Math Study Guide for the PSAT/NMSQT Exam

General Information

The math section of the PSAT/NMSQT exam is divided into two portions. For one of them, you may use an approved **calculator (See “Extra Tips” at the end of this guide.). During the other section, you may not use a calculator and it will be necessary to do all calculations on your own.

Regardless of which section you’re working on, there are some concept understandings and skills necessary to correctly answer the questions. The material is taken from four areas of math, as described by the test creators:

The Heart of Algebra
Problem Solving and Data Analysis
Passport to Advanced Math
Additional Topics in Math (only 2 questions from this area)

You generally do not need to memorize formulas to do well on this test. Most formulas will be provided for you and you just need to know which formula to use.

Terms to Know

Ratio, proportion, similar, congruent, percentage, units, measurement, line, linear equation, slope, $x$-intercept, $y$-intercept, rise, run, quadratic, parabola, systems of equations, combination, substitution, elimination, graphing, absolute value, equality, inequality, open circle, closed circle, dotted line, bold line, perimeter, area, surface area, volume, circumference, radius, diameter, acute, right, obtuse, mean, median, mode

Ratio and Proportion

A ratio is a quantitative comparison between objects. It can be written as a fraction, using a colon, or with the word “to”. For example, the fraction $\frac{2}{3}$, represents a ratio of $2$ to $3$, and can also be written as $2 \text{:}3$.

As an example, consider a classroom of ten students that has six girls and four boys. We can describe the ratio of boys to girls as “$4 \text{:}6$”, “$\frac{4}{6}$”, or “$4$ to $6$”. We can also reduce the fraction and state that the ratio is $2 \text{:}3$. This means that for every $2$ boys there are $3$ girls, or that for every $3$ girls there are $2$ boys.

Closely linked with the concept of ratios is proportionality. Two objects are said to be in proportion with each other when a constant change in one of the objects is matched with a constant change in the other object.

As an example, consider a student that earns $\$15$ per hour. In the first hour, the student earns $\$15$, and in the second hour the student earns another $\$15$ for a total of $\$30$. Dividing the total wage earned by the total hours worked, a constant $\$15$ is earned per hour. Graphing this example would show a straight line. The slope of this line would be $\frac{15}{1}=15$.

This idea of proportionality extends to relationships between similar objects. Knowing that two objects are similar enables an equation to be made between them.

For example, consider four objects: $a$, $b$, $c$, and $d$. If the ratio of $a$ to $b$ is proportional to that of $c$ to $d$, the equation $\frac{a}{b}=\frac{c}{d}$ can be established. In the case of three known values and one unknown value, the one unknown value can be found by using cross-multiplication.

Answer:

We have the equation $\frac{a}{b}=\frac{c}{d}$. Suppose we know the values of $a$, $b$, and $c$. With cross-multiplication, we can solve for $d$:

\[\frac{a}{b}=\frac{c}{d}\] \[a \times d = b \times c\] \[ad = bc\] \[d = \frac{bc}{a}\]

If we knew the other three variables and had to solve for the remaining one, we would follow the exact same process.

Percentage

Percentages can be used for the comparison between two values. Percentages represent parts per 100. For example, $5\%$ is equal to $\frac{5}{100}$, or $0.05$; $100\%$ is equal to $\frac{100}{100}$, or $1.00$.

Often, it is useful to know what percentage one number is of another one. For example, what number is $30\%$ of $100$?

We can rephrase this question as an equation. We know that the relationship between the $\%$ and $100$ is equal to the relationship between the part and the whole: $\frac{\text{part}}{\text{whole}}=\frac{\%}{100}$. It can be seen from the problem that the unknown value is the $\%$, and that the known values are the part and the whole.

Setting up the equation:

\[\frac{30}{100}=\frac{x}{100}\]

Cross-multiply and solve to verify that $30$ is the answer.

Units of Measurement

Units of measurement provide a meaningful way to compare experimental data. Dimensional analysis is a useful method for converting between units of measurement. To use dimensional analysis, it is necessary to learn the standard units of measurement.

The three types of measurement to learn first are the names and symbols for measurements of mass, length, and time.

Mass has a standard measurement of kilograms, or $kg$; length is in meters, or $m$; time in seconds, or $s$.

Within and across these categories, is a description of these standard measurements as the same quantity of a different measurement.

For example, $1$ meter is the same as $100$ centimeters is the same as $1 \text{,}000$ millimeters; there are $1 \text{,}000$ meters in $1$ kilometer. The prefixes centi-, milli-, and kilo- are crucial to know and understand.

Because $1$ meter is the same as $100$ centimeters, we can rewrite their relationship as $1$ meter = $100$ centimeters. This is like saying that “there are $100$ centimeters in $1$ meter”, or that “there is $1$ meter for every $100$ centimeters”.

Mathematically, we may express this as a ratio or fraction, which is actually the conversion factor for the meter-centimeter relationship:

$\frac{\text{100 } cm}{1 \text{ m}}$ or $\frac{\text{1 } m}{\text{100 } cm}$

So if we need to convert $3$ meters to centimeters, we simply multiply $3 m$ with the appropriate conversion factor.

\[3\;m \cdot \frac{\text{100 } cm}{\text{1 } m} = \text {300 } cm\]

Take note that we chose the conversion factor $\frac{100 \;cm}{1 \;m}$ over the other because we wanted the $m$ units to cancel out and leave the $cm$ unit intact.

It is highly advised that the technique of dimensional analysis be mastered. The best way to develop familiarity with the unit names and values, as well as the technique of dimensional analysis is to practice many, many problems.