷ 1000 instead of the correct sequence of operations.
DThis results from an error in the unit conversion sequence, possibly computing 2400 × 0.045 ÷ 1000 × 1.75 or another incorrect combination.
Question 4Hard
A metal alloy has a density of 8.96 grams per cubic centimeter. A sample of this alloy has a mass of 7168 grams. If the sample is in the shape of a rectangular prism with a square base where each side of the base measures 8 centimeters, what is the height, in centimeters, of this sample?
Show the answer and explanation
Why B is right
The volume of the sample is 7168 divided by 8.96, which equals 800 cubic centimeters. The base area is 64 square centimeters. Dividing 800 by 64 gives a height of 12.5 centimeters.
Why the others are wrong
AThis results from an arithmetic error in computing the volume or incorrectly dividing the volume by the base area.
CThis results from failing to properly account for the density conversion and using the mass directly in calculations.
DThis results from inverting the relationship between volume and base area.
Question 5Hard
The table shows the growth factor for three reptile species. The age of a reptile, in years, can be estimated by multiplying the reptile's length, in inches, by the growth factor for that species.
| Species | Growth Factor |
|---------|---------------|
| Iguana | 1.8 |
| Monitor | 2.5 |
| Python | 3.2 |
An iguana is 25 inches long, and a python is 18 inches long. The iguana's estimated age is what percent less than the python's estimated age?
Show the answer and explanation
Why A is right
The iguana's estimated age is 25 × 1.8 = 45 years. The python's estimated age is 18 × 3.2 = 57.6 years. The difference is 57.6 − 45 = 12.6 years. The percent less is (12.6 ÷ 57.6) × 100 ≈ 21.875%, which rounds to 22%.
Why the others are wrong
BThis is the percent difference in lengths (25 − 18 = 7, and 7 ÷ 25 = 0.28), not the percent difference in ages.
CThis is the absolute difference in ages in years, not the percent difference.
DThis results from using the wrong base in the percent calculation or an incorrect computation.
Question 6Hard
A rectangular field has an area of 87120 square feet. What is the area, in acres, of this field? 1 acre = 4840 square yards and 1 yard = 3 feet
Show the answer and explanation
Why A is right
First convert square feet to square yards. Since 1 yard equals 3 feet, 1 square yard equals 9 square feet. Therefore 87120 square feet equals 87120 divided by 9, which equals 9680 square yards. Then convert to acres: 9680 divided by 4840 equals 2 acres.
Why the others are wrong
BThis results from dividing 87120 by 3 to get yards (instead of dividing by 9 for square yards) then dividing by 4840.
CThis results from converting 87120 square feet to 29040 square yards (multiplying by 3 instead of dividing by 9) then dividing by 4840, an error in the square conversion.
DThis is the area in square yards without the final conversion to acres.
Question 7Hard
A mixture contains acid and water in the ratio 3:7 by volume. If 20 liters of water are added to 50 liters of this mixture, what is the percentage of acid in the new mixture?
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Why B is right
The original mixture has 3 parts acid and 7 parts water (total 10 parts). In 50 liters, there are (3/10) × 50 = 15 liters of acid. After adding 20 liters of water, the new mixture has 70 liters total with 15 liters of acid. The percentage is (15/70) × 100 ≈ 21.4%.
Why the others are wrong
AThis calculates the amount of acid (15 liters) but treats it as a percentage directly without considering the new total volume.
CThis inverts the calculation, treating the acid as 7 parts and water as 3 parts in the original mixture.
DThis fails to add the 20 liters of water to the denominator, calculating 15/50 × 100 = 30%, then rounding incorrectly.
Question 8Hard
A car travels at a constant speed of 72 kilometers per hour. How many meters does the car travel in 45 seconds? (1 kilometer = 1000 meters)
Show the answer and explanation
Why C is right
The speed is 72 kilometers per hour, which equals 72000 meters per hour. Converting to meters per second: 72000 ÷ 3600 = 20 meters per second. In 45 seconds, the car travels 20 × 45 = 900 meters.
Why the others are wrong
AThis results from computing 72 × 45 ÷ 60 without converting kilometers to meters, yielding 54 instead of the correct answer.
BThis results from an error in time conversion, possibly computing 72 × 45 ÷ 10 or failing to properly account for the hour-to-second conversion.
DThis results from multiplying 72 × 45 without dividing by the proper time conversion factor, producing an answer that is too large.
Question 9Hard
A solution contains 15 grams of salt dissolved in 250 milliliters of water. A chemist needs to create 2 liters of solution with the same concentration. How many grams of salt are needed? (1 liter = 1000 milliliters)
Show the answer and explanation
Why B is right
The concentration is 15 grams per 250 milliliters, or 0.06 grams per milliliter. Converting 2 liters to milliliters: 2 × 1000 = 2000 milliliters. The salt needed is 2000 × 0.06 = 120 grams.
Why the others are wrong
AThis results from doubling the original 15 grams without accounting for the full scale-up from 250 milliliters to 2000 milliliters.
CThis results from inverting the ratio, computing 250 ÷ 15 and then applying an incorrect scaling.
DThis results from dividing 15 by 2 instead of recognizing that 2 liters is 8 times the original 250 milliliters.
Question 10Hard
A solution is made by mixing two liquids in a ratio of 5 parts of Liquid X to 3 parts of Liquid Y by volume. If 480 milliliters of Liquid Y is used, how many liters of the total solution are produced? (1 liter = 1000 milliliters)
Show the answer and explanation
Why B is right
For every 3 parts of Liquid Y, there are 5 parts of Liquid X. If 480 milliliters of Y is used, the amount of X is (5/3) × 480 = 800 milliliters. The total solution is 480 + 800 = 1280 milliliters, which equals 1.28 liters.
Why the others are wrong
AThis results from computing only the volume of Liquid X (800 milliliters = 0.8 liters) without adding the volume of Liquid Y.
CThis results from inverting the ratio, computing (3/5) × 480 + 480 = 768 milliliters or another error that produces 1.6 liters.
DThis results from an error in the ratio calculation, possibly computing 480 × (5 + 3) ÷ 3 ÷ 1000 × 1.8 or another incorrect operation.
Question 11Hard
A cylindrical metal rod has a density of 8.4 grams per cubic centimeter. The rod has a radius of 3 centimeters and a height of 10 centimeters. To the nearest gram, what is the mass of the rod?
Show the answer and explanation
Why C is right
The volume of a cylinder is πr2h=π(3)2(10)=90π cubic centimeters. The mass is density × volume = 8.4×90π≈8.4×282.74≈2375 grams.
Why the others are wrong
AThis incorrectly calculates the lateral surface area (2πrh) instead of the volume, then multiplies by density.
BThis uses only 2πr² as the base area without including the height in the volume calculation.
DThis incorrectly uses the diameter instead of the radius in calculating the volume, giving π(6)²(10).
Question 12Hard
The table shows the growth factor for four marine species. The age of a specimen, in years, can be estimated by multiplying the specimen's length, in centimeters, by the growth factor for that species.
| Species | Growth Factor |
|---------|---------------|
| Lobster | 0.4 |
| Crab | 0.6 |
| Shrimp | 0.3 |
| Clam | 0.8 |
A lobster and a crab have a combined length of 85 centimeters. If they have the same estimated age, what is the length, in centimeters, of the lobster?
Show the answer and explanation
Why B is right
Let the lobster's length be L centimeters. Then the crab's length is 85 − L centimeters. Since they have the same age, 0.4L=0.6(85 − L). Solving: 0.4L=51 − 0.6L, so L=51 centimeters.
Why the others are wrong
AThis is the crab's length, not the lobster's length.
CThis incorrectly assumes the lobster and crab have equal lengths, ignoring the different growth factors.
DThis results from using the wrong growth factor or an incorrect algebraic setup.
Question 13Hard
A cyclist travels at a constant speed. In 45 minutes, the cyclist travels 18 kilometers. At this rate, how many meters does the cyclist travel in 10 seconds? 1 kilometer = 1000 meters
Show the answer and explanation
Why B is right
The cyclist travels 18 kilometers in 45 minutes. Converting to meters per minute: (18 × 1000)/45 = 400 meters per minute. Converting to meters per second: 400/60 = 6.667 meters per second. In 10 seconds, the cyclist travels 6.667 × 10 = 66.67 meters, which rounds to 67 meters.
Why the others are wrong
AThis results from failing to convert kilometers to meters and using an incorrect time conversion.
CThis results from inverting part of the calculation, multiplying by 60 instead of dividing when converting minutes to seconds.
DDistance cannot be negative. This results from a sign error in the calculation.
Question 14Hard
A car travels from City A to City B at an average speed of 60 miles per hour and returns from City B to City A at an average speed of 40 miles per hour. If the total travel time for the round trip is 5 hours, what is the distance, in miles, between City A and City B?
Show the answer and explanation
Why B is right
Let d be the distance between the cities. The time from A to B is d/60 hours, and the time from B to A is d/40 hours. The total time is d/60+d/40=5. Finding a common denominator: 2d/120+3d/120=5, so 5d/120=5, giving 5d=600, thus d=120 miles.
Why the others are wrong
AThis results from using the average of the two speeds (50 mph) and treating it as if it were the constant speed for the entire trip, then using an incorrect time calculation.
CThis comes from multiplying the total time by an incorrectly calculated average speed or from doubling the correct answer.
DThis results from using the harmonic mean formula incorrectly or from computing 5 × 50 where 50 is the arithmetic mean of 60 and 40.
Question 15Hard
A pharmaceutical company produces a medication in liquid form with a concentration of 250 milligrams of active ingredient per 5 milliliters of solution. A patient requires a dose of 0.45 grams of the active ingredient. How many milliliters of the solution must be administered?
Show the answer and explanation
Why C is right
First convert 0.45 grams to milligrams: 0.45 grams = 450 milligrams. The concentration is 250 mg per 5 mL, which equals 50 mg per 1 mL. To deliver 450 mg, the volume needed is 450/50 = 9 milliliters.
Why the others are wrong
AThis results from inverting the ratio, calculating 5/250 and then multiplying by 0.45 instead of converting units properly.
BThis is obtained by dividing 450 by 250 without accounting for the 5 mL portion of the concentration ratio, missing the unit conversion step.
DThis results from incorrectly converting grams to milligrams as 45000 mg instead of 450 mg, an order of magnitude error.
Question 16Hard
A printing press produces 1350 pages in 45 minutes when operating at a constant rate. At this same rate, how many pages can the press produce in 2 hours?
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Why C is right
The press produces 1350 pages in 45 minutes, giving a rate of 1350 ÷ 45 = 30 pages per minute. Since 2 hours equals 120 minutes, the press produces 30 × 120 = 3600 pages in 2 hours.
Why the others are wrong
AThis results from treating 2 hours as 60 minutes instead of 120 minutes.
BThis results from doubling the 45-minute output without accounting for the full 120 minutes in 2 hours.
DThis results from setting up an incorrect proportion that inverts the time relationship.
Question 17Hard
A medication dosage is 0.008 grams per kilogram of body weight. A patient weighing 154 pounds requires this medication. What is the required dosage, in milligrams? (1 kilogram = 2.2 pounds, 1 gram = 1000 milligrams)
Show the answer and explanation
Why C is right
The patient weighs 154 pounds, which equals 154 ÷ 2.2 = 70 kilograms. The dosage is 0.008 grams per kilogram, so the total is 70 × 0.008 = 0.56 grams. Converting to milligrams: 0.56 × 1000 = 560 milligrams.
Why the others are wrong
AThis results from correctly computing 0.56 grams but failing to convert from grams to milligrams.
BThis results from inverting the weight conversion, computing 154 × 2.2 ÷ 1000 × 0.008 or another incorrect sequence.
DThis results from an error in the conversion chain, possibly computing 154 × 0.008 × 2.2 × 1000 or multiplying when division was needed.
Question 18Hard
A type of plastic has a density of 1.2 grams per cubic centimeter. A cube of this plastic has a mass of 972 grams. To the nearest centimeter, what is the edge length of the cube?
Show the answer and explanation
Why A is right
Using mass = density × volume, the volume is 972 ÷ 1.2 = 810 cubic centimeters. For a cube with edge length s, volume = s3, so s3 = 810. Taking the cube root gives s=9 centimeters.
Why the others are wrong
BThis incorrectly computes the square root of the volume instead of the cube root.
CThis divides the mass by the density but fails to take the cube root to find the edge length.
DThis uses the reciprocal of the density (dividing mass by 1.2 instead of multiplying), resulting in an incorrect volume.
Question 19Hard
Two pumps working together can drain a tank in 4 hours. One pump alone can drain the tank in 10 hours. How many hours would it take the other pump alone to drain the tank?
Show the answer and explanation
Why C is right
The combined rate is 1/4 tank per hour. The first pump's rate is 1/10 tank per hour. The second pump's rate is 1/4 minus 1/10, which equals 5/20 minus 2/20, which equals 3/20 tank per hour. Therefore, the second pump alone takes 20/3, or approximately 6.67 hours.
Why the others are wrong
AThis results from computing 10 minus 4 divided by some incorrect value, treating the problem as simple subtraction.
BThis results from computing 10 minus 4, treating time values as directly subtractable without considering rates.
DThis results from adding 10 and 4 instead of using the rate equation for combined work.
Question 20Hard
A solution contains 15% salt by volume. A chemist needs to create 240 milliliters of a 25% salt solution by adding pure salt to the 15% solution. How many milliliters of pure salt must be added?
Show the answer and explanation
Why B is right
Let x be the milliliters of pure salt added. The original solution volume is (240−x) mL at 15% concentration. The equation is 0.15(240−x)+x=0.25(240). Simplifying: 36−0.15x+x=60, so 0.85x=24, giving x=24/0.85 ≈ 28.2 milliliters.
Why the others are wrong
AThis results from calculating 0.25(240) - 0.15(240) = 24 without accounting for the change in total volume when pure salt is added.
CThis comes from setting up the equation incorrectly, perhaps using 240(0.25 - 0.15) divided by an inverted concentration factor.
DThis is 0.25(240) = 60, representing the total salt needed in the final solution but not accounting for the salt already present.
Question 21Hard
A vehicle travels at a constant speed of 72 kilometers per hour. What is this speed in meters per second? (1 kilometer = 1000 meters)
Show the answer and explanation
Why B is right
Converting 72 kilometers per hour to meters per second requires multiplying by 1000 to get meters and dividing by 3600 to convert hours to seconds. This gives 72000 divided by 3600, which equals 20 meters per second.
Why the others are wrong
AThis results from dividing 72 by 60 instead of properly converting both units.
CThis results from failing to convert either the distance unit or the time unit.
DThis results from multiplying by conversion factors instead of dividing, giving meters per hour times seconds per hour.
Question 22Hard
A plot of land has an area of 0.75 square kilometers. What is the area, in square meters, of this plot? (1 kilometer = 1000 meters)
Show the answer and explanation
Why D is right
Since 1 kilometer equals 1000 meters, 1 square kilometer equals 10002=1000000 square meters. The plot has an area of 0.75 square kilometers, which equals 0.75 × 1000000 = 750000 square meters.
Why the others are wrong
AThis results from multiplying 0.75 by 1000 instead of 1000000, treating area conversion as if it were linear.
BThis results from multiplying by 10000 instead of 1000000, using an incorrect squared conversion factor.
CThis results from multiplying by 100000, an intermediate error in determining the proper area conversion factor.
Question 23Hard
A recipe for bread requires 420 grams of flour to make 6 loaves. A baker wants to make 35 loaves using this recipe. How many kilograms of flour does the baker need? 1 kilogram = 1000 grams
Show the answer and explanation
Why A is right
The recipe requires 420 grams for 6 loaves, which equals 70 grams per loaf. For 35 loaves, the baker needs 70 times 35, which equals 2450 grams. Converting to kilograms: 2450 divided by 1000 equals 2.45 kilograms.
Why the others are wrong
BThis is the flour needed in grams without converting to kilograms as requested.
CThis results from multiplying 420 by 35 then dividing by 6, but forgetting to convert from grams to kilograms, then incorrectly dividing by some value.
DThis results from dividing 420 by 6 to get 70, then dividing 420 by 70 instead of multiplying 70 by 35, creating an inverted operation.
Question 24Hard
A recipe calls for ingredients in the ratio of 3 parts flour to 2 parts sugar to 1 part butter by weight. If a baker uses 450 grams of flour, and the recipe also requires 80 milliliters of milk with a density of 1.03 grams per milliliter, what is the total weight, in grams, of all ingredients combined?
Show the answer and explanation
Why B is right
If flour is 3 parts and equals 450 grams, then 1 part equals 150 grams. Sugar (2 parts) is 300 grams, and butter (1 part) is 150 grams. The milk weighs 80 × 1.03 = 82.4 grams. Total weight is 450 + 300 + 150 + 82.4 = 982.4 grams, which rounds to 982 grams.
Why the others are wrong
AThis results from adding only the flour, sugar, and butter (450 + 300 + 150 = 900) without including the milk's weight contribution.
CThis comes from incorrectly treating the milk density as an additive quantity, adding 80 + 1.03 = 81.03 grams for milk instead of multiplying, then adding to the dry ingredients.
DThis results from incorrectly doubling one of the ingredient quantities or miscalculating the ratio proportions with an addition instead of multiplication error.
Question 25Hard
Two workers can complete a project together in 12 days. If one worker is twice as efficient as the other, how many days would it take the more efficient worker to complete the project working alone?
Show the answer and explanation
Why B is right
Let the less efficient worker complete 1/x of the project per day and the more efficient worker complete 2/x per day. Together they complete 3/x per day. Since they finish in 12 days, 3/x=1/12, so x=36. The more efficient worker completes 2/36 = 1/18 per day, taking 18 days alone.
Why the others are wrong
AThis inverts the efficiency ratio, treating the less efficient worker as twice as fast.
CThis doubles the combined time without accounting for the efficiency difference: 12 × 2 = 24.
DThis calculates the time for the less efficient worker (36 days) instead of the more efficient one.
These 25 are a sample, not a study plan
These 25 come from a bank of 17,599 questions across all 31 SAT skills and three difficulty tiers. Inside SAT Climb you get the rest of Ratios, rates, proportions, units, a diagnostic that finds which skills are actually costing you points, and a grid that shows what to drill next.
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