SAT Math Problem-Solving and Data Analysis practice: 9 worksheets with answer keys

Ratios, rates and unit conversions

Ratios, rates and units · 20 questions

Worked Examples: SAT Strategy

Practice Questions

Level 1: Easy (Foundational SAT)

1. A machine produces 45 parts every 3 minutes. At this rate, how many parts will the machine produce in 20 minutes? (Grid-in)
2. A local theater has a ratio of children's tickets sold to adult tickets sold of $3 : 5$. If a total of 120 tickets were sold, how many children's tickets were sold?
   • A) $30$
   • B) $45$
   • C) $75$
   • D) $200$
3. A hose puts exactly 88 ounces of water into a bucket in $x$ seconds. Which expression represents the number of ounces of water the hose puts into the bucket in 1 second?
   • A) $88x$
   • B) $\frac{x}{88}$
   • C) $\frac{88}{x}$
   • D) $88 - x$
4. A recipe calls for 2 cups of sugar for every 3 cups of flour. If a baker uses 15 cups of flour to make a large batch, how many cups of sugar are needed? (Grid-in)
5. How many centimeters are equivalent to 4.5 meters? (1 meter = 100 centimeters). (Grid-in)

Level 2: Medium (Standard SAT)

1. A car travels at a constant speed of 75 miles per hour. How many miles does the car travel in 40 minutes?
   • A) $30$
   • B) $40$
   • C) $50$
   • D) $112.5$
2. On a map, 1 inch represents an actual distance of 15 miles. If two cities are 4.2 inches apart on the map, what is the actual distance between the two cities, in miles? (Grid-in)
3. Machine A can print 400 pages in 2 hours. Machine B can print 600 pages in 3 hours. If both machines work together at their respective constant rates, how many total pages will they print in 5 hours?
   • A) $1000$
   • B) $1500$
   • C) $2000$
   • D) $2500$
4. A traveler is converting US Dollars (USD) to Euros. The exchange rate is 1 USD = 0.85 Euros. If the traveler buys a jacket that costs 170 Euros, what is the cost of the jacket in USD? (Grid-in)
5. A swimming pool has a volume of 15 cubic meters. What is the volume of the swimming pool in cubic centimeters? (1 meter = 100 centimeters).
   • A) $1,500$
   • B) $15,000$
   • C) $1,500,000$
   • D) $15,000,000$

Level 3: Hard (Advanced SAT)

1. A liquid flows through a pipe at a rate of 12 gallons per minute. What is the rate of flow in quarts per second? (1 gallon = 4 quarts, 1 minute = 60 seconds).
   • A) $0.8$
   • B) $1.25$
   • C) $3$
   • D) $48$
2. Pump A can fill a water tank in 3 hours working alone. Pump B can fill the exact same tank in 6 hours working alone. If both pumps are turned on at the same time, how many hours will it take them to fill the tank together? (Grid-in)
3. A gear system consists of Gear X, which has 12 teeth, and Gear Y, which has 36 teeth. The gears are interlocked so that when Gear X turns, it turns Gear Y. If Gear X completes 15 full revolutions, how many full revolutions does Gear Y complete? (Grid-in)
4. A car travels $m$ miles in $h$ hours. Which of the following expressions represents the number of feet the car travels in $x$ minutes? (1 mile = 5,280 feet, 1 hour = 60 minutes).
   • A) $\frac{88mx}{h}$
   • B) $\frac{5280m}{60hx}$
   • C) $\frac{88hx}{m}$
   • D) $\frac{mx}{88h}$
5. A chemist has 50 liters of a solution that is $20%$ acid and $80%$ water. How many liters of pure water must the chemist add to the solution to dilute it so that the new solution is exactly $10%$ acid? (Grid-in)

Level 4: Very Hard (Slightly beyond SAT)

1. A student mixes 200 milliliters (mL) of a juice drink that is $30%$ sugar with 300 mL of a juice drink that is $80%$ sugar. What is the sugar percentage of the final 500 mL mixture?
   • A) $45%$
   • B) $50%$
   • C) $55%$
   • D) $60%$
2. A construction crew of 5 workers can build 3 identical brick walls in 4 days, assuming each worker works at the same constant rate. How many workers are needed to build 6 identical brick walls in exactly 2 days? (Grid-in)
3. A certain metal alloy has a density of 2.5 grams per cubic centimeter ($\text{g/cm}^3$). A solid block of this alloy has a volume of 0.04 cubic meters ($\text{m}^3$). What is the mass of the block, in kilograms (kg)? (1 meter = 100 cm, 1 kg = 1,000 g). (Grid-in)
4. A driver travels from City A to City B at an average speed of 40 miles per hour. The journey takes 2 hours. The driver then travels from City B to City C at an average speed of 60 miles per hour, which takes 3 hours. What is the driver's average speed for the entire journey from City A to City C, in miles per hour?
   • A) $48$
   • B) $50$
   • C) $52$
   • D) $54$
5. A factory produces $p$ widgets for every $q$ dollars spent on manufacturing. Which of the following expressions represents the cost, in cents, to manufacture $r$ widgets? (1 dollar = 100 cents).
   • A) $\frac{100pr}{q}$
   • B) $\frac{100qr}{p}$
   • C) $\frac{qr}{100p}$
   • D) $\frac{pr}{100q}$

Ratios, rates, proportions and units

Ratios, rates and units · 20 questions

Worked Examples: SAT Strategy

Practice Questions

Level 1: Easy (Foundational SAT)

1. A recipe for cookies calls for 2 cups of sugar for every 3 cups of flour. If a baker uses 9 cups of flour, how many cups of sugar are needed? (Grid-in)
2. What is $15%$ of $300$? (Grid-in)
3. If $40%$ of a number is $80$, what is $100%$ of the number?
   • A) $120$
   • B) $160$
   • C) $200$
   • D) $320$
4. A runner completes a 5-kilometer race in 25 minutes. What is the runner's average speed in kilometers per hour? (Grid-in)
5. The ratio of boys to girls in a school club is 4 to 5. If there are 36 students in the club, how many are girls?
   • A) $16$
   • B) $20$
   • C) $25$
   • D) $30$

Level 2: Medium (Standard SAT)

1. A store buys a jacket for $\$40$ and sells it for $\$60$. What is the percentage markup on the cost of the jacket?
   • A) $20%$
   • B) $33.3%$
   • C) $50%$
   • D) $66.7%$
2. An architect is using a scale of 1 inch = 5 feet for a blueprint. If a room is 15 feet by 20 feet in reality, what is the area of the room on the blueprint, in square inches? (Grid-in)
3. A liquid is leaking from a tank at a rate of 12 milliliters per second. How many liters of liquid leak from the tank in 1 hour? (1 liter = 1,000 milliliters)
   • A) $0.72$
   • B) $7.2$
   • C) $43.2$
   • D) $720$
4. The price of an item was decreased by $20%$. By what percentage must the new price be increased to return it to its original price?
   • A) $20%$
   • B) $25%$
   • C) $40%$
   • D) $80%$
5. Machine A can produce 120 widgets in 3 hours. Machine B can produce 150 widgets in 5 hours. If both machines run continuously, how many total widgets will they produce in 8 hours? (Grid-in)

Level 3: Hard (Advanced SAT)

1. A painter can cover $x$ square meters of wall in $y$ minutes. At this rate, how many hours will it take the painter to cover $z$ square meters of wall?
   • A) $\frac{yz}{60x}$
   • B) $\frac{60yz}{x}$
   • C) $\frac{xz}{60y}$
   • D) $\frac{60xz}{y}$
2. A population of bacteria increases by $40%$ every 3 hours. If the initial population is 500, what is the best estimate of the population after 9 hours?
   • A) $1,100$
   • B) $1,372$
   • C) $1,500$
   • D) $1,960$
3. In a certain chemical process, $p$ grams of substance A react with $q$ grams of substance B to produce $r$ grams of substance C. If the ratio of $p$ to $q$ is strictly 5 to 2, and no material is lost in the reaction ($p+q=r$), how many grams of substance A are needed to produce 140 grams of substance C? (Grid-in)
4. A rectangular plot of land has a length that is $25%$ greater than its width. If the width is decreased by $20%$ and the length is increased by $20%$, what is the overall percentage change in the area of the plot?
   • A) It decreases by $4%$
   • B) It decreases by $5%$
   • C) It remains the same
   • D) It increases by $4%$
5. The density of a certain metal is 8.5 grams per cubic centimeter. A solid cube of this metal has a mass of 544 grams. What is the side length of the cube in centimeters? (Grid-in)

Level 4: Very Hard (Slightly beyond SAT)

1. A water tank can be filled by Pipe X in 4 hours and drained by Pipe Y in 6 hours. If both pipes are opened simultaneously when the tank is empty, how many hours will it take to completely fill the tank? (Grid-in)
2. On a map, 1 centimeter represents $k$ kilometers. A region has an area of 50 square centimeters on the map. If the actual area of the region is 1,800 square kilometers, what is the value of $k$? (Grid-in)
3. The currency in Country A is the Alpha, and the currency in Country B is the Beta. The exchange rate is 1 Alpha = 1.25 Betas. A trader exchanges $x$ Alphas for Betas, spends $20%$ of the Betas, and then exchanges the remaining Betas back into Alphas at a new exchange rate of 1 Alpha = 1.20 Betas. What fraction of the original $x$ Alphas does the trader have left?
   • A) $\frac{5}{6}$
   • B) $\frac{19}{24}$
   • C) $\frac{1}{1.2}$
   • D) $\frac{4}{5}$
4. A car travels from City A to City B at an average speed of 40 miles per hour, and immediately returns from City B to City A along the same route at an average speed of 60 miles per hour. What is the car's average speed for the entire round trip?
   • A) $48$ mph
   • B) $50$ mph
   • C) $52$ mph
   • D) $54$ mph
5. The population of a city grew by $r%$ in 2021 and then grew by $s%$ in 2022. If the population at the end of 2022 was exactly $21%$ greater than the population at the beginning of 2021, and $r$ and $s$ are positive integers, what is the value of $r \times s$? (Grid-in)

Percentages and proportions

Percentages · 20 questions

Worked Examples: SAT Strategy

Practice Questions

Level 1: Easy (Foundational SAT)

1. What is $20%$ of $440$? (Grid-in)
2. Out of $300$ seeds that were planted, $80%$ sprouted. How many of these seeds sprouted?
   • A) $24$
   • B) $80$
   • C) $240$
   • D) $280$
3. A pair of shoes originally costs $\$80$. If the shoes are on sale for $25%$ off, what is the sale price of the shoes?
   • A) $\$20$
   • B) $\$55$
   • C) $\$60$
   • D) $\$100$
4. In a poll of $200$ voters, $130$ voted for Candidate A and the rest voted for Candidate B. What percentage of the polled voters voted for Candidate B? (Grid-in)
5. {|l|c|c|c|} & Right-Handed & Left-Handed & Total
   Male & 42 & 8 & 50
   Female & 45 & 5 & 50
   Total & 87 & 13 & 100
   Based on the table above, what is the probability of randomly selecting a female who is left-handed from the entire group?
   • A) $\frac{5}{100}$
   • B) $\frac{5}{50}$
   • C) $\frac{5}{13}$
   • D) $\frac{13}{100}$

Level 2: Medium (Standard SAT)

1. A store purchases a jacket from a supplier for $\$40$. The store marks up the price by $50%$ to sell to customers. During a clearance sale, the store discounts the jacket by $20%$ off the selling price. What is the clearance price of the jacket?
   • A) $\$48$
   • B) $\$52$
   • C) $\$58$
   • D) $\$60$
2. $72$ is $p%$ of $120$. What is the value of $p$? (Grid-in)
3. The price of a plane ticket increased by $15%$ from March to April. If the ticket cost $\$345$ in April, what was the price of the ticket in March?
   • A) $\$293.25$
   • B) $\$300.00$
   • C) $\$330.00$
   • D) $\$396.75$
4. {|l|c|c|c|} & Cat & Dog & Total
   Has a Yard & 12 & 38 & 50
   No Yard & 24 & 6 & 30
   Total & 36 & 44 & 80
   Based on the table, if a person who owns a dog is chosen at random, what is the probability that this person has a yard? (Grid-in)
5. A number $x$ is increased by $25%$ to get $y$. Then, $y$ is decreased by $40%$ to get $z$. If $z = 60$, what is the value of $x$? (Grid-in)

Level 3: Hard (Advanced SAT)

1. The regular price of a shirt at a store is $\$11.70$. The sale price of the shirt is $80%$ less than the regular price, and the sale price is $30%$ greater than the store's cost for the shirt. What was the store's cost, in dollars, for the shirt? (Grid-in)
2. $150$ is $p%$ greater than $40$. What is the value of $p$? (Grid-in)
3. In a company, $60%$ of the employees are female, and $40%$ are male. If $30%$ of the female employees and $25%$ of the male employees take the bus to work, what percentage of ALL employees take the bus to work?
   • A) $27.5%$
   • B) $28%$
   • C) $55%$
   • D) $58%$
4. {|l|c|c|c|} & Passed & Failed & Total
   Studied & $x$ & 10 & 90
   Did not Study & 15 & $y$ & 60
   Total & 95 & 55 & 150
   According to the partially completed table above, what is the value of $x - y$? (Grid-in)
5. If $a$ is $50%$ of $b$, and $b$ is $120%$ of $c$, what percentage of $c$ is $a$?
   • A) $60%$
   • B) $70%$
   • C) $170%$
   • D) $240%$

Level 4: Very Hard (Slightly beyond SAT)

1. The price of an investment portfolio decreases by $x%$ in Year 1. In Year 2, the portfolio's new value increases by $2x%$. If the value of the portfolio at the end of Year 2 is exactly equal to its original value before Year 1, and $x > 0$, what is the value of $x$? (Grid-in)
2. A population of a town increased by $p%$ from 2010 to 2015, and then decreased by $p%$ from 2015 to 2020. If the town's population in 2020 was $9% $ less than its population in 2010, what is the value of $p$? (Grid-in)
3. In a certain survey, $70%$ of respondents answered "Yes" to Question 1. Of those who answered "Yes" to Question 1, $40%$ answered "Yes" to Question 2. If exactly $140$ respondents answered "Yes" to BOTH questions, how many total people responded to the survey?
   • A) 200
   • B) 350
   • C) 500
   • D) 700
4. An electronics retailer buys a computer for $C$ dollars. The retailer marks up the price by $25%$ to create the list price. Later, the retailer offers a discount of $d%$ off the list price. If the discounted price is exactly $10%$ less than the original cost $C$, what is the value of $d$? (Grid-in)
5. The total cost of an item, including an $8%$ sales tax, is $\$135$. A customer returns the item and receives a full refund, but the store charges a $15%$ restocking fee based on the item's price before tax. How much money, in dollars, does the store keep as a restocking fee? (Grid-in)

Percentages, probability and statistics

Percentages · 20 questions

Worked Examples: SAT Strategy

Practice Questions

Level 1: Easy (Foundational SAT)

1. What is $20%$ of $440$? (Grid-in) [6]
2. $13$ is $p%$ of $25$. What is the value of $p$? (Grid-in) [7]
3. $4, 10, 18, 4, 4, 5, 6, 5$
   What is the median of the data set shown? [8]
   • A) $4$
   • B) $5$
   • C) $7$
   • D) $14$
4. On a street with 7 houses, 2 houses are blue. If a house from this street is selected at random, what is the probability of selecting a house that is blue? [9]
   • A) $\frac{1}{7}$
   • B) $\frac{2}{7}$
   • C) $\frac{5}{7}$
   • D) $\frac{2}{5}$
5. The table summarizes the distribution of color and shape for 100 tiles of equal area. There are 30 total red tiles. If one of these 100 tiles is selected at random, what is the probability of selecting a red tile? (Express your answer as a decimal or fraction). (Grid-in) [10]

Level 2: Medium (Standard SAT)

1. To estimate the proportion of a population that has a certain characteristic, a random sample was selected. The proportion is estimated to be $0.49$, with an associated margin of error of $0.04$. Which conclusion is most appropriate? [11]
   • A) It is plausible that the proportion is between $0.45$ and $0.53$.
   • B) It is plausible that the proportion is less than $0.45$.
   • C) The proportion is exactly $0.49$.
   • D) It is plausible that the proportion is greater than $0.53$.
2. Scott selected 20 employees at random from all 400 employees at a company. He found that 16 of the employees in this sample are enrolled in exactly three professional development courses. Based on this, what is the best estimate of the total number of employees at the company enrolled in exactly three courses? [12]
   • A) $16$
   • B) $320$
   • C) $380$
   • D) $384$
3. $210$ is $p%$ greater than $30$. What is the value of $p$? (Grid-in) [13]
4. The regular price of a shirt at a store is $\$11.70$. The sale price of the shirt is $80%$ less than the regular price, and the sale price is $30%$ greater than the store's cost for the shirt. What was the store's cost, in dollars, for the shirt? (Grid-in) [14]
5. A table summarizes members of an organization by age. There are $135$ total members, and $107$ of them are at least 40 years old. If a member is selected at random, what is the probability that the selected member is at least 40 years old? [15]
   • A) $\frac{28}{135}$
   • B) $\frac{35}{135}$
   • C) $\frac{100}{135}$
   • D) $\frac{107}{135}$

Level 3: Hard (Advanced SAT)

1. Five sea turtles each have a nest. The original data set of the number of eggs laid is: $149, 144, 148, 136, 139$. A sixth nest with $121$ eggs is added to create a new data set. Which correctly compares the means of the two data sets? [16]
   • A) The mean of the original data set is greater than the mean of the new data set.
   • B) The mean of the original data set is less than the mean of the new data set.
   • C) The means of both data sets are equal.
   • D) There is not enough information to compare the means.
2. An analyst estimated the mean price of a carton of tomatoes in Utah to be $\$4.23$, with an associated margin of error of $\$0.08$. Which statement is plausible about the mean price for all locations in Utah? [4]
   • A) It is between $\$4.15$ and $\$4.31$.
   • B) It is either less than $\$4.15$ or greater than $\$4.31$.
   • C) It is exactly $\$4.23$.
   • D) It is greater than $\$4.31$.
3. Two dot plots represent the number of glue sticks brought in by students for Class A and Class B. Both dot plots are perfectly symmetrical and have the exact same shape, though Class B's values are centered at a higher number. Which statement best compares their standard deviations? [17]
   • A) Class A's standard deviation is less than Class B's.
   • B) Class A's standard deviation is equal to Class B's.
   • C) Class A's standard deviation is greater than Class B's.
   • D) There is not enough information.
4. A poll of 803 randomly selected voters showed 483 would vote for Angel Cruz and 320 for Terry Smith. If 6,424 people vote in the actual election, by how many votes would Angel Cruz be expected to win? [18]
   • A) $163$
   • B) $1,304$
   • C) $3,864$
   • D) $5,621$
5. A weight of 39 pounds is added to an original data set of 71 tortoise weights. If the original mean was 17 pounds, which statement best compares the mean and median of the new data set to the original? [19]
   • A) The mean of the new data set is greater, and the median of the new data set is greater.
   • B) The mean of the new data set is greater, and the medians of the two data sets could be equal.
   • C) The mean of the new data set is less, and the median is less.
   • D) The mean of the new data set is less, and the medians are equal.

Level 4: Very Hard (Slightly beyond SAT)

1. A store marks up the wholesale cost of a jacket by $40%$. During a clearance sale, the store applies a $25%$ discount to the marked-up price. If the final sale price of the jacket is $\$126$, what was the original wholesale cost, in dollars? (Grid-in)
2. Class X has 20 students with a mean test score of 80. Class Y has 30 students. If the combined mean test score for all 50 students in both classes is 86, what is the mean test score for Class Y? (Grid-in)
3. At a high school, $60%$ of students participate in sports, $40%$ participate in music, and $20%$ participate in both sports and music. If a student is selected at random, what is the probability, expressed as a decimal, that the student participates in neither sports nor music? (Grid-in)
4. Data set $P$ consists of the values $\{2, 2, 2, 8, 8, 8\}$. Data set $Q$ consists of the values $\{4, 4, 4, 6, 6, 6\}$. Which of the following statements is true regarding the standard deviations of the two data sets?
   • A) The standard deviation of $P$ is greater than that of $Q$.
   • B) The standard deviation of $P$ is less than that of $Q$.
   • C) The standard deviations of $P$ and $Q$ are equal.
   • D) The standard deviations cannot be compared without calculating the exact means.
5. A researcher conducts a survey to estimate a population proportion. If the researcher wants to reduce the margin of error to exactly half of its original size while keeping the same confidence level, by what factor must the researcher multiply the original sample size? (Grid-in)

One-variable statistics and probability

One-variable data and probability · 20 questions

Worked Examples: SAT Strategy

Practice Questions

Level 1: Easy (Foundational SAT)

1. The mean of 3 numbers is 40. If two of the numbers are 35 and 45, what is the third number? (Grid-in)
2. A dataset consists of the following values: 4, 7, 7, 9, 10, 15. What is the median of this dataset?
   • A) 7
   • B) 8
   • C) 9
   • D) 10
3. A bag contains 4 red marbles, 5 blue marbles, and 3 yellow marbles. If one marble is drawn at random, what is the probability that it is NOT blue?
   • A) $\frac{5}{12}$
   • B) $\frac{7}{12}$
   • C) $\frac{1}{2}$
   • D) $\frac{7}{5}$
4. The ages of 5 children at a party are 6, 6, 7, 8, and 10. If an 11-year-old child joins the party, what will be the new median age? (Grid-in)
5. Data Set X has a standard deviation of 0. Which of the following could be Data Set X?
   • A) 1, 2, 3, 4, 5
   • B) -5, -5, 0, 5, 5
   • C) 8, 8, 8, 8, 8
   • D) 0, 10, 100, 1000

Level 2: Medium (Standard SAT)

1. A student's average (mean) score on 4 tests is an 82. What score must the student earn on the 5th test to raise their average to an 85? (Grid-in)
2. A frequency table shows the number of pets owned by 15 families. {|c|c|} Number of Pets & Frequency
   0 & 3
   1 & 5
   2 & 4
   3 & 2
   4 & 1
   What is the median number of pets owned by these families? (Grid-in)
3. A list of numbers contains 10 values. The largest value is removed from the list. Which of the following statistics MUST decrease?
   • A) The mean
   • B) The median
   • C) The standard deviation
   • D) The range
4. In a class of 30 students, 18 play an instrument and 12 play a sport. If 5 students play both an instrument and a sport, what is the probability that a randomly selected student plays neither?
   • A) $\frac{1}{6}$
   • B) $\frac{1}{5}$
   • C) $\frac{5}{30}$
   • D) $\frac{1}{4}$
5. Data Set A: 10, 12, 14, 16, 18
   Data Set B: 10, 14, 14, 14, 18
   Which of the following statements comparing the standard deviations of the two data sets is true?
   • A) The standard deviation of A is greater than B.
   • B) The standard deviation of B is greater than A.
   • C) The standard deviations are equal.
   • D) There is not enough information to compare them.

Level 3: Hard (Advanced SAT)

1. A dataset of 20 integers has a mean of 50 and a median of 48. If the highest number in the dataset is increased by 10, what is the new mean of the dataset? (Grid-in)
2. A factory produces lightbulbs. The probability that a randomly selected lightbulb is defective is $0.04$. If the factory produces 5,000 lightbulbs in a day, what is the expected number of non-defective lightbulbs? (Grid-in)
3. The mean of $a, b, c,$ and $d$ is 15. The mean of $a, b, c, d,$ and $e$ is 18. What is the value of $e$? (Grid-in)
4. Data Set 1: A normally distributed set of 1,000 values with a mean of 50 and a range of 20.
   Data Set 2: The identical set of 1,000 values from Data Set 1, but the number 5 is added to every single value.
   Which of the following statements about the standard deviation of the two datasets is true?
   • A) The standard deviation of Set 2 is 5 greater than Set 1.
   • B) The standard deviation of Set 2 is 25 greater than Set 1.
   • C) The standard deviations of the two sets are exactly the same.
   • D) The standard deviation of Set 2 is 5 times the standard deviation of Set 1.
5. A dataset of 9 consecutive even integers has a median of 20. What is the mean of this dataset?
   • A) 18
   • B) 20
   • C) 22
   • D) It cannot be determined from the information given.

Level 4: Very Hard (Slightly beyond SAT)

1. The average (mean) weight of a group of 10 players is 160 pounds. Two new players join the group, and the new average weight of all 12 players becomes 165 pounds. If one of the new players weighs 180 pounds, what is the weight, in pounds, of the other new player? (Grid-in)
2. A distribution of test scores has a mean of 75 and a standard deviation of 8. If every score in the distribution is multiplied by 1.5, what is the standard deviation of the new distribution? (Grid-in)
3. The probability of Event A occurring is $0.4$, and the probability of Event B occurring is $0.5$. If Event A and Event B are independent, what is the probability that Event A occurs but Event B does NOT occur?
   • A) $0.20$
   • B) $0.40$
   • C) $0.50$
   • D) $0.90$
4. A dataset has 15 identical values. If one of the values is changed from 10 to 40, the new mean is 12. What was the original mean of the dataset? (Grid-in)
5. Two classes take the same exam. Class X has 20 students and a mean score of 80. Class Y has 30 students and a mean score of 90. What is the combined mean score of all 50 students?
   • A) $84$
   • B) $85$
   • C) $86$
   • D) $88$

Two-variable data: scatterplots and models

Two-variable data and statistical claims · 20 questions

Worked Examples: SAT Strategy

Practice Questions

Level 1: Easy (Foundational SAT)

1. The line of best fit for a scatterplot is given by the equation $y = 8.4x + 12$. What is the predicted value of $y$ when $x = 5$? (Grid-in)
2. A mechanic tracks the number of hours, $h$, it takes to repair an engine based on the engine's age, $a$, in years. The line of best fit is $h = 1.2a + 2.5$. What is the best interpretation of $1.2$ in this context?
   • A) The predicted repair time for a brand-new engine.
   • B) The estimated increase in repair time, in hours, for each additional year of an engine's age.
   • C) The estimated increase in an engine's age for each additional hour of repair time.
   • D) The total number of hours required to repair an engine.
3. A scatterplot shows the relationship between the outside temperature and the number of cups of hot chocolate sold at a cafe. The data shows a strong negative linear association. Which of the following values could be the correlation coefficient, $r$, for this data?
   • A) $-0.92$
   • B) $-0.15$
   • C) $0.15$
   • D) $0.92$
4. A line of best fit is given by $y = -3x + 40$. For a given data point, the actual $x$-value is 8 and the actual $y$-value is 18. Does the line of best fit overpredict or underpredict the $y$-value for this point, and by how much?
   • A) It overpredicts by 2.
   • B) It underpredicts by 2.
   • C) It overpredicts by 16.
   • D) It underpredicts by 16.
5. A scatterplot exhibits an exponential trend where the $y$-values decrease rapidly at first and then gradually level out as $x$ increases. Which type of model would best fit this data?
   • A) Decreasing linear
   • B) Increasing linear
   • C) Exponential growth
   • D) Exponential decay

Level 2: Medium (Standard SAT)

1. A scatterplot contains 15 data points. The line of best fit is calculated and drawn on the plot. For exactly 9 of the data points, the actual $y$-value is strictly less than the predicted $y$-value. How many data points lie on or above the line of best fit? (Grid-in)
2. The table below summarizes data collected on the length of a metal spring, $L$, in millimeters, when a weight of $w$ grams is attached to it. {|c|c|c|c|c|} $w$ (grams) & $10$ & $20$ & $30$ & $40$
   $L$ (mm) & $45$ & $55$ & $65$ & $75$
   Which of the following equations best models the relationship between $w$ and $L$?
   • A) $L = 10w + 35$
   • B) $L = w + 35$
   • C) $L = 10w + 45$
   • D) $L = 1.1w^2 + 35$
3. The line of best fit for a set of data is $y = 0.8x - 12$. A data point has an actual $y$-value of $20$. If the line of best fit overpredicts the $y$-value for this data point by exactly $4$, what is the $x$-value of the data point? (Grid-in)
4. An exponential model of the form $y = a(b)^x$ is fit to a scatterplot. The parameters are found to be $a = 150$ and $b = 1.08$. What is the predicted percentage increase in $y$ for every increase of $1$ in $x$? (Grid-in)
5. A researcher creates a scatterplot comparing the population of a city (in thousands), $y$, to the number of years since 1990, $x$. The line of best fit is $y = -2.4x + 85$. According to the model, in what year is the population predicted to reach 25 thousand?
   • A) 2005
   • B) 2010
   • C) 2015
   • D) 2025

Level 3: Hard (Advanced SAT)

1. The line of best fit for a data set relating $x$ and $y$ is given by $y = mx + c$, where $m > 0$ and $c < 0$. Which of the following must be true about the scatterplot of the data?
   • A) The data shows a negative correlation.
   • B) As $x$ increases, $y$ tends to increase, and the line intersects the $y$-axis below the origin.
   • C) As $x$ increases, $y$ tends to decrease, and the line intersects the $y$-axis above the origin.
   • D) All data points in the scatterplot must have negative $y$-values.
2. The number of active users on a platform, $U$, in millions, $t$ years after its launch is modeled by the exponential function $U(t) = 4(1.5)^t$. A linear model is also created for the same data during the first 2 years, resulting in the line of best fit $U_L(t) = 2.5t + 4$. What is the difference, in millions, between the predictions made by the exponential model and the linear model when $t = 2$? (Grid-in)
3. A real estate agent modeled the price of houses, $P$, in dollars, based on their distance from the city center, $d$, in miles, using the equation $P = -15,000d + 450,000$. The agent later decides to rewrite the model so that distance is measured in kilometers, $k$. If 1 mile is approximately 1.6 kilometers, which equation best represents the rewritten model?
   • A) $P = -24,000k + 450,000$
   • B) $P = -9,375k + 450,000$
   • C) $P = -15,000k + 720,000$
   • D) $P = -15,000k + 281,250$
4. A data set has a line of best fit given by $y = 5x + 10$. An extreme outlier is removed from the data set, and a new line of best fit is calculated as $y = 2x + 10$. Which of the following best describes the outlier that was removed?
   • A) A point with a large positive $x$-value and a very large positive $y$-value.
   • B) A point with a large positive $x$-value and a very small negative $y$-value.
   • C) A point with an $x$-value of 0 and a very large positive $y$-value.
   • D) A point with an $x$-value of 0 and a very small negative $y$-value.
5. A scatterplot has a line of best fit with the equation $y = 4.5x - 10$. If every $x$-value in the original data set is multiplied by 2 and every $y$-value remains the same, what will be the slope of the new line of best fit? (Grid-in)

Level 4: Very Hard (Slightly beyond SAT)

1. A linear regression model $y = mx + c$ was fitted to a set of data points predicting a company's revenue, $y$, in millions, $x$ years after 2015. The model predicted revenues of $14$ million in 2018 and $26$ million in 2022. What is the predicted revenue, in millions, in 2026? (Grid-in)
2. A scatterplot shows a non-linear relationship between $x$ and $y$. However, when the researcher plots $x$ against the natural logarithm of $y$ ($\ln(y)$), the data forms a perfect straight line modeled by the equation $\ln(y) = 3x + 2$. Which of the following equations models the original relationship between $x$ and $y$?
   • A) $y = e^{3x} + e^2$
   • B) $y = e^2(e^3)^x$
   • C) $y = 3e^x + 2$
   • D) $y = 2(3)^x$
3. Model A predicts the population of Town A, in thousands, $t$ years after 2000 with the equation $P_A = 12t + 45$. Model B predicts the population of Town B with $P_B = 18t + 15$. According to the models, in what year will the population of Town B exceed the population of Town A by exactly 21 thousand? (Grid-in)
4. The line of best fit for a set of data is $y = 3x + 5$. There are three data points: $(2, a)$, $(4, b)$, and $(6, c)$. If the sum of the residuals (the differences between the actual $y$-values and the predicted $y$-values) for these three points is exactly $0$, what is the value of $a + b + c$? (Grid-in)
5. A researcher models the relationship between $x$ and $y$ using the exponential function $y = p(q)^x$. The data shows that when $x$ increases by $2$, the predicted value of $y$ increases by $44%$. If $q > 0$, what is the value of $q$? (Round to the nearest hundredth). (Grid-in)

Two-variable data and evaluating claims

Two-variable data and statistical claims · 20 questions

Worked Examples: SAT Strategy

Practice Questions

Level 1: Easy (Foundational SAT)

1. A poll of 500 likely voters in a city found that $52%$ support a new tax measure. The poll has a margin of error of $\pm 3%$. Based on this poll, which of the following statements is true?
   • A) Exactly $52%$ of all city residents support the tax measure.
   • B) The true percentage of likely voters who support the measure is likely between $49%$ and $55%$.
   • C) $3%$ of the voters surveyed did not understand the question.
   • D) If the poll were conducted again, exactly $52%$ of voters would support it.
2. A line of best fit on a scatterplot predicts a value of $y = 25$ when $x = 4$. The actual data point on the scatterplot at $x = 4$ is $(4, 30)$. What is the residual for this data point? (Grid-in)
3. A biologist studies the relationship between the temperature of an incubator and the growth rate of a specific bacteria. The scatterplot of the data shows a line of best fit with a negative slope. What does this indicate?
   • A) As temperature increases, the growth rate increases.
   • B) As temperature increases, the growth rate decreases.
   • C) Temperature and growth rate are not related.
   • D) The bacteria died before the experiment finished.
4. The equation of a line of best fit is $y = 3.2x + 15$, where $x$ represents years of experience and $y$ represents hourly wage in dollars. What is the best interpretation of the number 3.2 in this context?
   • A) The starting hourly wage for a worker with zero years of experience.
   • B) The estimated increase in hourly wage for each additional year of experience.
   • C) The exact hourly wage of a worker with 1 year of experience.
   • D) The number of years required to earn 15 dollars an hour.
5. A researcher randomly selects 100 dogs from a specific animal shelter to participate in a dietary study. Can the results of this study be generalized to all dogs in the world?
   • A) Yes, because the dogs were randomly selected.
   • B) No, because the sample size is too small.
   • C) No, because the sample was only selected from one specific shelter.
   • D) Yes, because dogs generally share the same dietary needs.

Level 2: Medium (Standard SAT)

1. A botanist estimates the mean height of a species of plant in a forest to be 45 cm, with a margin of error of 4 cm. Which of the following is a plausible value for the true mean height of this plant species in the forest?
   • A) 39 cm
   • B) 40 cm
   • C) 43 cm
   • D) 50 cm
2. The line of best fit for a set of data is $y = -0.5x + 20$. A data point in the set has an $x$-value of 6 and a residual of $-2$. What is the actual $y$-value of this data point? (Grid-in)
3. Two surveys are conducted to estimate the proportion of students who drive to school. Survey A samples 50 students and has a margin of error of $8%$. Survey B samples 500 students. Assuming both surveys used the same random sampling methods, what is most likely true about Survey B?
   • A) Survey B will have a margin of error greater than $8%$.
   • B) Survey B will have a margin of error less than $8%$.
   • C) Survey B will have a margin of error of exactly $8%$.
   • D) Survey B will be completely free of any error.
4. A gym owner wants to know if members are satisfied with the new equipment. She asks the first 40 members who walk into the gym on a Monday morning. She finds that $90%$ are satisfied. What is the most significant flaw in her study design?
   • A) She didn't establish a control group.
   • B) The sample was not randomly selected, so it may not represent all gym members.
   • C) The survey question was likely too confusing.
   • D) She didn't randomly assign members to different pieces of equipment.
5. A scatterplot maps the number of hours studied ($x$) against test scores ($y$). The line of best fit always passes exactly through which specific coordinate point?
   • A) $(0, 0)$
   • B) The $y$-intercept
   • C) The mean of the $x$-values and the mean of the $y$-values: $(\bar{x}, \bar{y})$
   • D) The median of the $x$-values and the median of the $y$-values

Level 3: Hard (Advanced SAT)

1. A medical researcher is testing a new allergy medication. She recruits 200 volunteers who suffer from severe allergies. She randomly assigns 100 to take the new medication and 100 to take a placebo. The group taking the new medication reports a significant reduction in symptoms. Which of the following is the most appropriate conclusion?
   • A) The new medication causes a reduction in allergy symptoms for all people.
   • B) The new medication causes a reduction in allergy symptoms for people with severe allergies similar to the volunteers.
   • C) There is a positive association between the medication and symptom reduction, but causation cannot be proven because they were volunteers.
   • D) The medication is guaranteed to work for the volunteers in the future.
2. In a study of house prices, the line of best fit is determined to be $y = 150x + 50,000$, where $x$ is the area of the house in square feet and $y$ is the predicted price in dollars. A house with an area of 2,000 square feet is sold for $\$320,000$. What is the absolute value of the residual for this house? (Grid-in)
3. Which of the following research designs would best allow a scientist to determine whether a new type of fertilizer causes plants to grow taller?
   • A) Observe two different farms, one that uses the fertilizer and one that does not, and measure the plants.
   • B) Randomly select 100 plants from a greenhouse that already uses the fertilizer and measure their heights.
   • C) Take 100 identical plants, randomly assign 50 to receive the fertilizer and 50 to receive no fertilizer, and measure their heights.
   • D) Ask farmers who use the fertilizer to report how tall their plants grow compared to previous years.
4. A wildlife biologist estimates the average weight of an adult male black bear in a specific national park to be 250 lbs, with a margin of error of 15 lbs at a $95%$ confidence level. Which of the following statements is mathematically true regarding this estimate?
   • A) $95%$ of all adult male black bears in the park weigh between 235 lbs and 265 lbs.
   • B) It is highly plausible that the true average weight of the population of adult male black bears in the park is between 235 lbs and 265 lbs.
   • C) If the biologist caught one more bear, there is a $95%$ chance it would weigh exactly 250 lbs.
   • D) The biologist is absolutely certain the average weight is at least 235 lbs.
5. A scatterplot shows a set of data points $(x,y)$ and a line of best fit. If every single $y$-value in the dataset is increased by 10, but the $x$-values remain identical, what will happen to the residuals of the data points?
   • A) The residuals will all increase by 10.
   • B) The residuals will all decrease by 10.
   • C) The residuals will remain exactly the same.
   • D) The residuals will be multiplied by 10.

Level 4: Very Hard (Slightly beyond SAT)

1. A data scientist creates a line of best fit mapping the relationship between daily ice cream sales ($x$) and daily umbrella sales ($y$) in a city. The line of best fit has a strong positive slope. The data scientist concludes that buying ice cream causes people to buy umbrellas. What is the fatal flaw in this conclusion?
   • A) The sample size of the days measured wasn't provided.
   • B) It confuses correlation with causation; a third confounding variable, like hot weather, likely causes both.
   • C) Ice cream and umbrellas are unrelated products, so the line of best fit must be calculated incorrectly.
   • D) The margin of error for umbrella sales is likely too high.
2. The line of best fit for predicting a student's final exam score ($y$) based on the number of hours they studied ($x$) is $y = 4.5x + 40$. A student who studied for 8 hours received a final exam score of 72. What is the residual for this student? (Grid-in)
3. A university surveys a random sample of 200 freshmen about their sleep habits. The university reports that freshmen sleep an average of 6.2 hours per night, with a margin of error of 0.5 hours. To decrease the margin of error to 0.25 hours (cut it in half) while maintaining the same confidence level, what must the university do?
   • A) Divide the sample size in half (survey 100 students).
   • B) Double the sample size (survey 400 students).
   • C) Quadruple the sample size (survey 800 students).
   • D) Survey only students who sleep more than 6 hours.
4. An exponential regression curve $y = a(b)^x$ is used to model population growth over time. For the data point at $x = 2$, the predicted $y$-value is 100. If the residual for this point is 15, what is the actual population measured at $x=2$? (Grid-in)
5. A researcher randomly assigns 60 high schools across the state into two groups to test a new math curriculum. Group A uses the new curriculum, and Group B uses the old curriculum. At the end of the year, Group A scores significantly higher on state tests. Which of the following is the most scientifically sound conclusion?
   • A) The new curriculum will cause higher math scores for any high school student in the world.
   • B) The new curriculum caused the higher math scores for the high schools in the state.
   • C) There is a correlation, but not causation, because the students within the high schools were not individually randomized.
   • D) The old curriculum causes students' math abilities to decrease over time.

Ratios, percentages, probability and inference

Stretch sheets · 10 questions · 10 SAT-style stats & data questions that connect to HL AA's discrete probability, conditional probability, and confidence-interval work.

Worked Examples: SAT Strategy

Practice Questions

Level 1: Easy (Foundational)

1. A recipe uses flour and sugar in the ratio $5:3$. If $200$ g of flour is used, determine the exact grams of sugar required. (SPR)
2. What is $15\%$ of $240$?
   A) $24$
   B) $30$
   C) $36$
   D) $45$
3. A fair coin is tossed twice. What is the exact probability of exactly one head?
   A) $\tfrac{1}{4}$
   B) $\tfrac{1}{2}$
   C) $\tfrac{3}{4}$
   D) $1$

Level 2: Medium (Standard)

1. A price is discounted $20\%$ from $\$50$, then reduced a further $10\%$. Determine the exact final price in dollars.
   A) $\$34$
   B) $\$36$
   C) $\$38$
   D) $\$40$
2. The mean of $5$ integers is $12$. When a $6^{\text{th}}$ integer is added the mean becomes $14$. Determine the exact value of the added integer.
   A) $14$
   B) $18$
   C) $20$
   D) $24$
3. A bag contains $3$ red, $2$ green, $5$ blue marbles. Determine the exact probability, as a decimal, of drawing a red marble. (SPR)
4. Events $A$ and $B$ are independent with $P(A)=0.4$ and $P(B)=0.5$. Determine $P(A \cap B)$.
   A) $0.1$
   B) $0.2$
   C) $0.4$
   D) $0.9$

Level 3: Hard (Advanced)

1. In a survey, $70\%$ of students study daily; among daily studiers, $80\%$ score $\ge 7$. What is the probability a random student studies daily AND scores $\ge 7$?
   A) $0.14$
   B) $0.36$
   C) $0.50$
   D) $0.56$
2. A dataset has mean $50$ and standard deviation $8$. By the empirical rule, approximately what percent of values fall in $[42, 58]$?
   A) $50\%$
   B) $68\%$
   C) $95\%$
   D) $99.7\%$
3. A $95\%$ confidence interval for a mean is $(48, 52)$. Determine the exact value of the sample mean $\bar{x}$. (SPR)

Ratios, rates, probability and descriptive statistics

Stretch sheets · 10 questions · 10 SAT-style data questions tuned to HL AI — bivariate correlation, expected value, and confidence-interval intuition.

Practice Questions

Level 1: Easy (Foundational)

1. A car travels $180$ km in $3$ hours. Determine its exact average speed in km/h. (SPR)
2. What is $25\%$ of $80$?
   A) $15$
   B) $20$
   C) $25$
   D) $40$
3. The mode of $\{2, 4, 4, 5, 7, 7, 7, 9\}$ is:
   A) $4$
   B) $5$
   C) $7$
   D) $9$

Level 2: Medium (Standard)

1. A phone that cost $\$500$ is on sale for $\$400$. Determine the exact percent decrease.
   A) $20\%$
   B) $25\%$
   C) $30\%$
   D) $40\%$
2. A spinner has $4$ equal sectors: red, blue, green, yellow. Determine the exact probability of NOT landing on red.
   A) $\tfrac{1}{4}$
   B) $\tfrac{1}{2}$
   C) $\tfrac{3}{4}$
   D) $1$
3. A car's fuel efficiency is $12$ km per litre. Determine the exact litres to travel $180$ km. (SPR)
4. The correlation coefficient of a scatterplot is $r = -0.85$. Which best describes the relationship?
   A) Strong positive linear
   B) Strong negative linear
   C) Weak positive linear
   D) No correlation

Level 3: Hard (Advanced)

1. A fair six-sided die is rolled twice. Determine the exact probability that the sum is $7$.
   A) $\tfrac{1}{6}$
   B) $\tfrac{1}{9}$
   C) $\tfrac{5}{36}$
   D) $\tfrac{7}{36}$
2. A discrete random variable $X$ has $P(X=x)=\tfrac{x}{10}$ for $x \in \{1,2,3,4\}$. Determine the exact expected value $E(X)$.
   A) $2.5$
   B) $2.8$
   C) $3.0$
   D) $3.5$
3. A dataset of $100$ values has mean $50$ and standard deviation $6$. Determine the approximate number of values within $\pm 2\sigma$ (empirical rule). (SPR)