Addition, Subtraction, Multiplication, and Division

Everything in math is built on four operations. Each has an inverse that undoes it: subtraction undoes addition, and division undoes multiplication.

Order of Operations

When an expression has more than one operation, do them in this order:

1. Grouping: parentheses ( ), brackets [ ], fraction bars and roots, innermost first
2. Exponents
3. Multiplication and division, from left to right
4. Addition and subtraction, from left to right

You may know this as PEMDAS. The important detail: multiplication and division are on the same level, so you go left to right. The same is true of addition and subtraction.

Example.

$$8 + 2 \times (5 - 3)^2 = 8 + 2 \times 2^2 = 8 + 2 \times 4 = 8 + 8 = 16$$

Common mistake. $12 \div 3 \times 2$ is $4 \times 2 = 8$, not $12 \div 6 = 2$. Left to right!

Signs. Multiplying or dividing two numbers with the same sign gives a positive result; different signs give a negative result. Subtracting a negative is adding: $7 - (-3) = 10$.

Long Division, Remainders, and Fractions

To divide $157$ by $6$, work digit by digit:

• $6$ goes into $15$ two times ($12$), leaving $3$. Bring down the $7$ to make $37$.
• $6$ goes into $37$ six times ($36$), leaving $1$.

So $157 \div 6 = 26$ with remainder $1$. You can always check:

$$\text{dividend} = \text{divisor} \times \text{quotient} + \text{remainder}: \qquad 157 = 6 \times 26 + 1$$

The remainder is the part that didn't divide evenly, so you can write it as a fraction of the divisor:

$$\frac{157}{6} = 26\tfrac{1}{6}$$

Working with fractions

• Simplify by dividing top and bottom by a common factor: $\frac{18}{24} = \frac{3}{4}$.
• Add or subtract using a common denominator: $\frac{2}{3} + \frac{3}{4} = \frac{8}{12} + \frac{9}{12} = \frac{17}{12}$.
• Multiply straight across: $\frac{2}{3} \cdot \frac{5}{7} = \frac{10}{21}$.
• Divide by multiplying by the reciprocal: $\frac{3}{5} \div \frac{9}{10} = \frac{3}{5} \cdot \frac{10}{9} = \frac{30}{45} = \frac{2}{3}$.