Fractions, Signs and Order of Operations, SAT math lesson

Picture a bar 5 units long and cut off three and a half units. What is left is one and a half units, so $5 - \frac{7}{2} = \frac{3}{2}$ . Arithmetic like this sits inside almost every SAT question, and one sign slip or one step done in the wrong order is enough to land on a wrong answer choice. The rules below fix how an expression is read, so that you get the value the question writer meant.

See also Powers, Roots, and Signs in Foundations for Calculus.

Key terms

Negative number

Definition A negative number is a real number less than $0$ . For every real number $a$ there is exactly one number $-a$ , its opposite, with $a + (-a) = 0$ . Subtracting means adding the opposite,

a - b = a + (-b).

For products and quotients of nonzero numbers, two numbers with the same sign give a positive result and two numbers with opposite signs give a negative result. In particular $-(-a) = a$ and $(-1)a = -a$ .

In plain words A negative number sits to the left of $0$ on the number line. Subtracting a negative moves you to the right, and multiplying by a negative flips the sign.

Example Work out $-3 - (-8) + (-4)(-2)$ . Subtracting $-8$ is adding $8$ , and the product of two negatives is positive, so

\begin{aligned} &-3 - (-8) + (-4)(-2) \\[0.4em] &\quad = -3 + 8 + 8 = 13. \end{aligned}

Fraction

Definition For real numbers $a$ and $b$ with $b \ne 0$ , the fraction $\frac{a}{b}$ is the quotient $a \div b$ , the unique number $q$ with $bq = a$ . Two fractions are equal when they name the same number, and for $b \ne 0$ and $d \ne 0$

\frac{a}{b} = \frac{c}{d} \quad \text{exactly when} \quad ad = bc.

The basic rules, again with every denominator nonzero, are

\begin{aligned} \frac{a}{b} + \frac{c}{d} &= \frac{ad + bc}{bd}, \\[0.4em] \frac{a}{b} \cdot \frac{c}{d} &= \frac{ac}{bd}, \\[0.4em] \frac{a}{b} \div \frac{c}{d} &= \frac{a}{b} \cdot \frac{d}{c}, \quad c \ne 0. \end{aligned}

A negative sign may sit in front, on top or on the bottom, since $-\frac{a}{b} = \frac{-a}{b} = \frac{a}{-b}$ .

In plain words A fraction is a division waiting to happen. To add fractions you first give them a common denominator, but to multiply them you simply multiply tops and bottoms.

Example Back to the bar. Write $5$ as $\frac{10}{2}$ , then

5 - \frac{7}{2} = \frac{10}{2} - \frac{7}{2} = \frac{3}{2}.

On a number line marked in halves, start at $5$ and step $7$ halves to the left. You land on $1.5$ .

Numerator

Definition In the fraction $\frac{a}{b}$ the number or expression $a$ above the bar is the numerator. The fraction bar acts as a grouping symbol, so the whole numerator is one quantity. For example

\frac{x + 6}{2} = \frac{1}{2}(x + 6) = \frac{x}{2} + 3,

which is not the same as $x + \frac{6}{2}$ unless $x = 0$ .

In plain words Everything above the bar is divided by the bottom, not just the last term.

Example A common trap is canceling part of a numerator. With $x = 4$ ,

\frac{x + 6}{2} = \frac{10}{2} = 5,

while the wrong shortcut $x + 3$ gives $7$ . The correct split is $\frac{x}{2} + 3 = 2 + 3 = 5$ .

Denominator

Definition In the fraction $\frac{a}{b}$ the number or expression $b$ below the bar is the denominator. Division by $0$ is undefined, so a fraction with denominator $0$ has no value, and an expression such as $\frac{1}{x - 2}$ is defined only for $x \ne 2$ . Like the numerator, the denominator is grouped as a whole, so

\frac{12}{2 \cdot 3} = \frac{12}{6} = 2,

even though $12 \div 2 \cdot 3$ read from left to right is $18$ . When you type a fraction on one line, for example in a calculator, put parentheses around the whole numerator and the whole denominator.

In plain words The denominator says what you divide by, and it can never be zero. When a variable sits in the denominator, the values that make it zero are simply not allowed.

Example The expression

\frac{x + 1}{x^2 - 9}

is undefined when $x^2 - 9 = 0$ , that is at $x = 3$ and at $x = -3$ . At $x = -1$ the numerator is $0$ and the denominator is $-8$ , so the value is $0$ , which is perfectly fine. A zero on top is allowed, a zero on the bottom is not.

Parentheses

Definition Parentheses $( \; )$ , brackets $[ \; ]$ and other grouping symbols mark a part of an expression that is evaluated first, as a single quantity, before it is combined with anything outside. Nested groups are evaluated from the inside out. A fraction bar, an absolute value bar and the bar of a square root group in the same way.

In plain words Parentheses tell you what to do first. They also decide what a minus sign or an exponent applies to.

Example Compare these two ways of squaring negative three.

\begin{aligned} (-3)^2 &= (-3)(-3) = 9, \\[0.4em] -3^2 &= -(3 \cdot 3) = -9. \end{aligned}

In $-3^2$ the exponent applies only to $3$ , and the minus sign is applied afterward. The same holds with a variable. For any real $x$ , $(-x)^2 = x^2$ , while $-x^2$ is the opposite of $x^2$ . So when $x = 5$ , $-x^2 = -25$ and $(-x)^2 = 25$ . When you substitute a negative number, put it in parentheses first. With $x = -5$ the expression $-x^2$ becomes $-(-5)^2 = -25$ .

Order of operations

Definition An expression is evaluated in this order.

Grouping symbols, from the innermost outward, including fraction bars.
Exponents and roots.
Multiplication and division, from left to right as they appear.
Addition and subtraction, from left to right as they appear.

Multiplication does not come before division, and addition does not come before subtraction. Within each of the last two levels, you simply go from left to right.

In plain words Groups first, then powers, then times and divide, then plus and minus. Ties go left to right.

Example Here is a believable error and its fix. A student evaluates

12 - 6 \div 3 \cdot 2 + (-1)^3

as $12 - 6 \div 6 - 1 = 10$ , doing the multiplication before the division. Division and multiplication have equal rank, so they are done left to right, $6 \div 3 = 2$ and then $2 \cdot 2 = 4$ . Also $(-1)^3 = -1$ . The correct value is

12 - 4 - 1 = 7.

Fractions inside parentheses follow the same rules. To evaluate $\frac{1}{2}\left(\frac{2}{3} - 2\right)^2$ , start inside the parentheses, where $\frac{2}{3} - 2 = -\frac{4}{3}$ . Squaring gives $\frac{16}{9}$ , and half of that is $\frac{8}{9}$ .

Check yourself

What is the value of $-x^2 + 3x$ when $x = -2$ ? (A) $-10$ (B) $-2$ (C) $2$ (D) $10$

Show the answer

The answer is (A). Substitute with parentheses, $-(-2)^2 + 3(-2) = -4 - 6 = -10$ . Choice (B) comes from reading $-x^2$ as $(-x)^2 = 4$ , which gives $4 - 6$ . Choice (C) keeps $-4$ but drops the sign of $3x$ , and choice (D) makes both mistakes.

What is the value of the expression below? Enter your answer as a fraction or a decimal.

\frac{3}{4} - \frac{1}{2}\left(\frac{1}{3} - 1\right)

Show the answer

Inside the parentheses, $\frac{1}{3} - 1 = -\frac{2}{3}$ . Half of that is $-\frac{1}{3}$ , and subtracting a negative adds, so

\frac{3}{4} + \frac{1}{3} = \frac{9}{12} + \frac{4}{12} = \frac{13}{12}.

Enter 13/12 or the decimal 1.083, which is $1.0833\ldots$ cut off where the five characters run out. A mixed number such as 1 1/12 is not accepted.

For which value of $x$ is the expression $\frac{x + 4}{x - 3}$ undefined? (A) $-4$ (B) $-3$ (C) $0$ (D) $3$

Show the answer

The answer is (D). The expression is undefined only where the denominator $x - 3$ equals $0$ , which is at $x = 3$ . At $x = -4$ the numerator is $0$ , so the value is $0$ , which is defined. Choice (B) uses the wrong sign of $3$ , and choice (C) wrongly assumes that $x = 0$ causes trouble, although the value there is $-\frac{4}{3}$ .

Fractions, Signs and Order of Operations

Theory, part 1

Practice, part 2

Key terms

Negative number

Fraction

Numerator

Denominator

Parentheses

Order of operations

Check yourself

How this lesson connects

Used later in