PSAT 10 Math — Problem-Solving & Data Analysis Study Notes

Ratios, rates, proportional relationships, and units

A lot of “data” and “real-world” math starts with comparing quantities. The PSAT often hides the key relationship in words—your job is to translate those words into a structure (a ratio, a rate, or a proportional relationship) and keep the units under control.

Ratios and equivalent ratios

A ratio compares two quantities by division. If a recipe uses flour and sugar in a ratio of $3:2$, that means “for every $3$ parts flour, there are $2$ parts sugar.” Ratios can be written as:

• $3:2$ (colon form)
• $\frac{3}{2}$ (fraction form)
• “$3$ to $2$” (words)

Two ratios are equivalent if they represent the same multiplicative comparison. You create equivalent ratios by multiplying or dividing both parts by the same nonzero number.

Why this matters: equivalent ratios are the backbone of scaling—doubling a recipe, resizing a map, or converting between units.

Worked example (equivalent ratios):
If the ratio of cats to dogs in a shelter is $4:5$ and there are $20$ dogs, how many cats are there?

Set up an equivalent ratio:

$\frac{\text{cats}}{\text{dogs}}=\frac{4}{5}$

We want cats when dogs $=20$:

$\frac{x}{20}=\frac{4}{5}$

Solve:

$x=20\cdot\frac{4}{5}=16$

So there are $16$ cats.

Common pitfall: swapping the order. If you accidentally use $\frac{\text{dogs}}{\text{cats}}$, you’ll get the reciprocal and the wrong answer.

Rates and unit rates

A rate is a ratio comparing quantities with different units, like dollars per hour or miles per gallon. A unit rate is a rate “per $1$ unit,” like $\$18$ per hour.

Why this matters: unit rates let you compare deals and speeds directly.

Worked example (unit rate):
A car travels $150$ miles using $6$ gallons of gas. What is the fuel efficiency in miles per gallon?

$\text{mpg}=\frac{150\,\text{miles}}{6\,\text{gallons}}=25\,\text{miles/gallon}$

Proportional relationships and constant of proportionality

Two quantities $x$ and $y$ are in a proportional relationship if they change by a constant multiplicative factor. The relationship has the form:

$y=kx$

where $k$ is the constant of proportionality (also called the unit rate in many contexts).

How you recognize proportionality:

• In a table, the ratio $\frac{y}{x}$ is constant.
• On a graph, the relationship is a straight line through the origin.
• In an equation, it can be written as $y=kx$.

Worked example (finding $k$):
A store charges $\$12$ for $3$ pounds of apples. If cost $C$ is proportional to pounds $p$, find the equation.

Compute $k$:

$k=\frac{C}{p}=\frac{12}{3}=4$

So:

$C=4p$

Interpretation: $4\,\text{dollars per pound}$.

Common pitfall: assuming “linear” means “proportional.” A line like $y=5x+2$ is linear but not proportional because it does not pass through the origin.

Units and dimensional analysis (unit conversions)

Dimensional analysis is the idea that you can treat units like factors that cancel, ensuring your final unit makes sense. This is one of the safest ways to do PSAT conversion problems.

Worked example (conversion using canceling units):
Convert $72$ miles per hour to feet per second.

Use the facts $1\,\text{mile}=5280\,\text{feet}$ and $1\,\text{hour}=3600\,\text{seconds}$.

$72\,\frac{\text{miles}}{\text{hour}}\times\frac{5280\,\text{feet}}{1\,\text{mile}}\times\frac{1\,\text{hour}}{3600\,\text{seconds}}$

Cancel miles and hours:

$=72\times\frac{5280}{3600}\,\frac{\text{feet}}{\text{second}}$

Simplify:

$\frac{5280}{3600}=\frac{22}{15}$

So:

$72\times\frac{22}{15}=\frac{1584}{15}=105.6$

Final:

$105.6\,\text{feet/second}$

Common pitfall: converting only the number and forgetting the unit, or using the conversion factor upside down.

Exam Focus

• Typical question patterns:
  • Identify whether a relationship is proportional from a table, graph, or description, then find $k$.
  • Compare unit rates (best buy, fastest speed, cheapest per ounce).
  • Multi-step unit conversions using chained conversion factors.
• Common mistakes:
  • Mixing up ratio order (e.g., using $\frac{\text{dogs}}{\text{cats}}$ instead of $\frac{\text{cats}}{\text{dogs}}$).
  • Assuming any straight-line relationship is proportional (forgetting “through the origin”).
  • Not canceling units—ending with a number that has the wrong meaning.

Percentages

A percentage is a ratio “per $100$.” Thinking of percent as a rate helps: $35\%$ means $35$ per $100$, or $\frac{35}{100}$.

Percent problems often test whether you can correctly identify the whole (the base) and how the percent is being applied.

Converting between forms

• Percent to decimal: divide by $100$.
• Decimal to percent: multiply by $100$.
• Percent to fraction: write over $100$ and simplify.

For example:

$18\%=0.18=\frac{18}{100}=\frac{9}{50}$

Percent of a quantity

“$p\%$ of $N$” means multiply $N$ by $\frac{p}{100}$.

Worked example (percent of):
Find $15\%$ of $80$.

$0.15\times 80=12$

Percent increase and decrease

A percent change compares how much something changes relative to its original value:

$\text{percent change}=\frac{\text{new}-\text{original}}{\text{original}}\times 100\%$

But on test day, it’s often faster to use multipliers:

• Increase by $r\%$: multiply by $1+\frac{r}{100}$
• Decrease by $r\%$: multiply by $1-\frac{r}{100}$

Why this matters: many PSAT questions chain changes (discount then tax, or two increases). Multipliers make chaining straightforward.

Worked example (discount then tax):
A jacket costs $\$60$. It’s discounted by $25\%$, then a $10\%$ sales tax is applied to the discounted price. What is the final price?

Discount multiplier:

$1-0.25=0.75$

Tax multiplier:

$1+0.10=1.10$

Final price:

$60\times 0.75\times 1.10=60\times 0.825=49.5$

So the final price is $\$49.50$.

Common pitfall: adding or subtracting percents directly (thinking $-25\%$ then $+10\%$ means net $-15\%$). The second percent is applied to a different base.

Percent problems with “is/of” and unknown base

A reliable translation is:

• “is” $\rightarrow$ the part
• “of” $\rightarrow$ the whole
• “what percent” $\rightarrow$ multiply by $100\%$ at the end

Worked example (finding the whole):
$18$ is $30\%$ of what number?

Set up:

$18=0.30\times x$

Solve:

$x=\frac{18}{0.30}=60$

Percent vs percentage points

If a survey result goes from $40\%$ to $55\%$, that is an increase of $15$ percentage points, but the percent increase is:

$\frac{55-40}{40}=\frac{15}{40}=0.375$

which is $37.5\%$.

The PSAT sometimes tests whether you confuse these.

Exam Focus

• Typical question patterns:
  • Compute percent change and interpret what it means in context.
  • Reverse percent problems (find original price from sale price, find whole from part).
  • Apply multiple successive percent changes using multipliers.
• Common mistakes:
  • Using the wrong “whole” in the denominator when finding percent change.
  • Treating successive percent changes as additive.
  • Confusing percent increase with percentage-point increase.

One-variable data: distributions and measures of center and spread

One-variable data means you measure a single quantity for many individuals—like heights of students or number of texts sent per day. The PSAT focuses on reading and interpreting distributions, not heavy computation.

Distributions: shape, center, and spread

A distribution shows how values are spread out. You might see it as a dot plot, histogram, or box plot.

When you describe a distribution, you’re usually describing:

• Shape: symmetric, skewed left, skewed right
• Center: typical value (mean or median)
• Spread: how variable the data are (range or IQR)
• Outliers: unusually far values

Why this matters: choosing the right summary depends on the shape. Skew and outliers can pull the mean, so the median may represent “typical” better.

Measures of center: mean vs median

The mean is the arithmetic average:

$\text{mean}=\frac{\text{sum of values}}{n}$

The median is the middle value when data are ordered.

• Mean uses all values and is sensitive to outliers.
• Median is resistant to outliers.

Worked example (mean vs median):
Data: $2,3,3,4,20$

Mean:

$\frac{2+3+3+4+20}{5}=\frac{32}{5}=6.4$

Median is the third value (ordered list):

$3$

Interpretation: the mean is much larger because $20$ is an outlier.

Measures of spread: range and interquartile range

The range is:

$\text{max}-\text{min}$

The interquartile range (IQR) is:

$\text{IQR}=Q_3-Q_1$

where $Q_1$ is the first quartile (about the $25\%$ point) and $Q_3$ is the third quartile (about the $75\%$ point).

Why IQR matters: like the median, it’s resistant to outliers. Box plots are built from quartiles and show IQR visually.

Worked example (IQR from a list):
Data (already ordered): $4,5,7,8,9,10,12,13$

Lower half: $4,5,7,8$ so:

$Q_1=\frac{5+7}{2}=6$

Upper half: $9,10,12,13$ so:

$Q_3=\frac{10+12}{2}=11$

Then:

$\text{IQR}=11-6=5$

Common pitfall: including the median in both halves when $n$ is even/odd without being consistent. On the PSAT, if a box plot is given, you can often read $Q_1$ and $Q_3$ directly and avoid this issue.

Interpreting box plots and histograms

A box plot shows five key numbers: minimum, $Q_1$, median, $Q_3$, maximum. The box length is IQR.

A histogram groups data into bins. The height of each bar indicates frequency (or sometimes relative frequency). Pay attention to whether bins have equal widths; on the PSAT they typically do.

A crucial interpretation skill: “more spread out” means a larger range or IQR, not a larger mean.

Exam Focus

• Typical question patterns:
  • Compare two distributions using center and spread (often from box plots).
  • Decide whether mean or median is a better “typical” measure based on skew/outliers.
  • Interpret what quartiles and IQR say about the data.
• Common mistakes:
  • Saying the “highest bar” in a histogram is the greatest value (it’s the most frequent bin).
  • Assuming a larger maximum means “more variable” without comparing IQR/range.
  • Using mean-based reasoning when the distribution is skewed with outliers.

Two-variable data: models and scatterplots

Two-variable data records two measurements for each individual, like hours studied and test score. The PSAT emphasizes interpreting scatterplots, fitting/using linear models, and understanding what association does (and doesn’t) imply.

Scatterplots and association

A scatterplot graphs points $\left(x,y\right)$ where $x$ is the explanatory variable and $y$ is the response variable.

You look for:

• Direction: positive (upward) or negative (downward)
• Form: linear or nonlinear
• Strength: tight cluster vs widely scattered
• Outliers: points far from the pattern

Why this matters: deciding whether a linear model makes sense comes before calculating anything.

Linear models and slope meaning

A common model is a linear equation:

$y=mx+b$

• $m$ is the **slope**: change in $y$ for each $1$ unit increase in $x$
• $b$ is the **y-intercept**: predicted $y$ when $x=0$ (may or may not be meaningful in context)

Worked example (interpreting slope):
Suppose a model for cost $C$ (in dollars) vs number of tickets $t$ is:

$C=12t+5$

Here $12$ means each additional ticket adds $\$12$. The $5$ is a fixed fee.

Common pitfall: interpreting $b$ even when $x=0$ is outside the realistic domain (e.g., a model for adult height at age $0$).

Making predictions: interpolation vs extrapolation

If you predict within the range of observed $x$ values, that’s interpolation and is usually more reliable. Predicting beyond the data range is extrapolation, which is riskier because trends can change.

Worked example (extrapolation warning):
If your data include ages $14$ to $18$, using the line to predict at age $30$ is extrapolation—treat it cautiously.

Residuals (prediction errors)

A residual is:

$\text{residual}=\text{actual}-\text{predicted}$

• Positive residual: model underpredicted.
• Negative residual: model overpredicted.

Residuals help you judge fit. If residuals show a curved pattern, a linear model may be inappropriate.

Correlation vs causation

A strong association does not prove that $x$ causes $y$. There could be confounding variables, or the direction of influence could differ.

• Correlation: variables move together.
• Causation: changing one variable produces a change in the other.

Causal conclusions require a well-designed experiment (typically random assignment). Observational studies can show association but not definitive causation.

Exam Focus

• Typical question patterns:
  • Interpret slope and intercept of a linear model in context.
  • Use a line of best fit to estimate/predict and compute a residual.
  • Decide whether a linear model is appropriate from the scatterplot.
• Common mistakes:
  • Claiming causation from a scatterplot or survey.
  • Extrapolating far beyond the data without noticing.
  • Mixing up residual sign (remember: actual minus predicted).

Probability and conditional probability

Probability questions test careful counting and careful interpretation of “and,” “or,” and “given.” On the PSAT, you often see probabilities from tables (including two-way tables) rather than complicated formulas.

Basic probability ideas

For a simple situation where outcomes are equally likely:

$P(A)=\frac{\text{number of favorable outcomes}}{\text{number of total outcomes}}$

Probabilities are between $0$ and $1$.

“And” vs “Or”

• $A \text{ and } B$ means both happen: $A\cap B$.
• $A \text{ or } B$ means at least one happens: $A\cup B$.

If events are mutually exclusive (cannot both happen), then:

$P(A\cup B)=P(A)+P(B)$

If they are not mutually exclusive, you must subtract the overlap:

$P(A\cup B)=P(A)+P(B)-P(A\cap B)$

Common pitfall: adding probabilities for “or” when overlap exists.

Conditional probability

Conditional probability is the probability of $A$ given that $B$ has occurred:

$P(A\mid B)=\frac{P(A\cap B)}{P(B)}$

A practical way to think about it: the condition “given $B$” shrinks the sample space down to only cases where $B$ is true.

Two-way tables

A two-way table organizes counts for two categorical variables (e.g., grade level and plays a sport).

Worked example (conditional probability from a table):
In a school:

Find $P(\text{plays sport}\mid \text{10th grade})$.

Condition on “10th grade,” so the denominator is the total number of 10th graders:

$P(\text{sport}\mid \text{10th})=\frac{30}{50}=0.6$

Independence

Events $A$ and $B$ are **independent** if knowing $B$ does not change the probability of $A$:

$P(A\mid B)=P(A)$

Equivalently:

$P(A\cap B)=P(A)P(B)$

On the PSAT, independence is often tested conceptually: “Are these events independent?” You check whether the conditional probability matches the overall probability.

Worked example (checking independence):
From the table above:

$P(\text{sport})=\frac{55}{100}=0.55$

But:

$P(\text{sport}\mid \text{10th})=0.6$

Since $0.6\neq 0.55$, playing a sport is not independent of being in 10th grade (in this sample).

Exam Focus

• Typical question patterns:
  • Compute probabilities from two-way tables, including conditional probabilities.
  • Use the “or” rule with overlap: $P(A\cup B)$.
  • Determine whether events are independent by comparing $P(A\mid B)$ and $P(A)$.
• Common mistakes:
  • Using the wrong denominator for conditional probability (forgetting to restrict to the “given” group).
  • Treating “or” as always-add without subtracting intersection.
  • Confusing independence with “mutually exclusive” (they are different ideas).

Inference from sample statistics

Inference means using data from a sample to make a conclusion about a population. On the PSAT, inference is usually informal: you interpret sample statistics, evaluate sampling methods, and reason about margin of error and bias.

Population vs sample; parameter vs statistic

• Population: the entire group you care about.
• Sample: the subset you actually measure.

A parameter is a number describing the population (usually unknown). A statistic is a number computed from the sample.

Example: the population proportion of students who prefer online homework is a parameter; the proportion in your survey sample is a statistic.

Sampling methods and bias

A sample is most useful when it’s representative of the population.

• A random sample gives each member of the population a chance (ideally equal chance) to be selected.
• A biased sample systematically overrepresents or underrepresents part of the population.

Common sources of bias:

• Convenience sampling: surveying whoever is easiest to reach.
• Nonresponse bias: certain types of people are less likely to respond.
• Wording bias: leading questions influence answers.

Why this matters: a large biased sample can still give a bad conclusion. Size alone doesn’t fix bias.

Experiments vs observational studies (and causation)

• In an observational study, you observe what already happens; you can find association but not strong causal claims.
• In an experiment, you assign treatments (ideally with random assignment). Random assignment helps support cause-and-effect conclusions.

PSAT inference questions often ask what conclusions are justified based on the design.

Margin of error (MOE) and confidence ideas

The PSAT may describe a survey result like: “$52\%$ of voters support Candidate A with a margin of error of $\pm 3\%$.”

Interpretation: the true population proportion is expected to be within:

$52\%-3\%=49\%$

to

$52\%+3\%=55\%$

So an interval of:

$[49\%,55\%]$

Key idea: if two results have intervals that overlap a lot, the difference may not be meaningful based on that data alone.

Common pitfall: interpreting MOE as guaranteeing the true value is in the interval. In reality, it’s a statement about the method’s typical accuracy under repeated sampling (the PSAT keeps this informal—focus on using the interval correctly).

Making and evaluating claims from sample results

When a question asks whether a sample supports a claim, you typically check:

1. Was the sample random/representative?
2. Is the sample size reasonably large (larger tends to reduce variability)?
3. Does the claim fall inside a plausible interval (often using MOE)?

Worked example (using MOE to evaluate a claim):
A poll of a random sample reports $46\%$ of students prefer the new lunch menu with MOE $\pm 4\%$. Can you conclude that a majority (more than $50\%$) prefer it?

Construct the interval:

$46\%\pm 4\%\rightarrow [42\%,50\%]$

A majority would require the true proportion to be greater than $50\%$, but the interval does not exceed $50\%$. So the poll does not provide strong support that a majority prefer it.

Sample size and variability (qualitative understanding)

As sample size increases, sample statistics tend to vary less from sample to sample—so estimates become more precise (often reflected as a smaller margin of error, if the method is the same).

Common pitfall: thinking doubling sample size halves margin of error. The exact relationship isn’t typically tested on the PSAT; what matters is the direction: bigger sample generally means more precision, but bias is a separate issue.

Exam Focus

• Typical question patterns:
  • Decide whether a conclusion about a population is justified from a sample (focus on representativeness and design).
  • Interpret a reported value with margin of error by building the interval.
  • Distinguish when causation is justified (random assignment) versus only association.
• Common mistakes:
  • Confusing random sampling (for representativeness) with random assignment (for causation).
  • Believing a large convenience sample eliminates bias.
  • Misusing margin of error (adding it in the wrong direction or ignoring that “majority” means strictly above $50\%$).