PSAT 8/9 Math — Problem-Solving and Data Analysis (Deep-Dive Study Notes)

Ratios, rates, proportional relationships, and units

What ratios and rates are (and how they’re different)

A ratio compares two quantities by division. It tells you how much of one thing there is compared to another. You’ll see ratios written in several equivalent ways:

• Colon form: $a:b$
• Fraction form: $\frac{a}{b}$
• Word form: “$a$ to $b$”

A rate is a ratio that compares quantities with different units, like dollars per hour or miles per minute. Rates matter because they connect two different measurements and help you predict or scale situations.

A very important special rate is the unit rate—a rate “per $1$.” Unit rates make comparisons easy because everything is standardized. For example, comparing prices is easier when you convert both to $\text{dollars per }1\,\text{ounce}$ (or per $1\,\text{pound}$).

Why proportional relationships matter

Two quantities have a proportional relationship when they scale together at a constant multiplicative rate. Intuitively: if you double one quantity, the other doubles; if you triple one, the other triples.

Proportional relationships show up constantly—recipes, maps, similar figures, speed, pricing, and conversions. On the PSAT 8/9, they’re a core tool for solving real-world problems efficiently.

Mathematically, a relationship is proportional if it can be written as:

$y = kx$

where $k$ is the **constant of proportionality** (also called the **unit rate** or **scale factor**, depending on context). The key feature is that the graph is a straight line **through the origin** $\left(0,0\right)$.

A common confusion: not every linear relationship is proportional. If you have:

$y = mx + b$

and $b \neq 0$, then the relationship is linear but not proportional, because it doesn’t pass through the origin.

How to solve proportion problems

Many PSAT 8/9 questions rely on the idea that equivalent ratios are equal. If a relationship is proportional, then:

$\frac{a}{b} = \frac{c}{d}$

You can solve by cross-multiplying:

$ad = bc$

This works because both fractions represent the same number.

Example 1: Using a unit rate

A store sells $12\,\text{ounces}$ of trail mix for $\$3.60$. What is the unit price in dollars per ounce?

Step 1: Set up the rate.

$\frac{\$3.60}{12\,\text{oz}}$

Step 2: Divide to get “per $1$ ounce.”

$\frac{3.60}{12} = 0.30$

Answer:

$\$0.30\,\text{per}\,\text{oz}$

A typical mistake here is rounding too early (or mixing up numerator and denominator and accidentally finding ounces per dollar instead of dollars per ounce).

Example 2: Solving a proportion with cross multiplication

A map uses a scale where $1\,\text{inch}$ represents $5\,\text{miles}$. If two towns are $3.5\,\text{inches}$ apart on the map, how far apart are they in miles?

Step 1: Write a proportion.

$\frac{1\,\text{in}}{5\,\text{mi}} = \frac{3.5\,\text{in}}{x\,\text{mi}}$

Step 2: Cross-multiply.

$1 \cdot x = 5 \cdot 3.5$

Step 3: Compute.

$x = 17.5$

Answer:

$17.5\,\text{miles}$

Units and unit conversions (dimensional analysis)

Units are not decoration—they are part of the mathematics. One of the best habits you can build is to treat units like factors that can cancel.

For example, converting minutes to hours uses the fact that:

$1\,\text{hour} = 60\,\text{minutes}$

So if you have $150\,\text{minutes}$ and want hours, multiply by a “conversion factor” that equals $1$:

$150\,\text{min} \times \frac{1\,\text{hr}}{60\,\text{min}} = 2.5\,\text{hr}$

Notice how $\text{min}$ cancels, leaving hours.

Example 3: Multi-step rate with conversion

A runner goes $3\,\text{miles}$ in $30\,\text{minutes}$. What is the speed in miles per hour?

Step 1: Compute miles per minute.

$\frac{3\,\text{mi}}{30\,\text{min}} = 0.1\,\text{mi/min}$

Step 2: Convert minutes to hours.

$0.1\,\text{mi/min} \times 60\,\text{min/hr} = 6\,\text{mi/hr}$

Answer:

$6\,\text{miles per hour}$

A common error is to divide by $60$ instead of multiply (because you’re thinking “minutes to hours” but not tracking what needs to happen to the number).

Exam Focus

• Typical question patterns:
  • Find a unit rate from a table, graph, or story (price per item, miles per hour, etc.).
  • Decide whether a relationship is proportional and identify $k$ in $y = kx$.
  • Solve a scale/conversion problem with consistent units.
• Common mistakes:
  • Treating any straight-line relationship as proportional (forgetting the origin requirement).
  • Flipping a rate (finding $\text{hours per mile}$ when asked for $\text{miles per hour}$).
  • Forgetting to convert units before computing a rate.

Percentages

What a percent really means

A percent means “per hundred.” So $35\%$ is the same as $\frac{35}{100}$ and the decimal $0.35$.

Thinking this way helps you avoid memorizing tricks. Percent problems are mostly about interpreting “out of $100$” and scaling up or down.

Converting between forms (fraction, decimal, percent)

• Percent to decimal: divide by $100$.
  • $45\% = 0.45$
• Decimal to percent: multiply by $100$.
  • $0.08 = 8\%$
• Percent to fraction: write over $100$ and simplify.
  • $12\% = \frac{12}{100} = \frac{3}{25}$

A frequent slip is placing the decimal two places in the wrong direction.

Finding a percent of a quantity

“$p\%$ of $N$” means:

$\frac{p}{100} \times N$

Example 1: Percent of a number

What is $18\%$ of $250$?

Step 1: Convert to a decimal or fraction.

$18\% = 0.18$

Step 2: Multiply.

$0.18 \times 250 = 45$

Answer:

$45$

Percent increase and percent decrease

A percent change compares the amount of change to the original value.

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

• If the result is positive, it’s a percent increase.
• If the result is negative, it’s a percent decrease.

A common misconception is to divide by the new value instead of the original value. The base is the original.

Example 2: Percent increase

A jacket’s price goes from $\$40$ to $\$50$. What is the percent increase?

Step 1: Find the change.

$50 - 40 = 10$

Step 2: Divide by the original.

$\frac{10}{40} = 0.25$

Step 3: Convert to percent.

$0.25 = 25\%$

Answer:

$25\%$

Using multipliers (a powerful shortcut)

Percent changes can be handled quickly with multipliers.

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

For example, a $15\%$ discount means multiply by:

$1 - 0.15 = 0.85$

This method reduces arithmetic mistakes and helps with multi-step changes.

Example 3: Successive percent changes

A phone costs $\$200$. It is discounted by $20\%$, then sales tax of $5\%$ is applied to the discounted price. What is the final price?

Step 1: Apply the discount multiplier.

$200 \times 0.80 = 160$

Step 2: Apply the tax multiplier to the new price.

$160 \times 1.05 = 168$

Answer:

$\$168$

Important idea: a $20\%$ decrease and then a $20\%$ increase does not return you to the original. The second percent is applied to a different base.

Percent vs percentage points

Sometimes data is reported as “the percent went from $40\%$ to $55\%$.”

• The change in percentage points is $55 - 40 = 15$ percentage points.
• The percent increase relative to the original is:

$\frac{55 - 40}{40} = \frac{15}{40} = 0.375 = 37.5\%$

PSAT questions may test whether you know which interpretation is being asked.

Exam Focus

• Typical question patterns:
  • Compute percent of a quantity (discounts, tax, tips, markups).
  • Find percent change given old and new values.
  • Interpret percent information in tables/graphs (including “percentage points”).
• Common mistakes:
  • Using the wrong base for percent change (must be the original).
  • Adding/subtracting percent values instead of using multipliers for successive changes.
  • Confusing percentage points with percent increase.

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

What one-variable data means

One-variable data is a list of values from a single measurement—heights of students, number of texts sent in a day, quiz scores, and so on. Your goal is to describe the data in a way that is informative, not just a pile of numbers.

Two big ideas guide this:

1. Center: what value is “typical”?
2. Spread: how much do the values vary?

You also pay attention to the shape of the distribution and whether there are outliers.

Displaying distributions

A distribution shows how often values occur.

Common displays include:

• Dot plots: great for small to medium data sets.
• Histograms: group data into intervals (bins) for larger sets.
• Box plots: emphasize quartiles, median, and potential outliers.

A frequent mistake is reading a histogram like a bar chart: in a histogram, the horizontal axis is a number line, and bars represent intervals, not categories.

Measures of center

The most common measures of center are:

• Mean: the arithmetic average.

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

• Median: the middle value when data is ordered. If there are an even number of values, it’s the mean of the two middle values.
• Mode: the most frequent value (may be more than one, or none).

Why this matters: the mean uses every value, so it is sensitive to outliers; the median is resistant to outliers and often better for skewed data.

Example 1: Mean vs median with an outlier

Data set (minutes spent on homework):

$5,\,6,\,6,\,7,\,8,\,40$

Mean:

$\frac{5 + 6 + 6 + 7 + 8 + 40}{6} = \frac{72}{6} = 12$

Median: average of the 3rd and 4th values:

$\frac{6 + 7}{2} = 6.5$

The mean $12$ is pulled upward by the outlier $40$, while the median $6.5$ better reflects a “typical” value.

Measures of spread

Spread tells you how variable the data is.

• Range: max minus min.

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

Range is easy, but it uses only two values, so it can be overly influenced by an outlier.

• Interquartile range (IQR): the spread of the middle $50\%$ of the data.

$\text{IQR} = Q_3 - Q_1$

where $Q_1$ is the first quartile and $Q_3$ is the third quartile.

• Mean absolute deviation (MAD) (often used in middle school/early high school contexts): the average distance from the mean.

$\text{MAD} = \frac{\sum |x - \bar{x}|}{n}$

Here $\bar{x}$ is the mean, $x$ is a data value, and $n$ is the number of values.

MAD matters because it captures typical distance from the mean, not just extremes.

Example 2: Finding IQR

Data (sorted):

$2,\,3,\,3,\,5,\,6,\,9,\,10,\,12$

There are $8$ values.

• Median is average of 4th and 5th:

$\frac{5 + 6}{2} = 5.5$

Lower half: $2,3,3,5$

$Q_1 = \frac{3 + 3}{2} = 3$

Upper half: $6,9,10,12$

$Q_3 = \frac{9 + 10}{2} = 9.5$

So:

$\text{IQR} = 9.5 - 3 = 6.5$

Answer:

$6.5$

A common error is including the overall median in both halves when splitting data. For even-sized data sets, you split cleanly into two halves.

Shape, skew, and outliers

• Symmetric distributions have left and right sides that roughly mirror.
• Skewed right means a long tail to the right (a few large values).
• Skewed left means a long tail to the left (a few small values).

When data is skewed, the mean is pulled toward the tail. That’s why median is often preferred for skewed distributions.

Box plots are especially useful for comparing distributions quickly—medians, IQRs, and overall ranges.

Exam Focus

• Typical question patterns:
  • Compute and interpret mean/median/range/IQR (often from a small list).
  • Compare two groups using center and spread (especially with box plots).
  • Identify skew and explain which measure of center is more appropriate.
• Common mistakes:
  • Calculating the median without sorting the data.
  • Miscomputing quartiles by splitting the data incorrectly.
  • Choosing mean as “typical” even when an extreme outlier makes median more meaningful.

Two-variable data: models and scatterplots

What two-variable data is

Two-variable data records paired values $\left(x,y\right)$ for each individual—like (hours studied, test score) or (temperature, ice cream sales). The main question becomes: does $y$ tend to change as $x$ changes?

A scatterplot is the standard way to display this. Each point represents one pair.

Association, direction, and strength

When you look at a scatterplot, you describe:

• Direction: positive (upward trend) or negative (downward trend).
• Strength: how tightly points cluster around a pattern.
• Form: linear pattern or curved pattern.

A crucial idea: correlation does not prove causation. Even if two variables move together, that doesn’t mean one causes the other. PSAT questions sometimes ask for cautious interpretation.

Modeling with a line (line of best fit)

If the scatterplot shows a roughly linear pattern, you can model it with a line:

$y = mx + b$

• $m$ is the **slope**: the predicted change in $y$ for a one-unit increase in $x$.
• $b$ is the **y-intercept**: the predicted value of $y$ when $x = 0$.

The line is an approximation—real data usually does not land exactly on it.

Interpreting slope in context

Slope is where many students lose points, not because it’s hard, but because they forget to interpret it with units.

If $x$ is hours and $y$ is dollars, then $m$ has units $\text{dollars per hour}$.

Example 1: Interpreting a model

A model for weekly earnings is:

$E = 15h + 20$

where $h$ is hours worked and $E$ is dollars earned.

• The slope $15$ means $15\,\text{dollars per hour}$.
• The intercept $20$ suggests a base amount earned even if $h = 0$ (maybe a bonus or fixed stipend).

A common mistake is to say “the intercept is the starting hours” or to ignore that $x = 0$ might not be meaningful in the real situation.

Predictions and interpolation vs extrapolation

Using a model to estimate values is called prediction.

• Interpolation: predicting within the range of the data (usually safer).
• Extrapolation: predicting beyond the data range (riskier because the pattern may change).

PSAT questions may ask whether a prediction is reasonable, which is often about recognizing extrapolation.

Example 2: Making a prediction

A line of best fit is:

$y = 2.5x + 10$

Predict $y$ when $x = 8$.

Substitute:

$y = 2.5 \times 8 + 10 = 20 + 10 = 30$

Answer:

$30$

Residuals (how far off the model is)

A residual is:

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

Residuals tell you whether the model overestimates or underestimates.

• Positive residual: actual is above the line.
• Negative residual: actual is below the line.

Example 3: Computing a residual

Model:

$y = 3x + 2$

At $x = 4$, the predicted value is:

$y = 3 \times 4 + 2 = 14$

If the actual value is $18$, then:

$\text{residual} = 18 - 14 = 4$

Residual:

$4$

Two-way tables as two-variable data

Sometimes two-variable data is categorical (not numerical), like “owns a pet” vs “is in a sports club.” A two-way table organizes counts for combinations of categories, and it becomes a bridge to conditional probability (covered later).

Exam Focus

• Typical question patterns:
  • Describe the relationship in a scatterplot (positive/negative, strong/weak).
  • Interpret slope and intercept of a linear model in context.
  • Use a line to predict values and recognize interpolation vs extrapolation.
• Common mistakes:
  • Interpreting slope without units or mixing up what changes (confusing $\Delta x$ and $\Delta y$).
  • Treating extrapolated predictions as equally reliable as interpolated ones.
  • Thinking points must lie on the line for the model to be “correct” (real data is noisy).

Probability and conditional probability

What probability measures

Probability measures how likely an event is, from $0$ (impossible) to $1$ (certain).

If all outcomes are equally likely, then:

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

Probability matters in PSAT data analysis because it connects counting, proportions, and interpreting real-world uncertainty (surveys, risk, predictions).

Experimental vs theoretical probability

• Theoretical probability is based on reasoning about equally likely outcomes (like a fair die).
• Experimental probability is based on data:

$P(\text{event}) \approx \frac{\text{number of times event occurs}}{\text{number of trials}}$

Experimental probability often gets closer to theoretical probability as the number of trials grows, but small samples can be misleading.

Complementary events

The complement of event $A$ is “not $A$.”

$P(\text{not }A) = 1 - P(A)$

This is especially useful when “not $A$” is easier to count.

Example 1: Using the complement

A bag has $3$ red marbles and $7$ blue marbles. Probability of not red?

Total marbles: $10$.

$P(\text{red}) = \frac{3}{10}$

So:

$P(\text{not red}) = 1 - \frac{3}{10} = \frac{7}{10}$

Answer:

$\frac{7}{10}$

Independent vs dependent events

Two events are independent if one happening does not change the probability of the other. A typical example is flipping a fair coin twice.

Events are dependent if one affects the other—often when you do not replace an item after selecting it.

A frequent mistake is assuming events are independent just because they are described separately. The key question is: does the first outcome change the situation?

Conditional probability

Conditional probability is the probability of event $A$ given that event $B$ has already occurred.

The formal definition is:

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

This matters because many real-world probabilities are conditional: probability of rain given cloudy skies, probability a student is in band given that they are in eighth grade, etc.

A practical way to think about conditional probability: “Given $B$, restrict your world to only outcomes where $B$ happens, then compute the probability of $A$ inside that restricted world.”

Two-way tables and conditional probability

Two-way tables make conditional probability much easier because you can see the “restricted world” as a row or column total.

Example 2: Conditional probability from a two-way table

A school surveys students about whether they play a sport.

Find the probability that a student plays a sport given that the student is in 8th grade.

Step 1: Restrict to 8th grade. The relevant total is $30$.

Step 2: Count favorable outcomes within that group. In 8th grade, $18$ play a sport.

So:

$P(\text{plays sport} \,|\, \text{8th grade}) = \frac{18}{30} = \frac{3}{5}$

Answer:

$\frac{3}{5}$

A common mistake is dividing by the grand total $60$ instead of the conditional total $30$.

Multiplication rule (including conditional form)

For any events $A$ and $B$:

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

If the events are independent, then $P(B|A) = P(B)$, giving:

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

Example 3: Dependent events (no replacement)

A jar contains $4$ green and $6$ yellow beads. You pick two beads without replacement. What is the probability both are green?

Step 1: Probability first bead is green.

$P(G_1) = \frac{4}{10}$

Step 2: Probability second bead is green given first was green. Now there are $3$ green left out of $9$ beads.

$P(G_2|G_1) = \frac{3}{9}$

Step 3: Multiply.

$P(G_1 \cap G_2) = \frac{4}{10} \times \frac{3}{9} = \frac{12}{90} = \frac{2}{15}$

Answer:

$\frac{2}{15}$

The classic error is using $\frac{4}{10}$ for both picks, which would incorrectly assume replacement (independence).

Exam Focus

• Typical question patterns:
  • Compute simple probabilities from equally likely outcomes or from data.
  • Use two-way tables to find conditional probabilities and interpret them in words.
  • Decide whether events are independent and compute combined probabilities (with and without replacement).
• Common mistakes:
  • Using the overall total instead of the conditional total in $P(A|B)$.
  • Assuming independence when the first event changes the sample space.
  • Forgetting to use the complement rule when it simplifies counting.