Mathematics Study Guide for the ParaPro Assessment

Operations

There are 4 major operations in arithmetic: addition, subtraction, multiplication, and division and there are special names for the numbers used in these operations.

Addition—Consider the equation \(a + b = c\)
\(a\) and \(b\) are called addends and \(c\) is called the sum.

Subtraction—Consider the equation \(a - b = c\)
\(a\) is the minuend, \(b\) is the subtrahend, and \(c\) is the difference.

Multiplication—Consider the equation \(a \cdot b = c\)
\(a\) and \(b\) are called factors, and \(c\) is called the product.
Other ways to write this equation: \(ab = c,\; a\) x \(b = c,\; a \cdot b = c,\; \text{and}\; (a)(b) = c\)

Division—Consider the equation \(a \div b = c\)
\(a\) is the dividend, \(b\) is the divisor, and \(c\) is the quotient.
Other ways to write the equation: \(\frac{a}{b} = c, a/b = c\)

Inverse operations—Addition and subtraction are inverse operations because one “undoes” the other. Example: take a number. Now add \(3\) to it. Next, subtract \(3\) from that value. You should be at the number you started with. Similarly, multiplication and division are inverse operations of one another.

Inequalities—Sometimes values are not equal, but we still might want to show how they are related. For this, we use inequality symbols. Commonly-used inequality symbols are:

\(\lt\) less than
\(\le\) less than or equal to, or at most
\(\gt\) greater than
\(\ge\) greater than or equal to, or at least
\(\neq\) not equal to

Examples of true inequalities: \(3 \gt -1,\; 4 \le 4,\; 5 \neq 10,\; -2 \ge -7\)

Note: A combination of numbers, values, and symbols of mathematical operations is called an expression. Once the symbol “=” is involved, the expression becomes an equation. Expressions with inequality symbols are called inequalities.
Examples: \(4x - 3\) is an expression, \(4x - 3 = 2\) is an equation, and \(4x - 3 \le 2\) is an inequality.

Operations with Whole Numbers

Addition and subtraction with whole numbers are as easy as moving right (addition) or left (subtraction) on a number line.

Multiplication of whole numbers is just repeated addition. Example: \(3 \cdot 4\) is just \(3\) added to itself \(4\) times:

\(3 + 3 + 3 + 3 = 12\).

Division is just repeated subtraction. Example: \(15 \div 5\) means how many times I can subtract \(5\) from \(15\) to get to \(0\)?

\(15 - 5 - 5 - 5 = 0\), so the answer is \(3\).

Note: It is generally assumed that you know how to do these operations already, but you should seek extra practice if you’re unsure about any of them.

Operations with Fractions

A fraction (\(\frac{a}{b}\)) or ratio is just a comparison of two numbers by division. In this example \(a\) is the numerator and \(b\) is the denominator. If \(a \gt b\), like in \(\frac{5}{3}\), we say the fraction is improper. You can rewrite \(\frac{5}{3}\) as a combination of a whole number and a part: \(1 \frac{2}{3}\). We call numbers in this form mixed numbers.

It will be important to be able to go back and forth between the two forms. Here is how that’s done:

Improper to Mixed Number

To write \(\frac{14}{3}\) as a mixed number: \(3\) goes into \(14\) four times, with remainder \(2\), so the answer is \(4 \frac{2}{3}\).

Mixed Number to Improper

To write \(5 \frac{1}{4}\) as an improper fraction: \(4\) times \(5\) is \(20\), add \(1\) to it and you’ll get \(21\). The answer is \(\frac{21}{4}\).

Reciprocal

The reciprocal (or inverse) of a fraction \(\frac{a}{b}\) is the fraction \(\frac{b}{a}\).

Addition and Subtraction

Addition and subtraction of fractions rely on one simple concept: common multiples. A multiple of a number can be evenly divided by that number.

So, \(5,10, 15, …\) are all multiples of \(5\). Note the multiples of \(3\) are \(3, 6, 9, 12, 15, …\)
Since both \(5\) and \(3\) have the multiple \(15\), we would say \(15\) is a common multiple of \(3\) and \(5\).
Note: There are infinitely many common multiples of \(3\) and \(5\), \(15\) happens to be the least common multiple.

To add (or subtract) two fractions, follow these steps:

Ensure the fractions are not mixed numbers. (If so, change to improper fractions.)
Identify the least common multiple of the denominators (This is also called the least common denominator or LCD).
Rewrite each fraction with those denominators.
Add (or subtract) the top values and leave the denominator unchanged.

Example: \(3 \frac{2}{3} + \frac{3}{4}\)

\(3 \frac{2}{3} = \frac{11}{3}\)
so the new problem is \(\frac{11}{3} + \frac{3}{4}\).
\(3\) has multiples \(3, 6, 9,\) \(\bf{12}\) \(, 15, …\)
\(4\) has multiples \(4, 8, \bf{12}\) \(, 16, …\)
So, the LCD is \(12\).
Rewrite each with denominator \(12\): \(\frac{11}{3} \cdot \frac{4}{4} = \frac{44}{12}\) and \(\frac{3}{4} \cdot \frac{3}{3} = \frac{9}{12}\)
Perform the operation: \(\frac{44}{12} + \frac{9}{12} = \frac{53}{12}\)

Note: You will probably need to reduce this answer to lowest terms or simplify before you are finished. See the section on “Equivalent Forms of Numbers,” above.

Multiplication

Multiplication of fractions is simple. After ensuring no mixed numbers, just multiply the numerators and then multiply the denominators.

Example: \(\frac{4}{3} \cdot \frac{6}{7} = \frac{4 \cdot 6}{3 \cdot 7} = \frac{24}{21}\)

Division

To divide fractions, simply multiply by the reciprocal.

Example: \(\frac{2}{3} \div \frac{5}{9} = \frac{2}{3} \cdot \frac{9}{5} = \frac{18}{15}\)

Lowest Terms

Again, it is important to be able to simplify or reduce your answer to lowest terms. (Reference “Equivalent Forms of Numbers” above for a quick refresher on reducing fractions.) Usually, answers are expected to be in lowest terms, but be aware that an unreduced answer might still be correct if the reduced choice is not present.

Operations with Decimals

A decimal number is just another way to represent a fraction or mixed number using the base of our number system, \(10\). We’ve already gone over how to convert between the two.

Addition and Subtraction

To add (or subtract) decimals, simply line up the decimal points and add vertically.
Note: Lining up the decimal points in addition and subtraction is critical.

Example:

\[\begin{align} 12.34& \\ \underline{+\;\,1.25}& \\ 13.59& \\ \end{align}\]

Multiplication

To multiply decimals, multiply vertically like normal. Then count up the total number of digits to the right of the decimal point in the two factors. Count from the right that many digits in the product and place your decimal point there.

Example:

\[\begin{align} 2.3& \\ \underline{\times 1.4}& \\ (Note: There \;are \;2 \;digits \;to \;the\; right\; of \;the \;decimals) \quad 92& \\ \underline{+230} \\ 3.22 \end{align}\]

Note: we moved two places from the right and put the decimal point between \(3\) and \(2\).

Division

To divide decimals follow these steps:

Move the decimal point of the divisor all the way to the right. Count the digits along the way.
Now, move the decimal point of the dividend the same number of places to the right.
Divide (using long division).
Put the decimal point in the quotient directly above its location in the dividend.
Note: You can always verify your answer by doing multiplication.

Example: \(1.44 \div 1.2\)

\(1.2\) (the divisor) becomes \(12\). You had to move the decimal point one place to the right.
Move the decimal one place to the right in \(1.44\) (the dividend). \(1.44\) becomes \(14.4\)
Now, do \(14.4 \div 12\).
You should get \(1.2\)[.]
Verify: \(12 \cdot 1.2 = 14.4\)

Operations with Positive and Negative Numbers

A negative number is a real number less than \(0\). Negative numbers lie to the left of \(0\) on the number line. Reference this image:

Addition and Subtraction

In general, addition means you’re moving to the right on the number line, whereas subtraction means you’re moving left. For the example \(3 - 5\), start at \(3\) on the number line and move \(5\) to the left. You’ll end up at \(-2\).

Adding a Negative is just subtraction. Example: \(4 + (-2) = 4 - 2 = 2\)
Subtracting a Negative is just addition. Example: \(-3-(-8) = -3+8=5\)

Note: Addition is commutative (remember: \(a + b = b + a\)). In the last example \(- 3 + 8\), instead of thinking about starting at \(-3\) on the number line and moving \(8\) to the right, turn it into \(8+(-3)=8-3=5\).

Multiplication and Division

For multiplication and division, follow this rule: If the signs are the same, the answer is positive. Otherwise, the answer is negative.

Example: \((-3) \cdot (-5) = 15\) because the signs are the same (both negative).

Example: \(2 \cdot (-5) = -10\) because the signs aren’t the same (positive and negative).

The Order of Operations

If I give you the problem \(4 + 10 \div 2\), I should assume that you and I would simplify the expression to the same value. Let’s examine two ways to do this problem:

You:
\(4 + 10 \div 2\)
\(4 + 5\)
\(9\)

Me:
\(4 + 10 \div 2\)
\(14 \div 2\)
\(7\)

You got \(9\) and I got \(7\). So, who’s right? We’ve developed a system called the Order of Operations to solve this debate. I was wrong because I didn’t follow the order of operations.

Remember to follow the order of operations (PEMDAS):
Do everything in parentheses (P), left to right.
Evaluate any exponents (E), left to right.
Do all multiplication and division MD), in order, left to right.
Then do all addition and subtraction (AS), in order, from left to right.
A good way to remember is: “Please Excuse My Dear Aunt Sally”