3.3 Order of Operations in Formulas

Narrated lesson: Order of Operations in Formulas

Synthetic narration with English captions, a transcript, and time to think before checking the answer.

Read the transcript Download the video Narration credits and licences

Simplify Expressions with a Fraction Bar

Where does the negative sign go in a fraction? Usually, the negative sign is placed in front of the fraction, but you will sometimes see a fraction with a negative numerator or denominator. Remember that fractions represent division. The fraction $- \frac{1}{3}$ could be the result of dividing $\frac{−1}{3},$ a negative by a positive, or of dividing $\frac{1}{−3},$ a positive by a negative. When the numerator and denominator have different signs, the quotient is negative.

Negative 1 over positive 3 is equal to negative one third. Negative over positive equals negative. Positive 1 over negative 3 is equal to negative one third. Positive over negative equals negative.

If both the numerator and denominator are negative, then the fraction itself is positive because we are dividing a negative by a negative.

$\frac{−1}{−3} = \frac{1}{3} \frac{negative}{negative} = positive$

Fraction bars act as grouping symbols. The expressions above and below the fraction bar should be treated as if they were in parentheses. For example, $\frac{4 + 8}{5 - 3}$ means $(4 + 8) \div (5 - 3) .$ The order of operations tells us to simplify the numerator and the denominator first—as if there were parentheses—before we divide.

We’ll add fraction bars to our set of grouping symbols to have a more complete set here.

Publisher clarification. This lesson practices numeric expressions with fraction bars. The same grouping rule applies when a formula has a numerator and denominator: evaluate each complete group before dividing.

Publisher clarification. Before dividing, check that the denominator is not zero. Division by zero is undefined. A zero numerator gives zero when the denominator is nonzero.

Example 3

Simplify: $\frac{4 - 2 (3)}{2^{2} + 2} .$

Show solution

	$\frac{4 - 2 (3)}{2^{2} + 2}$
Use the order of operations. Multiply in the numerator and use the exponent in the denominator.	$\frac{4 - 6}{4 + 2}$
Simplify the numerator and the denominator.	$\frac{−2}{6}$
Simplify the fraction.	$- \frac{1}{3}$

Example 4

Simplify: $\frac{{(8 - 4)}^{2}}{8^{2} - 4^{2}} .$

Show solution

	$\frac{{(8 - 4)}^{2}}{8^{2} - 4^{2}}$
Use the order of operations (parentheses first, then exponents).	$\frac{{(4)}^{2}}{64 - 16}$
Simplify the numerator and denominator.	$\frac{16}{48}$
Simplify the fraction.	$\frac{1}{3}$

Practice Makes Perfect

Simplify Expressions with a Fraction Bar

In the following exercises, identify the equivalent fractions.

Which of the following fractions are equivalent to $\frac{5}{−11} ?$
$\frac{−5}{−11}, \frac{−5}{11}, \frac{5}{11}, - \frac{5}{11}$

Which of the following fractions are equivalent to $\frac{−4}{9} ?$
$\frac{−4}{−9}, \frac{−4}{9}, \frac{4}{9}, - \frac{4}{9}$

Show solution

$\frac{−4}{9}, - \frac{4}{9}$

Which of the following fractions are equivalent to $- \frac{11}{3} ?$
$\frac{−11}{3}, \frac{11}{3}, \frac{−11}{−3}, \frac{11}{−3}$

Which of the following fractions are equivalent to $- \frac{13}{6} ?$
$\frac{13}{6}, \frac{13}{−6}, \frac{−13}{−6}, \frac{−13}{6}$

Show solution

$\frac{13}{−6}, \frac{−13}{6}$

In the following exercises, simplify.

$\frac{4 + 11}{8}$

$\frac{9 + 3}{7}$

Show solution

$\frac{12}{7}$

$\frac{22 + 3}{10}$

$\frac{19 - 4}{6}$

Show solution

$\frac{5}{2}$

$\frac{48}{24 - 15}$

$\frac{46}{4 + 4}$

Show solution

$\frac{23}{4}$

$\frac{−6 + 6}{8 + 4}$

$\frac{−6 + 3}{17 - 8}$

Show solution

$- \frac{1}{3}$

$\frac{22 - 14}{19 - 13}$

$\frac{15 + 9}{18 + 12}$

Show solution

$\frac{4}{5}$

$\frac{5 \cdot 8}{−10}$

$\frac{3 \cdot 4}{−24}$

Show solution

$- \frac{1}{2}$

$\frac{4 \cdot 3}{6 \cdot 6}$

$\frac{6 \cdot 6}{9 \cdot 2}$

Show solution

$\frac{4^{2} - 1}{25}$

$\frac{7^{2} + 1}{60}$

Show solution

$\frac{5}{6}$

$\frac{8 \cdot 3 + 2 \cdot 9}{14 + 3}$

$\frac{9 \cdot 6 - 4 \cdot 7}{22 + 3}$

Show solution

$\frac{26}{25}$

$\frac{15 \cdot 5 - 5^{2}}{2 \cdot 10}$

$\frac{12 \cdot 9 - 3^{2}}{3 \cdot 18}$

Show solution

$\frac{11}{6}$

$\frac{5 \cdot 6 - 3 \cdot 4}{4 \cdot 5 - 2 \cdot 3}$

$\frac{8 \cdot 9 - 7 \cdot 6}{5 \cdot 6 - 9 \cdot 2}$

Show solution

$\frac{5}{2}$

$\frac{5^{2} - 3^{2}}{3 - 5}$

$\frac{6^{2} - 4^{2}}{4 - 6}$

Show solution

−10

$\frac{2 + 4 (3)}{−3 - 2^{2}}$

$\frac{7 + 3 (5)}{−2 - 3^{2}}$

Show solution

−2

$\frac{7 \cdot 4 - 2 (8 - 5)}{9 \cdot 3 - 3 \cdot 5}$

$\frac{9 \cdot 7 - 3 (12 - 8)}{8 \cdot 7 - 6 \cdot 6}$

Show solution

$\frac{51}{20}$

$\frac{9 (8 - 2) −3 (15 - 7)}{6 (7 - 1) −3 (17 - 9)}$

$\frac{8 (9 - 2) −4 (14 - 9)}{7 (8 - 3) −3 (16 - 9)}$

Show solution

$\frac{18}{7}$

Corequisite Skills for Introductory Statistics is a custom course edition compiled for Introductory Statistics corequisite courses from OpenStax Contemporary Mathematics, Elementary Algebra 2e, Intermediate Algebra 2e, Prealgebra 2e. Content by OpenStax is reproduced as published, apart from any corrections recorded for this edition. Sections are renumbered to follow the course outline. The course outline is adapted from Larry Green's Support Course for Elementary Statistics (LibreTexts, https://stats.libretexts.org/Bookshelves/Introductory_Statistics/Support_Course_for_Elementary_Statistics), used under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). Lessons 2.4 and 6.7 are written for this edition by XYZ Homework. Every OpenStax lesson is abridged to the subsections and practice exercises this course uses. Lessons are headed with their course titles, not the source section titles. Access the complete originals free at openstax.org. License: CC-BY-NC-SA-4.0.

	$\frac{4 + 8}{5 - 3}$
Simplify the expression in the numerator.	$\frac{12}{5 - 3}$
Simplify the expression in the denominator.	$\frac{12}{2}$
Simplify the fraction.	6

	$\frac{4 (−3) + 6 (−2)}{−3 (2) −2}$
Multiply.	$\frac{−12 + (−12)}{−6 - 2}$
Simplify.	$\frac{−24}{−8}$
Divide.	$3$