These are 4 hard SAT Problem-Solving and Data Analysis questions from our question bank, all at the top of our difficulty scale. This domain is about 15% of the Math section, 5 to 7 questions: ratios, rates and units, percentages, one- and two-variable data, probability and statistical claims. The hard ones are usually careful-reading problems, where one wrong unit or one misread percent costs the point. Try each one, then open the solution.
Question 1 (Hard, Percent Change and Percents)
The speeds of particles A, B, and C are a meters per second, b meters per second, and c meters per second, respectively. If the speed of particle A is 6,600% of the speed of particle C and the speed of particle C is 0.004% of the speed of particle B, which expression represents the value of a + b in terms of c?- A. 25,066c
- B. 6,850c
- C. 6,604c
- D. 3,160c
Show answer and solution
Answer: A. 25,066c
Turn each percent into a multiplier. 6,600% of c is 66c, so a = 66c. 0.004% is 0.00004, so c = 0.00004b, and b = c / 0.00004 = 25,000c. Then a + b = 66c + 25,000c = 25,066c.
Tip: Write "x% of" as x/100 times. Very small percents like 0.004% become very large multipliers when you solve for the other quantity.
Question 2 (Hard, Rates, Unit Conversions and Ratios)
An object's speed is increasing at a rate of 10.7 meters per second squared. What is this rate in miles per minute squared, rounded to the nearest tenth? (Use 1 mile = 1,609 meters.)- A. 0.4
- B. 23.9
- C. 150.4
- D. 286.9
Show answer and solution
Answer: B. 23.9
Convert seconds squared to minutes squared: one minute is 60 seconds, so one minute squared is 3,600 seconds squared. 10.7 × 3,600 = 38,520 meters per minute squared. Divide by 1,609: 38,520 / 1,609 ≈ 23.9 miles per minute squared.
Tip: A squared unit needs its conversion factor squared. Per second squared to per minute squared means multiply by 602, not 60.
Question 3 (Hard, Statistics I)
A data set of the orbital periods. rounded to the nearest whole number of Earth days, for 13 of Jupiter’s moons is represented in the dot plot. An additional moon with an orbital period of 621 days is added to the original data set to create a new data set of 14 orbital periods. Which statement best compares Ure mean and median of the new data set lo the mean and median of the original data set?
A data set of the orbital periods, rounded to the nearest whole number of Earth days, for 13 of Jupiter’s moons is represented in the dot plot. An additional moon with an orbital period of 621 days is added to the original data set to create a new data set of 14 orbital periods. Which statement best compares the mean and median of the new data set to the mean and median of the original data set?- A. The mean of the new data set is equal to the mean of the original data set, and the median of the new data set is equal to the median of the original data set.
- B. The mean of the new data set is equal to the mean of the original data set, and the median of the new data set is less than the median of the original data set.
- C. The mean of the new data set is less than the mean of the original data set, and the median of the new data set is less than the median of the original data set.
- D. The mean of the new data set is less than the mean of the original data set, and the median of the new data set is equal to the median of the original data set.
Show answer and solution
Answer: D. The mean of the new data set is less than the mean of the original data set, and the median of the new data set is equal to the median of the original data set.
Reading the dot plot gives 13 values from 720 to 732, and the median (7th value) is 724. Adding 621 makes 14 values. The new median is the average of the 7th and 8th values, 724 and 724, which is still 724. The mean drops because 621 is far below every other value.
Tip: An outlier pulls the mean toward it but barely moves the median. Check where the middle values land after you add it.
Review: Statistics I notes
Want more at this level?
Targeted Practice builds sets from the skills you miss.
Question 4 (Hard, Linear vs. Exponential Models)
An acceptable noise criterion rating for the background noise in a laundry room is 50. For a noise criterion rating of 50, the equation y = 22(0.997)x-60 + 47 gives the estimated sound pressure level, y, in decibels, as a function of the octave band center frequency, x, in hertz, where x ≥ 60. Which of the following is the best interpretation of 47 in this context?
- A. 47 is 22 less than the estimated sound pressure level, in decibels, at an octave band center frequency of 60 hertz.
- B. 47 is 22 less than the estimated sound pressure level, in decibels, at an octave band center frequency of 0 hertz.
- C. 47 is the estimated sound pressure level, in decibels, at an octave band center frequency of 60 hertz.
- D. 47 is the estimated sound pressure level, in decibels, at an octave band center frequency of 0 hertz.
Show answer and solution
Answer: A. 47 is 22 less than the estimated sound pressure level, in decibels, at an octave band center frequency of 60 hertz.
When x = 60, the exponent is 0, so 22(0.997)0 = 22 and y = 22 + 47 = 69. So 47 is 22 less than the sound level at 60 hertz. Choices C and D treat 47 as a value of y at some frequency, but y is always 22(0.997)x-60 + 47, which is more than 47.
Tip: In a(b)x-h + k, plug in the x that makes the exponent zero. That shows what each constant means.
How to get better
- Learn the idea behind each solution. Each tip above is a pattern that shows up on many hard questions.
- Read the notes linked in each solution, then drill the skill in Targeted Practice.
- Watch it explained in the free Advanced Math video series with Mr. O.
- Test yourself under time with a free full-length practice test.
Want more? Try these hard SAT math questions, hard algebra questions and hard geometry questions.
Frequently asked questions
How many data analysis questions are on the SAT?
Problem-Solving and Data Analysis is about 15% of the Math section, about 5 to 7 of the 44 questions.
What statistics is on the SAT?
Mean, median, range and standard deviation, data displays like dot plots and scatterplots, margin of error, and evaluating statistical claims.
Is probability on the SAT?
Yes. College Board lists probability in this domain.
Hard SAT Math practice
This guide is part of Digital SAT Vocabulary and Advanced Math: Resource Hub.
Related guides:
- Hard SAT Math Questions: 5 Problems With Full Solutions
- Hard SAT Math Practice Test: 5 Timed Questions With Answers
- Hard SAT Geometry Questions: 5 Problems With Full Solutions
- Hard SAT Algebra Questions: 5 Problems With Full Solutions
- Hard SAT English Practice Test: 5 Reading Questions With Answers
- Released SAT Practice Tests: All 8 Official Tests
Explore every SAT hub
- Digital SAT Complete Guide Format, scoring and a full study plan
- Advanced Math and Vocabulary Hub Hard math topics and high-value words
- SAT Grammar Guide Every grammar and punctuation rule
- SAT Reading Guide Reading question types and strategies
- SAT Practice Tests Guide Official tests and how to use them
- Desmos Guide Calculator shortcuts for SAT Math
- SAT Scores by College Ranges and testing policies for 43 schools
- PSAT Guide PSAT/NMSQT, PSAT 10, PSAT 8/9 and National Merit
- Test Strategy Hub Timing, pacing and prep resources
- SAT Predictions Questions most likely to repeat
Sources
- College Board, The Math Section: Overview.
Frequently Asked Questions
How many data analysis questions are on the SAT?
Problem-Solving and Data Analysis is about 15% of the Math section, about 5 to 7 of the 44 questions.
What statistics is on the SAT?
Mean, median, range and standard deviation, data displays like dot plots and scatterplots, margin of error, and evaluating statistical claims.
Is probability on the SAT?
Yes. College Board lists probability in this domain.