Core math concepts

Order of operations

When an expression has multiple operations (like addition, multiplication, and division), you need a consistent order to solve it correctly. This prevents different people from getting different answers.

The order of operations is often remembered with the acronym PEMDAS:

P - Parentheses
E - Exponents
M & D - Multiplication and division
A & S - Addition and subtraction

Multiplication and division are the same “level,” so you work left to right and do whichever comes first. The same is true for addition and subtraction.

Follow this order until the expression is simplified. Let’s work through an example.

Solve.

$$(3+2÷2\times 6-4)\times 3$$

Step 1: Look at the expression and identify which operations appear: parentheses, addition, division, multiplication, subtraction, and another multiplication.

Step 2: Parentheses come first in PEMDAS, so focus on what’s inside:

$$(3+2÷2\times 6-4)$$

Step 3: Inside the parentheses, division and multiplication are the same level — work left to right. Divide first:

$$2÷2=1$$

$$(3+1\times 6-4)$$

Step 4: Now multiply:

$$1\times 6=6$$

$$(3+6-4)$$

Step 5: Add and subtract from left to right:

$$3+6=9$$

$$9-4=5$$

Step 6: Finally, multiply the result by $3$:

$$5\times 3=15$$

Knowledge check: Solve

$$(8+7)\times \frac{40}{5}+2$$

(spoiler)

$$122$$

Prime numbers

Definitions

Prime number

A whole number greater than $1$ that is only divisible by $1$ and itself.

Composite number

A whole number greater than $1$ that has more than two factors.

Examples of prime numbers are $2,3,5,7,$ and $11$. $6$ is composite because its factors are $1,2,3,$ and $6$.

Knowing prime numbers is useful for factoring because it helps you recognize when a number (or polynomial) is fully factored.

Inequalities

An inequality compares two values that are not necessarily equal. Instead of showing that two sides are the same, it shows that one side is larger or smaller than the other.

When working with inequalities, start by identifying the symbol:

• $<$ means less than
• $>$ means greater than
• Adding the line underneath means “or equal to,” so $<$ becomes $\le$, and $>$ becomes $\ge$

Example:

$$3<5$$

This means 3 is less than 5.

Sidenote

Memory trick

Sometimes it’s difficult to remember the exact name of the inequality symbol. A helpful trick is that the open side faces the larger number.

You solve most inequalities the same way you solve equations, with one key exception: if you multiply or divide by a negative number, you must flip the inequality sign.

Your goal is to get the variable by itself, just like solving an equation.

Example:

$$-2x>12$$

Start by isolating $x$. The $-2$ is multiplied by $x$, so divide both sides by $-2$.

$$\frac{\cancel{-2}x}{\cancel{-2}}>\frac{12}{-2}$$

Because you divided by a negative number, flip the inequality sign to keep the inequality true.

$$x<-6$$

Knowledge check: Solve

$-5(2x+6)+6\ge 136$

(spoiler)

$x\le -16$

Conversions

Because you can’t use a calculator on the ASVAB, you’ll want to be comfortable converting between decimals, percentages, and fractions by hand. Understanding place value will help you see how decimals relate to fractions and percents.

All three forms— decimal, fraction, and percent—represent the same value, just written in different ways.

For example:

$$0.51=\frac{51}{100}=51\%$$

• $0.51$ means 51 hundredths, which is why it can be written as $\frac{51}{100}$
• A percent means “out of 100,” so $\frac{51}{100}$ is the same as $51\%$

Let’s break down each type of conversion.

Decimal to percent

Percent means “per $100$,” or “out of $100$.” To convert a decimal to a percent, multiply by $100$.

A quick way to do this is to move the decimal two places to the right.

$$0.25\times 100=25\%$$

• $0.25$ means 25 hundredths
• Multiplying by $100$ turns it into 25 out of 100
• This is why it becomes $25\%$

To go back from percent to decimal, divide by $100$ (move the decimal two places to the left): $25\%=0.25$ and $7\%=0.07$.

Decimal to fraction

Decimals can be written as fractions because they represent parts of a whole based on place value. Use place value to determine the denominator.

The position of the decimal tells you what the denominator should be:

• One place to the right = tenths → denominator is $10$
• Two places to the right = hundredths → denominator is $100$
• Three places to the right = thousandths → denominator is $1000$

If the decimal is in the tenths place (one place value to the right of the decimal), write it over $10$:

$$0.2=\frac{2}{10}$$

This simplifies to $\frac{1}{5}$.

If the decimal is in the hundredths place (two places to the right of the decimal), write it over $100$:

$$0.25=\frac{25}{100}$$

This simplifies to $\frac{1}{4}$.

If the decimal is in the thousandths place (three places to the right of the decimal), write it over $1000$:

$$0.255=\frac{255}{1000}$$

This simplifies to $\frac{51}{200}$ (divide the top and bottom by $5$).

The number of digits after the decimal tells you how many zeros go in the denominator. Answer choices are usually given in lowest terms, so simplify your fraction once you’ve written it.

To convert a percent to a fraction, write it over $100$ and simplify: $25\%=\frac{25}{100}=\frac{1}{4}$.

Knowledge check: Write $0.47$ as a fraction.

(spoiler)

$$\frac{47}{100}$$

Fraction to percent

To convert a fraction to a percent, first convert it to a decimal by dividing the numerator by the denominator. Then convert the decimal to a percent by multiplying by $100$. This works because a fraction represents division.

If the denominator divides evenly into $100$ (like $2,4,5,10,20,25,$ or $50$), you can skip the division and just scale the fraction up to a denominator of $100$: $\frac{7}{20}=\frac{7\times 5}{20\times 5}=\frac{35}{100}=35\%$. This shortcut is usually faster than long division when it applies.

Sidenote

Long division

Don’t remember how to do long division? Not to worry! We’ll cover that in the next section, Pre-Algebra.

Example:

Start with the fraction:

$$\frac{1}{4}$$

Divide the numerator by the denominator:

$$1÷4=0.25$$

Convert the decimal to a percent:

$$0.25\times 100=25\%$$

So,

$$\frac{1}{4}=0.25=25\%$$

Because $4$ divides evenly into $100$, the shortcut works too: $\frac{1}{4}=\frac{25}{100}=25\%$.

Knowledge check

Convert $\frac{3}{5}$ to a percent.

(spoiler)

$60\%$