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Math: Problem-Solving and Data Analysis

The PSAT's biggest data domain: units and percents, distributions, models and probability, plus the two data skills that live only on the SAT.

30 topics · 113 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

How Big the Data Domain Is on the PSAT

  1. How it worksHow do 7 to 9 of the 40 operational math questions end up in problem-solving and data analysis?
  2. Worked exampleFind how many of the 40 operational PSAT math questions are data questions when the domain is 20 percent.
  3. Common mistakeWhy isn't the PSAT's smaller geometry share a reason to skip geometry practice altogether?
Your turneasy1 of 3
Problem-solving and data analysis is 20 percent of the 40 operational questions on a PSAT math section. A student answers 5 of those data questions correctly. How many of them did the student miss?

Ratios, Rates, and Unit Rates

  1. How it worksHow do you turn a ratio into a unit rate you can just multiply by?
  2. Worked exampleFind the better buy between a 12 ounce bottle at 4.20 dollars and a 20 ounce bottle at 6.60 dollars.
  3. Common mistakeWhy doesn't a red to blue ratio of 2 to 3 mean two thirds of the marbles are red?
  4. Worked exampleWalk me through splitting 96 trading cards between two students in a 5 to 7 ratio.
Your turneasy1 of 3
A bottle holding 19 ounces of juice sells for 7 dollars. Find the unit price in dollars per ounce.

Scale Drawings and Proportions in Science

  1. How it worksHow does setting up a proportion turn a map distance into a real distance?
  2. Worked exampleFind the real length of a wall measuring 3.5 inches on a plan drawn at 1 inch to 6 feet.
  3. Worked exampleFind the grams of salt in 250 millilitres of a solution holding 14 grams per 400 millilitres.
  4. Going furtherWhat changes about the scale factor when a drawing is enlarged on a copier set to 150 percent?
Your turneasy1 of 3
A floor plan is drawn at a scale of 1 inch to 8 feet. A wall measures 5.74 inches on the plan. Find the real length of the wall in feet.

Derived Units from Products and Quotients

  1. How it worksHow do you build a derived unit by multiplying or dividing two quantities together?
  2. Worked exampleFind the cost of running a 1500 watt heater for 6 hours at 14 cents per kilowatt-hour.
  3. Worked exampleFind the population density in people per square kilometre for 84000 people spread over 35 square kilometres.
Your turneasy1 of 3
A district of 44 square kilometres is home to 68749 people. Find the population density in people per square kilometre.

One-Step Unit Conversion

  1. How it worksHow do you pick which way up to write a conversion factor so the unit you want survives?
  2. Worked exampleWalk me through converting 2.5 hours into seconds with the units cancelling as you go.
  3. Common mistakeWhy doesn't dividing by 2.54 turn inches into centimetres?
Your turneasy1 of 3
Convert 4.98 hours into seconds.

Multistep and Multidimensional Unit Conversion

  1. How it worksHow does a chain of conversion factors carry you from miles per hour to metres per second?
  2. Worked exampleFind how many metres per second 72 kilometres per hour works out to be.
  3. Common mistakeWhy isn't one cubic foot equal to 12 cubic inches when one foot equals 12 inches?
  4. Worked exampleWalk me through converting 3 dollars per gallon into dollars per litre using 3.785 litres per gallon.
Your turneasy1 of 3
Convert 111 kilometres per hour into metres per second.

The Scale-Factor Property of a Proportional Relationship

  1. How it worksHow does the constant of proportionality force both quantities to scale by the same factor?
  2. Common mistakeWhy doesn't adding 5 to one quantity in a proportional relationship add 5 to the other?
  3. Worked exampleFind the new value of y when y equals 18 and x is then multiplied by 2.5 in a proportional relationship.
Your turneasy1 of 3
In a proportional relationship between x and y, y equals 31 at a particular value of x. That value of x is then multiplied by 3.45. Find the new value of y.

Percentages in Context: Discounts, Taxes, and Tips

  1. How it worksHow does one multiplication handle a discount and a sales tax in the same step?
  2. Worked exampleFind the final price of a 64 dollar jacket after 25 percent off and then 8 percent sales tax.
  3. Worked exampleFind the bill before the tip when a diner pays 51.75 dollars including an 18 percent tip.
  4. Common mistakeWhy isn't a 20 percent discount followed by a 20 percent tax the same as paying the sticker price?
Your turneasy1 of 3
A jacket priced at 96 dollars is marked 29 percent off, and 9 percent sales tax is then charged on the discounted price. Find the final price in dollars.

Interest: Simple and Compound

  1. How it worksHow does compounding turn last year's interest into this year's principal?
  2. Worked exampleFind the balance after 3 years on 2000 dollars at 4 percent annual interest compounded yearly.
  3. Going furtherWhat changes about the balance when that same 4 percent is compounded quarterly instead of yearly?
  4. Worked exampleFind the simple interest earned on 1500 dollars at 6 percent over 30 months.
Your turneasy1 of 3
Find the simple interest earned on 2393 dollars at 7 percent per year over 46 months.

Percent Increase and Percent Decrease

  1. How it worksHow do you compute a percent change from the old value rather than the new one?
  2. Worked exampleFind the percent increase when a town grows from 4800 people to 5640 people.
  3. Common mistakeWhy isn't a drop from 80 to 60 the same size percent change as a rise from 60 to 80?
  4. Worked exampleFind the original price of a laptop now selling for 714 dollars after a 15 percent markdown.
Your turneasy1 of 3
A town's population grows from 4696 people to 6149 people. Find the percent increase.

Percent Change and the Growth Factor

  1. How it worksHow does a growth factor let you apply the same percent change ten times in one multiplication?
  2. Worked exampleFind the growth factor for a 12 percent yearly decline and run it out over 5 years.
  3. Going furtherWhat growth factor undoes a 20 percent increase, and why does 0.80 fail to do it?
  4. Common mistakeWhy isn't a 3 percent monthly increase the same as a 36 percent increase across a year?
Your turneasy1 of 3
A quantity starts at 1545 and declines by 16 percent each year. Find its value after 7 years.

Percentages At or Above 100 Percent

  1. How it worksHow does a percent above 100 turn into a multiplier greater than 1?
  2. Worked exampleFind the number whose 175 percent is 84, and show why the answer comes out smaller than 84.
  3. Common mistakeWhy isn't a 200 percent increase the same thing as doubling?
Your turneasy1 of 3
It is given that 221 percent of a number n is equal to 118. Which statement about n is correct?

Frequency Tables and Histograms

  1. How it worksHow does a frequency table become the heights of the bars in a histogram?
  2. Worked exampleFind the median of a data set recorded as frequencies: three 2s, five 3s, eight 5s and four 9s.
  3. Common mistakeWhy isn't the tallest bar in a histogram always sitting where the median is?
  4. Going furtherWhat happens to the shape of a histogram when the bin width is doubled?
Your turneasy1 of 3
A frequency table records the value 2 appearing 5 times, 3 appearing 7 times, 5 appearing 12 times and 9 appearing 4 times. Find the mean of the data set.

Dot Plots and Box Plots

  1. How it worksHow do the five numbers of a box plot get picked out of a sorted list?
  2. How it worksHow does the length of the box tell you where the middle half of the data sits?
  3. Worked exampleFind the median and range of a dot plot holding 4, 4, 5, 7, 7, 7, 9 and 12.
  4. Common mistakeWhy doesn't a long whisker mean there are lots of data points out at that end?
Your turneasy1 of 3
A dot plot shows the values 3, 3, 6, 10, 10, 10, 13 and 18. Find the range of the data.

Calculating, Comparing, and Interpreting Mean, Median, and Range

  1. How it worksHow does the mean balance a data set the way a seesaw balances weights?
  2. Worked exampleFind the mean, the median and the range of 12, 15, 15, 18, 22, 40 and 41.
  3. Worked exampleFind the missing seventh value when six numbers average 24 and all seven average 26.
  4. Common mistakeWhy isn't the median simply the middle number of the list before you sort it?
Your turneasy1 of 3
Find the mean of 34, 32, 42, 18 and 10.

Comparing Distributions with the Same Spread

  1. How it worksHow does shifting every value up by 7 move the mean without touching the spread at all?
  2. Worked exampleCompare two dot plots of identical shape whose means sit at 30 and at 38, and name what differs.
  3. Going furtherWhat happens to the median and the range when every value in a data set is doubled?
Your turneasy1 of 3
Two dot plots have exactly the same shape. One is centred at 30 and the other at 41. What differs between the two data sets?

Comparing Distributions with Different Standard Deviations

  1. How it worksHow does standard deviation measure how far a typical value sits from the mean?
  2. How it worksHow do two dot plots sharing one centre still show different standard deviations through their shape?
  3. Common mistakeWhy isn't the data set with the bigger range always the one with the bigger standard deviation?
  4. Worked exampleCompare 10, 10, 10, 10, 10 against 2, 6, 10, 14, 18 and decide which spread is larger without computing it.
  5. Going furtherWhat happens to the standard deviation when 20 gets added to every single value?
Your turneasy1 of 3
Set A is 34, 34, 34, 34, 34. Set B is 34 minus 17, 34 minus 8, 34, 34 plus 8, 34 plus 17. Which set has the larger standard deviation, and why?

Outliers and Their Effect on Mean and Median

  1. How it worksHow does an extreme value drag the mean toward it while the median barely budges?
  2. Worked exampleFind the mean and median of 4, 5, 6, 7 and 8, then redo both after the 8 becomes 80.
  3. Common mistakeWhy isn't every unusually large value in a data set an outlier you should throw out?
  4. Ties togetherHow does the outlier story explain a box plot and a histogram disagreeing about where the centre is?
Your turneasy1 of 3
A data set is 4, 7, 11, 13 and 15. The largest value, 15, is replaced by 75. Find the mean of the new data set.

Reading a Scatterplot Without Predicting

  1. How it worksHow do you read direction, strength and shape off a scatterplot carrying no model?
  2. Worked exampleFind the point that breaks the pattern in 1 and 4, then 2 and 7, then 3 and 10, then 4 and 20, then 5 and 16.
  3. Common mistakeWhy isn't the point with the largest y value also the point that fits the trend worst?
Your turneasy1 of 3
A scatterplot of study hours against test score runs from about (2, 46) at the low end to about (12, 85) at the high end, with the points hugging one narrow path. Which description fits it?

Making Predictions from a Line of Best Fit

  1. How it worksHow do the slope and the intercept of a best-fit line each get read in the context's own units?
  2. Worked exampleFind the predicted y at x equal to 14 for a line of best fit written as y = 2.3x + 11.
  3. Worked exampleFind how far off a prediction lands when the model says 57 and the actual data point is 49.
  4. Going furtherWhat goes wrong when a best-fit line built on x from 1 to 10 gets used at x equal to 90?
Your turneasy1 of 3
A line of best fit is y = 3.59x + 15. Find the value it predicts for y when x = 24.

Fitting Linear Models to Data

  1. How it worksHow does a best-fit line balance the points above it against the points below it?
  2. Worked exampleFind the slope of the best-fit line through data rising steadily from 2 and 9 to 10 and 41.
  3. Common mistakeWhy isn't a best-fit line required to pass through any of the actual data points?
Your turneasy1 of 3
A best-fit line drawn through a scatterplot passes through (4, 11) and (19, 37). Find its slope.

Fitting Quadratic and Exponential Models

  1. How it worksHow do you tell a quadratic shape from an exponential shape just by watching the bend?
  2. How it worksHow does a table of values give away an exponential model through a constant ratio?
  3. Worked exampleFind the exponential model through 0 and 5, then 1 and 15, then 2 and 45, and say what the ratio means.
  4. Common mistakeWhy isn't every upward-curving scatterplot an exponential one?
Your turnmedium1 of 3
A colony starts at 10 thousand cells and multiplies by 2.57 every hour. Write an expression in terms of x for the number of thousands of cells after x hours.

Reading a Graph of a Modelled Relationship

  1. How it worksHow do you read an intercept, a maximum and a rate of change straight off a modelled graph?
  2. Worked exampleFind the maximum height and the landing time for the model h = -16t2 + 64t + 5.
  3. Going furtherWhat changes about a modelled graph when its vertical axis gets relabelled in thousands?
Your turneasy1 of 3
A ball's height in feet, t seconds after launch, is modelled by h = -16t2 + 81t + 9. Find the maximum height in feet.

Comparing Linear and Exponential Growth

  1. How it worksHow does adding the same amount each step differ from multiplying by the same factor each step?
  2. Worked exampleCompare 40 + 12t against 40 times 1.25t and find the first whole t where the second one wins.
  3. Worked exampleFind the doubling time for a quantity growing at 15 percent per year.
  4. Common mistakeWhy doesn't a huge fixed yearly increase eventually beat a small percentage increase?
  5. Ties togetherHow does the growth factor from percent change show up again inside an exponential model?
Your turnmedium1 of 3
A linear model gives 51 + 16t and an exponential model gives 51 times 1.57t, with t in years. Find the exponential model's value minus the linear model's at t = 6; a negative answer means the linear model is still ahead.

Relative Frequency from a One-Way Display

  1. How it worksHow does dividing a count by the total turn a frequency table into a probability?
  2. Worked exampleFind the probability of drawing a blue chip from a bag holding 14 blue, 21 red and 15 green.
  3. Worked exampleFind the relative frequency of the 30 to 39 age bin when it holds 96 of 640 surveyed people.
Your turneasy1 of 3
A bag holds 17 blue chips, 24 red chips and 21 green chips. Find the probability that a chip drawn at random is blue.

Two-Way Tables and Conditional Probability

  1. How it worksHow does one row of a two-way table become the entire universe for a conditional probability?
  2. How it worksHow do the row totals, the column totals and the grand total of a two-way table fit together?
  3. Worked exampleFind the probability a person prefers tea given they are under 30, when 150 are under 30, 90 of those prefer tea, and 310 were surveyed.
  4. Common mistakeWhy isn't the probability of A given B the same as the probability of B given A?
  5. Worked exampleFind what fraction of all 400 respondents are left-handed men when 24 of the 160 men are left-handed.
Your turneasy1 of 3
A survey of 384 people records that 179 are under 30 and that 107 of those under-30 respondents prefer tea. Find the probability that a person prefers tea GIVEN that they are under 30.

Area Models and Other Probability Displays

  1. How it worksHow does the area of a region inside a square become the probability of an event?
  2. Worked exampleFind the probability a dart lands inside a 3 by 4 rectangle painted on a 10 by 10 square target.
  3. Going furtherWhat happens to that probability when the whole target is scaled up but the inner rectangle is not?
Your turneasy1 of 3
A square target of side 14 centimetres carries a painted rectangle 4 centimetres wide and 7 centimetres tall. A dart lands at a uniformly random point on the target. Find the probability it lands inside the rectangle.

Working Back from a Probability to an Unknown Frequency

  1. How it worksHow do you turn a probability statement into an equation with the missing frequency inside it?
  2. Worked exampleFind how many red marbles a bag holds when 0.35 of its 260 marbles are red.
  3. Worked exampleFind the number of seniors surveyed when 42 people, or 0.24 of them, chose option B.
  4. Common mistakeWhy isn't the missing cell of a two-way table always found by subtracting from the grand total?
Your turneasy1 of 3
A bag holds 700 marbles, and x percent of them are red. Write an expression in terms of x for the number of red marbles.

Inference from Sample Statistics

  1. How it worksHow does a sample mean become the single best estimate of a population mean?
  2. Worked exampleFind the estimated number of left-handed people in a town of 24000 from 39 lefties in a sample of 300.
  3. Worked exampleFind the estimated total weight of 1800 crates when a random sample of 40 averages 22.5 kilograms.
  4. Common mistakeWhy isn't a sample proportion of 0.62 the same as knowing the population proportion is 0.62?
Your turneasy1 of 3
In a random sample of 370 residents of a town of 26490 people, x are left-handed. Write an expression in terms of x for the estimated number of left-handed residents in the town.

What PSAT Scope Drops: Margin of Error and Study Design

  1. How it worksHow does the framework leave PSAT inference with only estimating a population mean or proportion?
  2. How it worksHow does the SAT's data list run one testing point longer than the PSAT's, with a wider inference row?
  3. Common mistakeWhy isn't margin of error a PSAT topic even though College Board's own PSAT web page names it?
  4. Going furtherWhat disappears from PSAT prep once observational studies and random assignment are off the list?
  5. Ties togetherHow does dropping margin of error change what an inference question is able to ask on the PSAT?
Your turneasy1 of 3
A student puts 31 hours into an SAT data-analysis workbook, and 28 percent of that time goes to margin of error and to observational studies versus experiments. Find the number of hours spent on material the PSAT never tests.

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