PSAT 10 Math — Concepts, Skills, and Problem-Solving Methods

Algebraic Expressions, Equations, and Inequalities

Algebra is the language you use to describe relationships with symbols. On the PSAT 10, you’re rarely doing algebra “for its own sake”—you’re using it to model a situation, compare quantities, or predict an unknown. That’s why two habits matter more than memorizing tricks: (1) keeping track of what each variable represents, and (2) performing the same operation to both sides of an equation or inequality.

Expressions vs. equations (and why the difference matters)

An algebraic expression is a mathematical phrase like $3x+2$. It can be simplified or evaluated, but it is not “true” or “false” until you assign a value to $x$.

An equation sets two expressions equal, like $3x+2=11$. Equations have solutions—values of the variable(s) that make the statement true.

An inequality compares expressions using symbols like $<$, $\le$, $>$, or $\ge$, such as $3x+2<11$. Inequalities describe a set of solutions, often an interval.

A common PSAT theme is translating: a sentence, table, or context becomes an expression/equation/inequality.

Combining like terms and using distribution

Before solving, you often simplify.

• Like terms have the same variable part (same variables to the same powers). For example, $7x$ and $-2x$ are like terms; $7x$ and $7x^2$ are not.
• Distributive property means multiplying across parentheses: $a(b+c)=ab+ac$.

Why it matters: simplifying correctly reduces errors and reveals structure (like a common factor) that makes solving easier.

Example (simplify):

Simplify $4(2x-3)+5x$.

1. Distribute: $4(2x-3)=8x-12$.
2. Combine like terms: $8x-12+5x=13x-12$.

A frequent mistake is distributing only to the first term or losing a negative, like turning $4(2x-3)$ into $8x-3$.

Solving linear equations in one variable

A linear equation in one variable has the variable to the first power and no products of variables, such as $2x-7=9$. You solve by isolating the variable using inverse operations.

Example (solve):

Solve $5x-4=3x+10$.

1. Move variable terms together: subtract $3x$ from both sides.

$5x-3x-4=10$

2. Simplify:

$2x-4=10$

3. Add $4$ to both sides:

$2x=14$

4. Divide by $2$:

$x=7$

Notice you could also move constants first; the key is doing the same operation to both sides.

Solving linear inequalities (and the sign-flip rule)

Solving inequalities is like solving equations—until you multiply or divide by a negative.

If you multiply or divide both sides by a negative number, you must reverse the inequality sign. This rule exists because multiplying by a negative reverses order on the number line.

Example (solve an inequality):

Solve $-2x+5\ge 11$.

1. Subtract $5$:

$-2x\ge 6$

2. Divide by $-2$ and flip the sign:

$x\le -3$

A common mistake is forgetting to flip, leading to the exact opposite set of solutions.

Rearranging formulas (literal equations)

Sometimes you solve for a variable in a formula. This skill shows up in “solve for $x$” questions and in rearranging relationships from science-style contexts.

Example (solve for a variable):

Solve $A=\frac{1}{2}bh$ for $h$.

1. Multiply both sides by $2$:

$2A=bh$

2. Divide both sides by $b$ (assuming $b\ne 0$):

$h=\frac{2A}{b}$

The big idea: treat letters as numbers; isolate the target variable.

Unit conversions and dimensional reasoning

Many word problems become easier when you track units. If a quantity is $60\,\text{miles/hour}$ and you multiply by $2\,\text{hours}$, “hours” cancels and you get miles. This “unit cancellation” is a built-in error-check.

Exam Focus

• Typical question patterns:
  • Translate a phrase into an equation/inequality (e.g., “at least,” “no more than,” “increased by”).
  • Solve a linear equation with variables on both sides or with parentheses/fractions.
  • Solve an inequality and interpret the solution in context.
• Common mistakes:
  • Distributing incorrectly across parentheses, especially with negatives.
  • Forgetting to flip the inequality when dividing/multiplying by a negative.
  • Mixing up “less than” vs. “less than or equal to” in translations.

Linear Relationships and Graphs

A linear relationship is any situation where a constant change in one quantity produces a constant change in the other. Graphically, this shows up as a straight line. On the PSAT 10, linear models are everywhere: cost per item, speed, constant growth/decay, and comparing plans.

Slope as a rate of change

The slope measures how much $y$ changes when $x$ increases by $1$. It’s the “rate” in many real contexts.

If you have two points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$, then

$m=\frac{y_2-y_1}{x_2-x_1}$

Why it matters: slope connects graphs, tables, and contexts. If a taxi charges a base fee plus dollars per mile, the “dollars per mile” is the slope.

Example (slope from points):

Find the slope through $\left(2,3\right)$ and $\left(6,11\right)$.

$m=\frac{11-3}{6-2}=\frac{8}{4}=2$

A common error is reversing one difference but not the other (e.g., $\frac{3-11}{6-2}$), which changes the sign.

Forms of a line (and how to move between them)

Different forms highlight different features.

The intercepts are where the graph crosses axes:

• $y$-intercept: set $x=0$.
• $x$-intercept: set $y=0$.

Example (interpret slope-intercept):

If $y=3x+5$, then each increase of $1$ in $x$ increases $y$ by $3$, and when $x=0$ the output is $5$.

Graphing lines strategically

You can graph a line efficiently by:

1. Plotting the $y$-intercept $\left(0,b\right)$.
2. Using slope $m=\frac{\text{rise}}{\text{run}}$ to find a second point.
3. Drawing the line through the points.

If slope is negative, your “rise” is down when you move right.

Parallel and perpendicular lines

Two lines are:

• Parallel if they have the same slope.
• Perpendicular if their slopes are negative reciprocals.

If one slope is $m$, a perpendicular slope is

$-\frac{1}{m}$

(when $m\ne 0$). Horizontal lines have slope $0$ and are perpendicular to vertical lines (which have undefined slope).

Example (perpendicular slope):

A line with slope $\frac{2}{3}$ has a perpendicular slope

$-\frac{3}{2}$

Linear modeling from tables and word problems

If a situation has constant rate, you model it with

$y=mx+b$

• $m$ is “per 1” rate (dollars per item, miles per hour, etc.).
• $b$ is the starting value when $x=0$ (base fee, initial amount).

Example (build a model):

A streaming service charges $\$8$ plus $\$2$ per movie. Let $x$ be movies and $y$ be total cost.

• Rate is $2$ dollars per movie, so $m=2$.
• Base fee is $8$, so $b=8$.

$y=2x+8$

A frequent mistake is swapping the roles of $x$ and $y$ or misidentifying the base fee as the slope.

Exam Focus

• Typical question patterns:
  • Find slope from a graph, two points, or a table.
  • Interpret slope and intercept in context (what do they mean?).
  • Identify parallel/perpendicular relationships or write the equation of a line.
• Common mistakes:
  • Confusing $b$ (intercept) with “starting time” rather than the value at $x=0$.
  • Using slope formula with mismatched subtraction order.
  • Treating vertical lines as having slope $0$ (they are undefined).

Systems of Linear Equations and Inequalities

A system is a set of equations (or inequalities) that must be true at the same time. Systems are how you model “two constraints at once,” like comparing plans, mixing items, or finding a point that lies on two lines.

What a solution to a system means

For two linear equations in $x$ and $y$, a solution is an ordered pair $\left(x,y\right)$ that makes both equations true. Graphically, it’s where the lines intersect.

Three possibilities for two lines:

• One intersection point: one solution.
• Parallel distinct lines: no solution.
• Same line (equivalent equations): infinitely many solutions.

Solving by substitution

Use substitution when one equation is already solved for a variable (or easily can be).

Example (substitution):

Solve

$y=2x+1$

$x+y=10$

Substitute $y=2x+1$ into the second equation:

$x+(2x+1)=10$

$3x+1=10$

$3x=9$

$x=3$

Then

$y=2(3)+1=7$

So the solution is $\left(3,7\right)$.

A common mistake is substituting into the wrong place or forgetting parentheses: $x+2x+1$ is correct because $y$ is the entire expression $2x+1$.

Solving by elimination (linear combination)

Use elimination when you can add/subtract equations to cancel a variable. You may multiply one or both equations first.

Example (elimination):

Solve

$2x+y=11$

$2x-y=1$

Add the equations to eliminate $y$:

$(2x+y)+(2x-y)=11+1$

$4x=12$

$x=3$

Substitute into $2x+y=11$:

$6+y=11$

$y=5$

Solution: $\left(3,5\right)$.

A common error is adding when you should subtract (or vice versa), or multiplying incorrectly when preparing to eliminate.

Systems of inequalities (solution regions)

A system of inequalities describes a region on the coordinate plane. Each inequality creates a half-plane; the system solution is the overlap.

Key ideas:

• Boundary line is solid if inequality includes equality (like $\le$ or $\ge$).
• Boundary line is dashed if strict (like $<$ or $>$).
• To test which side to shade, use a test point (often $\left(0,0\right)$ if it’s not on the boundary).

You won’t always have to draw perfectly; often you interpret a graph or choose which region matches constraints.

Exam Focus

• Typical question patterns:
  • Solve a system and interpret the meaning of the intersection.
  • Decide whether a system has one, none, or infinitely many solutions.
  • Identify a feasible region given real constraints (budget, capacity, minimums).
• Common mistakes:
  • Forgetting to substitute back to find the second variable.
  • Sign errors when adding/subtracting equations.
  • Using dashed vs. solid boundaries incorrectly for inequalities.

Functions: Inputs, Outputs, and Notation

A function is a rule that assigns each input exactly one output. Functions matter because they’re a structured way to model how one quantity depends on another—and PSAT questions often ask you to interpret function notation, compare representations, or identify key features from a graph.

Function notation and meaning

If $f$ is a function, then $f(x)$ means “the output when the input is $x$.” It is not multiplication; it’s naming the output.

For example, if

$f(x)=2x-3$

then

$f(5)=2(5)-3=7$

A common mistake is reading $f(x)$ as $f\times x$ or treating $f(5)$ as a variable.

Domain and range

• The domain is the set of allowable inputs.
• The range is the set of possible outputs.

Domain can be restricted by context (you can’t buy $-3$ tickets) or by algebra (you can’t divide by zero).

Example (domain restriction):

If

$g(x)=\frac{1}{x-4}$

then $x\ne 4$, because division by $0$ is undefined.

Interpreting graphs of functions

Key features you’re often asked to read:

• Intercepts: where output is $0$ or input is $0$.
• Increasing/decreasing: where outputs rise or fall as inputs increase.
• Rate of change: for a line, it’s constant slope; for curves, it changes.

Linear functions as the foundation

Linear functions are the simplest functions: constant rate of change.

$f(x)=mx+b$

Understanding these deeply helps you later with systems, inequalities, and modeling.

Exam Focus

• Typical question patterns:
  • Evaluate $f(a)$ from a formula, table, or graph.
  • Identify domain restrictions from an expression or from context.
  • Interpret what $f(x)$ represents in a real situation.
• Common mistakes:
  • Confusing $f(x)$ with multiplication.
  • Mixing up domain and range.
  • Forgetting contextual restrictions (e.g., time cannot be negative).

Exponents, Radicals, and Exponential Relationships

Exponents and radicals show up in growth models, scientific notation, geometry formulas, and algebra manipulation. The PSAT 10 emphasizes understanding exponent rules and recognizing exponential vs. linear behavior.

Exponent rules (what they mean, not just how to do them)

Exponents represent repeated multiplication. This idea explains the rules.

If $a\ne 0$ and $m$ and $n$ are integers:

$a^m\cdot a^n=a^{m+n}$

$\frac{a^m}{a^n}=a^{m-n}$

$(a^m)^n=a^{mn}$

A classic mistake is adding exponents when multiplying different bases. For example, $2^3\cdot 5^3$ is not $10^6$; instead,

$2^3\cdot 5^3=(2\cdot 5)^3=10^3$

Negative exponents

A negative exponent means reciprocal:

$a^{-n}=\frac{1}{a^n}$

So

$2^{-3}=\frac{1}{2^3}=\frac{1}{8}$

Mistake to avoid: thinking $a^{-n}=-a^n$.

Radicals and rational exponents

A square root reverses squaring:

$\sqrt{a^2}=|a|$

That absolute value matters because both $3$ and $-3$ square to $9$.

Rational exponents connect roots and powers:

$a^{\frac{1}{2}}=\sqrt{a}$

$a^{\frac{1}{3}}=\sqrt[3]{a}$

$a^{\frac{m}{n}}=\left(\sqrt[n]{a}\right)^m$

Exponential growth and decay

An exponential function changes by a constant factor over equal input intervals.

A common form is

$y=a\cdot b^x$

• $a$ is the initial value (when $x=0$, because $b^0=1$).
• $b$ is the growth factor per unit of $x$.

If $b>1$, it’s growth; if $0<b<1$, it’s decay.

Example (interpret):

If $y=200\cdot 1.05^t$, then $200$ is the starting amount and each unit of $t$ multiplies the amount by $1.05$ (a $5\%$ increase).

A frequent PSAT confusion is mixing linear percent increase (adding the same amount) with exponential percent increase (multiplying by the same factor).

Exam Focus

• Typical question patterns:
  • Simplify expressions using exponent rules, including negative exponents.
  • Interpret an exponential model (identify initial value and growth factor).
  • Compare linear vs. exponential change given a table or description.
• Common mistakes:
  • Treating $a^{-n}$ as negative instead of reciprocal.
  • Applying exponent rules to sums (e.g., $\left(a+b\right)^2\ne a^2+b^2$).
  • Confusing percent points (linear) with percent growth factor (exponential).

Polynomials, Factoring, and Quadratic Functions

“Advanced math” on the PSAT 10 often means working with polynomials—especially quadratics. The point isn’t just to expand and factor; it’s to connect algebraic form to key features like zeros (x-intercepts), maximum/minimum values, and symmetry.

Polynomials and their structure

A polynomial is a sum of terms like $a_k x^k$ where exponents are nonnegative integers. Examples include $3x^2-5x+1$ or $x^4-2x^2+7$.

The degree is the highest exponent. Degree matters because it influences end behavior and how many turning points/zeros are possible, but PSAT questions usually focus on quadratic (degree $2$).

Multiplying polynomials

You distribute each term. FOIL is a special case for two binomials.

Example (multiply):

Multiply $(x+4)(x-3)$.

Distribute:

$(x+4)(x-3)=x(x-3)+4(x-3)$

$=x^2-3x+4x-12$

$=x^2+x-12$

Mistake to avoid: multiplying only the first terms and last terms and skipping the middle products.

Factoring as “undoing multiplication”

Factoring rewrites a polynomial as a product. It matters because products equal zero are easier to solve (via the zero-product property).

Greatest common factor (GCF)

First, look for a factor shared by all terms.

Example (GCF):

Factor $6x^2+9x$.

GCF is $3x$:

$6x^2+9x=3x(2x+3)$

Factoring quadratics of the form $x^2+bx+c$

You seek two numbers that multiply to $c$ and add to $b$.

Example:

Factor $x^2+5x+6$.

Numbers multiply to $6$ and add to $5$ are $2$ and $3$:

$x^2+5x+6=(x+2)(x+3)$

Factoring $ax^2+bx+c$ when $a\ne 1$

One reliable method is grouping after finding factors of $a\cdot c$.

Example (grouping):

Factor $2x^2+7x+3$.

1. Compute $a\cdot c=2\cdot 3=6$.
2. Find two numbers that multiply to $6$ and add to $7$: $6$ and $1$.
3. Split the middle term:

$2x^2+6x+x+3$

4. Group:

$(2x^2+6x)+(x+3)=2x(x+3)+1(x+3)$

5. Factor common binomial:

$2x^2+7x+3=(2x+1)(x+3)$

Mistake to avoid: splitting into numbers that multiply correctly but don’t add correctly.

Solving quadratics by factoring and the zero-product property

If

$(x+p)(x+q)=0$

then $x+p=0$ or $x+q=0$.

Example (solve):

Solve $x^2+x-12=0$.

Factor:

$x^2+x-12=(x+4)(x-3)$

Set each factor to zero:

$x+4=0$ gives $x=-4$.

$x-3=0$ gives $x=3$.

So the solutions are $x=-4$ and $x=3$.

Vertex form and the meaning of the vertex

A quadratic can be written as

$y=a(x-h)^2+k$

The vertex is $\left(h,k\right)$. If $a>0$ the parabola opens up (vertex is a minimum). If $a<0$ it opens down (vertex is a maximum).

Even if you don’t do full “completing the square” often on PSAT, you do need to interpret vertex form when it appears.

Example (interpret):

For $y=-2(x-3)^2+5$, the vertex is $\left(3,5\right)$ and the parabola opens downward, so the maximum value is $5$.

Using the discriminant idea (when solutions exist)

If you see a quadratic in standard form $ax^2+bx+c=0$, the expression

$b^2-4ac$

tells you about the number of real solutions: positive means two real solutions, zero means one real (repeated) solution, negative means none. You don’t always need to compute exact solutions to use this reasoning.

Exam Focus

• Typical question patterns:
  • Factor a quadratic and use it to solve an equation or find x-intercepts.
  • Match a quadratic’s form to features: zeros, vertex, maximum/minimum.
  • Interpret a quadratic model in context (height vs. time, revenue vs. price).
• Common mistakes:
  • Sign errors when factoring (especially with negative constants).
  • Forgetting the zero-product property requires the expression to equal $0$.
  • Misreading vertex form $a(x-h)^2+k$ as vertex $\left(-h,k\right)$ instead of $\left(h,k\right)$.

Rational Expressions and Equations (Fractions with Variables)

A rational expression is a fraction with polynomials in the numerator and/or denominator, like $\frac{x+1}{x-2}$. These show up in rate problems, proportional reasoning, and algebra manipulation.

Restrictions: values that make the expression undefined

You can’t divide by zero, so you must exclude inputs that make the denominator zero.

For $\frac{x+1}{x-2}$, you must have $x\ne 2$.

This matters on PSAT questions that ask for domain, simplification, or solutions—an algebraic cancellation may hide a restriction.

Simplifying by factoring and canceling

You can cancel only common factors, not terms.

Example (simplify):

Simplify

$\frac{x^2-9}{x-3}$

Factor the numerator as a difference of squares:

$x^2-9=(x-3)(x+3)$

Then

$\frac{(x-3)(x+3)}{x-3}=x+3$

But the original expression is undefined at $x=3$, so the simplified form is $x+3$ with the restriction $x\ne 3$.

A common mistake is canceling across addition, like trying to cancel the $x$ in

$\frac{x+2}{x+5}$

which is not valid.

Solving rational equations carefully

Typical approach:

1. State restrictions.
2. Multiply both sides by the least common denominator to clear fractions.
3. Solve the resulting equation.
4. Check solutions against restrictions.

Example (solve):

Solve

$\frac{1}{x-1}=2$

Restriction: $x\ne 1$.

Multiply both sides by $x-1$:

$1=2(x-1)$

$1=2x-2$

$3=2x$

$x=\frac{3}{2}$

Since $\frac{3}{2}\ne 1$, it’s valid.

Exam Focus

• Typical question patterns:
  • Simplify a rational expression by factoring.
  • Identify excluded values (domain restrictions).
  • Solve a rational equation from a word problem or algebraic setup.
• Common mistakes:
  • Canceling terms instead of factors.
  • Forgetting to exclude denominator-zero values.
  • Not checking for extraneous solutions after clearing denominators.

Ratios, Proportions, Percents, and Units (Problem Solving)

This domain is about reasoning with multiplicative relationships. Many PSAT 10 problems that “look hard” are actually ratio/percent problems in disguise.

Ratios and rates

A ratio compares quantities by division. A rate is a ratio with different units, like $\frac{\text{miles}}{\text{hours}}$.

When you write a ratio, be explicit about what’s being compared. If a recipe uses $2$ cups of flour for $3$ cups of sugar, then flour-to-sugar is $\frac{2}{3}$, but sugar-to-flour is $\frac{3}{2}$. Many mistakes come from flipping the ratio.

Proportions and scaling

A proportion is an equation stating two ratios are equal, often used when scaling up/down.

Example (scale):

If $3$ notebooks cost $\$7.50$, what is the cost of $8$ notebooks at the same price per notebook?

Unit rate:

$\frac{7.50}{3}=2.50$

Then

$8\times 2.50=20.00$

So the cost is $20.00$.

Percent as “per 100” and percent change

Percent means per $100$:

$p\% = \frac{p}{100}$

A percent increase of $r\%$ means multiply by

$1+\frac{r}{100}$

A percent decrease of $r\%$ means multiply by

$1-\frac{r}{100}$

Example (percent decrease):

A jacket price $\$80$ is discounted by $25\%$.

Multiply by $1-0.25=0.75$:

$80\times 0.75=60$

New price: $60$.

A common error is subtracting $25$ instead of $25\%$ of the price.

Weighted averages and mixtures

A weighted average occurs when different parts contribute unequally.

If you have values $v_1$ and $v_2$ with weights $w_1$ and $w_2$, the weighted average is

$\frac{w_1v_1+w_2v_2}{w_1+w_2}$

Example (class grade):

Tests are $70\%$ of a grade and homework is $30\%$. If test average is $84$ and homework average is $92$, then

$0.70\times 84+0.30\times 92=58.8+27.6=86.4$

Overall: $86.4$.

Exam Focus

• Typical question patterns:
  • Compute unit rates, convert units, and interpret rates.
  • Find percent increase/decrease and final amounts (discount, tax, tip).
  • Use proportions for scale drawings, recipes, or similar figures.
• Common mistakes:
  • Reversing ratios (part-to-whole vs. part-to-part).
  • Treating percent change as additive rather than multiplicative.
  • Mixing units (hours with minutes, dollars with cents) without converting.

Geometry and Measurement

Geometry on the PSAT 10 is less about proofs and more about using properties and formulas correctly in context. The key is understanding what each formula measures and what information you need to use it.

Angles, lines, and basic relationships

Important angle facts:

• A straight line is $180$ degrees.
• A full circle is $360$ degrees.
• Vertical angles are equal.
• Adjacent angles on a line sum to $180$.

If two lines are cut by a transversal, corresponding angles are equal when the lines are parallel, and alternate interior angles are equal.

Mistake to avoid: assuming lines are parallel just because a diagram “looks” parallel. Use given information.

Triangles: sum of angles and special right-triangle facts

In any triangle, interior angles sum to $180$ degrees.

A right triangle has a $90$ degree angle. For right triangles, the Pythagorean Theorem relates side lengths:

$a^2+b^2=c^2$

where $c$ is the hypotenuse (the side opposite the right angle).

Example (Pythagorean Theorem):

A right triangle has legs $6$ and $8$. Find the hypotenuse.

$6^2+8^2=c^2$

$36+64=c^2$

$100=c^2$

$c=10$

Mistake to avoid: using the theorem on a non-right triangle or labeling the hypotenuse incorrectly.

Area and perimeter (why “square units” matter)

Perimeter measures distance around a shape (linear units). Area measures surface covered (square units). Mixing these is a common conceptual error.

Key area formulas:

• Rectangle: $A=lw$
• Triangle: $A=\frac{1}{2}bh$
• Circle: $A=\pi r^2$

Circumference of a circle:

$C=2\pi r$

Example (circle):

If $r=5$, then

$A=\pi\times 25=25\pi$

and

$C=2\pi\times 5=10\pi$

Volume

Volume measures 3D space (cubic units).

Common volumes:

• Rectangular prism: $V=lwh$
• Cylinder: $V=\pi r^2h$

Interpreting volume problems often requires identifying radius vs. diameter and using consistent units.

Similarity and scale factor

Two figures are similar if corresponding angles are equal and corresponding side lengths are proportional.

If the scale factor from a smaller figure to a larger one is $k$:

• Side lengths multiply by $k$.
• Areas multiply by $k^2$.
• Volumes multiply by $k^3$.

This is a powerful PSAT concept: many questions hide the squared relationship for area.

Example (area scaling):

If a triangle’s side lengths are doubled (so $k=2$), its area becomes

$2^2=4$

times as large.

Exam Focus

• Typical question patterns:
  • Use Pythagorean Theorem to find a missing side length.
  • Compute area/circumference/volume from given dimensions.
  • Use similarity to find unknown lengths or compare areas.
• Common mistakes:
  • Confusing radius and diameter in circle problems.
  • Forgetting area scales with $k^2$ (not $k$).
  • Using perimeter formulas when the question asks for area (or vice versa).

Coordinate Geometry and Intro Trigonometry

Coordinate geometry connects algebra and geometry: shapes live on the coordinate plane, and you use algebraic tools (like slope and distance) to analyze them.

Distance and midpoint

The distance between $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ is

$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

This comes from the Pythagorean Theorem by treating horizontal and vertical changes as legs.

The midpoint is

$\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$

Example (distance):

Find distance from $\left(-1,2\right)$ to $\left(3,5\right)$.

$d=\sqrt{(3-(-1))^2+(5-2)^2}$

$d=\sqrt{4^2+3^2}$

$d=\sqrt{16+9}$

$d=5$

Mistake to avoid: forgetting to square the differences or mixing up signs.

Slope as a geometric tool

Slope helps you determine if segments are parallel or perpendicular (same slope or negative reciprocal). In coordinate geometry problems, you might be asked to prove a shape is a rectangle by showing adjacent sides are perpendicular and opposite sides are parallel.

Right-triangle trigonometry (basic)

Trig on PSAT 10 is usually limited to interpreting sine, cosine, and tangent in a right triangle.

For an angle $\theta$ in a right triangle:

$\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}$

$\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}$

$\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$

Why it matters: it’s another way to relate sides when you know an angle.

Example (tangent):

In a right triangle, relative to angle $\theta$, the opposite side is $9$ and adjacent side is $12$. Then

$\tan(\theta)=\frac{9}{12}=\frac{3}{4}$

Common confusion: mixing up opposite and adjacent; drawing the triangle and marking the angle helps.

Exam Focus

• Typical question patterns:
  • Find distance/midpoint between two points.
  • Use slope to classify lines or show relationships in figures.
  • Compute a trig ratio from a right triangle or interpret a given ratio.
• Common mistakes:
  • Sign errors in distance and midpoint calculations.
  • Using the wrong trig ratio for the given sides.
  • Confusing “relative to $\theta$” with the triangle’s absolute orientation.

Statistics, Data Analysis, and Probability

PSAT 10 data questions test whether you can read, summarize, and interpret information—not just calculate. The most important skill is connecting a numerical result back to the context: what does the statistic say, and what doesn’t it say?

Center: mean and median

• The mean is the average: sum divided by number of values.
• The median is the middle value when ordered (or average of the two middle values if there’s an even number).

Mean is sensitive to outliers; median is resistant. This matters when a single extreme value skews the average.

Example (outlier effect):

Data: $2,3,3,4,20$.

Mean:

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

Median is $3$.

The mean suggests a “typical” value larger than most data points because of $20$.

Spread: range and interquartile range (IQR)

• Range is max minus min.
• Interquartile range is $Q_3-Q_1$, measuring spread of the middle $50\%$.

PSAT questions often use box plots; you should know that the box spans $Q_1$ to $Q_3$ and the line inside is the median.

Two-way tables and relative frequency

A two-way table organizes counts for two categorical variables (like “plays sport” vs. “does not,” and “grade 9” vs. “grade 10”). You may be asked for:

• Joint frequency (a cell count)
• Marginal frequency (row/column total)
• Relative frequency (a proportion), sometimes conditional

A conditional relative frequency divides within a row or column total. The phrase “given that” signals conditional.

Example (conditional):

If $30$ out of $120$ students are in grade $10$ and play a sport, and there are $50$ grade $10$ students total, then

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

Mistake to avoid: dividing by the grand total when the question asks “given grade 10.”

Scatterplots and linear association

Scatterplots show relationships between two quantitative variables. Key ideas:

• Direction: positive or negative association.
• Strength: how tightly points cluster.
• Outliers: points far from the trend.

A line of best fit is a linear model that approximates the trend. On PSAT, you interpret slope and use the line to predict, while remembering predictions are approximate.

Basic probability

Probability is a number from $0$ to $1$.

If outcomes are equally likely:

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

For complements:

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

For “and” with independent events:

$P(A\text{ and }B)=P(A)\times P(B)$

“Independent” means one event doesn’t change the probability of the other.

Example (independent):

Flip a fair coin and roll a fair die. Probability of heads and rolling $6$:

$\frac{1}{2}\times\frac{1}{6}=\frac{1}{12}$

A common mistake is adding instead of multiplying for “and.”

Exam Focus

• Typical question patterns:
  • Interpret mean/median and choose which better describes “typical” for skewed data.
  • Compute conditional relative frequency from a two-way table.
  • Interpret scatterplots: trend, outliers, and meaning of slope in a fitted line.
• Common mistakes:
  • Using the wrong denominator for conditional probability.
  • Treating correlation as proof of causation (association does not imply cause).
  • Misreading a box plot (confusing whiskers with quartiles).

Modeling and Multi-Step Word Problems (Putting It All Together)

Modeling is the PSAT 10’s “glue skill”: you translate a real situation into math, solve, then interpret the result. The hardest part is usually the first and last step—choosing a model and stating what the answer means.

A reliable modeling process

When a problem feels messy, use this structure:

1. Define variables with units (for example, let $x$ be hours).
2. Write relationships (equations/inequalities) that match the wording.
3. Solve using algebra.
4. Check reasonableness (sign, size, units).
5. Interpret in a sentence tied to the context.

Translating common phrases into math

• “At least” means $\ge$.
• “At most” means $\le$.
• “No more than” means $\le$.
• “Increased by” means add.
• “Decreased by” means subtract.
• “Is” means equals.

Many incorrect answers come from translating one phrase incorrectly even when the algebra afterward is perfect.

Example: linear modeling with constraints

A gym charges $\$15$ to join and $\$10$ per month. You have at most $\$85$.

Let $m$ be the number of months, and total cost $C$ is

$C=10m+15$

“At most $85$” gives

$10m+15\le 85$

Solve:

$10m\le 70$

$m\le 7$

Interpretation: you can afford up to $7$ months.

A typical mistake is writing $10m+15=85$, which finds a boundary case but ignores “at most.”

Example: system modeling

A school sells adult tickets for $\$8$ and student tickets for $\$5$. If $120$ tickets were sold for $\$810$ total, how many of each?

Let $a$ be adult tickets and $s$ be student tickets.

Count equation:

$a+s=120$

Revenue equation:

$8a+5s=810$

Substitute $s=120-a$ into revenue:

$8a+5(120-a)=810$

$8a+600-5a=810$

$3a=210$

$a=70$

Then

$s=120-70=50$

So $70$ adult and $50$ student tickets.

Reasonableness check: $70\times 8=560$ and $50\times 5=250$, total $810$.

Example: quadratic modeling idea (maximum/minimum)

Quadratics often model situations with a peak or a lowest point. If a quadratic opens downward, it has a maximum at its vertex.

If revenue is modeled by

$R(p)=-p^2+20p$

then it opens downward (negative leading coefficient). The maximum occurs at the vertex. You might be given the vertex directly or asked to rewrite/recognize it. Even without rewriting, you can interpret that revenue rises then falls as $p$ increases.

A common mistake is assuming “bigger input gives bigger output,” which is false for downward-opening quadratics.

Exam Focus

• Typical question patterns:
  • Build an equation/inequality from a scenario and interpret the solution.
  • Combine domains: ratios with linear equations, or systems with pricing.
  • Decide which model type fits (linear vs. exponential vs. quadratic) from description or table.
• Common mistakes:
  • Defining variables unclearly and then mixing what they represent.
  • Solving correctly but giving the wrong quantity (answering $x$ when the question asks for $y$).
  • Ignoring context constraints (negative time, fractional people/items) when interpreting.