Does PEMDAS mean I always multiply before I divide?

No. Multiplication and division share the same rank, and so do addition and subtraction, so there are really four steps, not six. Within a rank you work left to right. In -12 3· 2 the division comes first only because it appears first, giving -8.

Is the fraction bar really the same as parentheses?

For grouping purposes, yes. A bar means divide, and it silently encloses the whole numerator and the whole denominator. So -8+2/-3 must be read as (-8+2)(-3)=2. This is why typing the same expression on one line requires you to add the parentheses yourself.

Why is -4^2 equal to -16 but (-4)^2 equal to 16?

The exponent applies only to the base written directly under it. In -4^2 the base is 4, so you square to get 16 and then take the opposite, -16. The parentheses in (-4)^2 make -4 the base, so (-4)(-4)=16.

Should I turn fractions into decimals or decimals into fractions?

Convert to whichever form stays exact. Terminating decimals like 0.5, 1.25, and 0.75 become clean fractions, but 1/3 and 2/3 become repeating decimals, and rounding them mid-problem makes the final answer wrong. If any denominator is not a factor-friendly value like 2, 4, 5, 8, 10, or 25, work in fractions.

M7MATH-2.4

Order of Operations with Rational Numbers

Learn to evaluate multi-step expressions with negative fractions and decimals: rank the operations, treat the fraction bar as grouping, and go left to right.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Order of Operations with Rational Numbers, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

Start Order of Operations with Rational Numbers free

What this lesson covers

You already know how to add, subtract, multiply, and divide rational numbers one step at a time. Now those skills get combined into a single expression — something like

\dfrac{-2^2+10}{-3}-\frac{1}{2}\left(\frac{2}{3}-1.25\right)

— where the answer depends entirely on the order you do things in. Change the order and you get a different number, so the order is not a suggestion; it is part of the definition of the expression.

This lesson focuses on the three places students most often slip: forgetting that a fraction bar acts like a set of parentheses around the whole numerator and the whole denominator, treating multiplication as if it always comes before division, and losing track of a negative sign when an exponent or a subtraction is involved. By the end you will have a reliable routine that works whether the numbers are integers, negative fractions, decimals, or a mix of all three.

The Ranking of Operations

Every numerical expression is evaluated by working through four ranks in order. Within a rank, you work left to right.

Rank	Operations	Note
1	Grouping: parentheses, brackets, absolute value bars, fraction bars	Innermost group first
2	Exponents	The exponent attaches only to its base
3	Multiplication and division	Same rank — left to right
4	Addition and subtraction	Same rank — left to right

The common memory device PEMDAS lists six letters, which tricks a lot of students into thinking there are six steps. There are four. Multiplication and division share one rank, and addition and subtraction share one rank. Reading it as "do all multiplication, then all division" produces wrong answers.

With rational numbers, one extra decision shows up: should you work in fractions or decimals? Either is legal, but pick whichever keeps the arithmetic exact. Terminating decimals like

1.25

convert cleanly to

\frac{5}{4}

, but

\frac{2}{3}

becomes a repeating decimal, so rounding it to

0.67

makes your final answer wrong. A good habit: if any number in the expression is a fraction with a denominator other than 2, 4, 5, 8, 10, 20, 25, or 50, convert the decimals to fractions instead of the other way around.

Write one line per step and change only one thing per line. Doing two operations in your head on the same line is where sign errors are born.

The Fraction Bar Is a Grouping Symbol

When an expression is written with a horizontal bar, the bar does two jobs at once: it means divide, and it invisibly wraps parentheses around everything above it and everything below it. So

\frac{-8+2}{-3}\quad\text{means}\quad (-8+2)\div(-3)=(-6)\div(-3)=2.

Simplify the numerator completely, simplify the denominator completely, and only then divide. A frequent error is dividing first:

-8+2\div(-3)

is a different expression entirely, and it is not what the bar says.

This matters most when the numerator or denominator has several terms, or when it contains an exponent. In

\dfrac{-2^2+10}{-3}

, first handle the exponent inside the numerator:

-2^2=-4

, so the numerator is

-4+10=6

, and

6\div(-3)=-2

.

Also watch where a negative sign lives. These three are all equal:

-\frac{3}{4}=\frac{-3}{4}=\frac{3}{-4}.

A single negative sign can sit in front of the bar, in the numerator, or in the denominator without changing the value. But two negatives, one in each place, cancel:

\dfrac{-3}{-4}=\dfrac{3}{4}

.

Absolute value bars work like parentheses too. In

|{-5}+2|

, combine inside first to get

|{-3}|=3

. Taking absolute values term by term to get

5+2=7

is a common wrong answer.

Same-Rank Operations Go Left to Right

Multiplication and division tie, so the tie is broken by position. Consider

-12\div 3\cdot 2

. Left to right:

-12\div 3=-4

, then

-4\cdot 2=-8

. If you multiplied first you would get

-12\div 6=-2

, which is wrong.

The same rule applies to addition and subtraction:

-1.5-0.5+2

becomes

-2+2=0

, not

-1.5-2.5=-4

.

A reliable way to avoid the trap is to rewrite subtraction as adding the opposite and division as multiplying by the reciprocal. Once every operation in a rank is the same, order truly stops mattering:

-12\div 3\cdot 2=-12\cdot\frac{1}{3}\cdot 2=-8.

-1.5-0.5+2=-1.5+(-0.5)+2=0.

Exponents need their own caution with negatives, because the exponent applies only to the base directly beneath it.

Expression	Base	Value
$(-3)^2$	$-3$	$9$
$-3^2$	$3$	$-9$
$\left(-\frac{1}{2}\right)^3$	$-\frac{1}{2}$	$-\frac{1}{8}$
$-\left(\frac{1}{2}\right)^3$	$\frac{1}{2}$	$-\frac{1}{8}$

Parentheses are what pull the negative sign into the base. Without them, you square the number and then apply the opposite. Notice that for an odd exponent the two versions happen to agree, which is why students who guess sometimes look right and then miss the next problem.

Building a Routine That Survives Fractions and Decimals

Here is a routine that works on any multi-step rational expression.

First, scan the whole expression and mark the groups: parentheses, absolute value bars, and each numerator and denominator of a fraction bar. Second, decide your number form — all fractions or all terminating decimals — and convert once, up front. Third, evaluate inside groups, using the four ranks again inside each group. Fourth, do exponents. Fifth, sweep left to right doing multiplications and divisions. Sixth, sweep left to right doing additions and subtractions.

Where students actually go wrong, in order of frequency: dropping a negative sign while copying the next line; dividing before finishing a numerator; treating

-x^2

as positive; and rounding

\frac{1}{3}

\frac{2}{3}

to a decimal partway through.

One more habit worth building: estimate before you compute. In

-2-\left(-\frac{7}{24}\right)

you can see instantly that subtracting a negative moves you up, so the answer should be a bit greater than

-2

but still negative. If your computed answer came out as

-2\frac{7}{24}

, the estimate catches the sign error immediately.

Finally, a note on notation you will meet in the next unit: a coefficient written next to parentheses, as in

\frac{1}{2}\left(\frac{2}{3}-1.25\right)

, means multiply. It has rank 3, so it happens after the inside of the parentheses is finished but before any addition or subtraction outside.

Reading Expressions Correctly Before You Compute

Two expressions can use the same numbers and symbols and still be different, so translating the written form correctly is half the work.

Expression	What it means	Value
$\dfrac{6}{2+1}$	$6\div(2+1)$	$2$
$\dfrac{6}{2}+1$	$(6\div 2)+1$	$4$
$-\dfrac{1}{2}\cdot 4^2$	$-\frac{1}{2}\cdot 16$	$-8$
$\left(-\dfrac{1}{2}\cdot 4\right)^2$	$(-2)^2$	$4$

When a problem is written in a single line, as it often is when you type it, the grouping must be made explicit with parentheses. The expression

\dfrac{-8+2}{-3}

has to be typed as

(-8+2)/(-3)

. Typing

-8+2/-3

changes the meaning, and this is a real source of wrong answers when students check work with a calculator.

A related habit: when you substitute a negative number into an expression later in the course, always put it in parentheses. If

x=-3

and you want

x^2

, write

(-3)^2=9

. Writing

-3^2

gives

-9

and quietly breaks the problem.

Finally, subtraction and negation look identical on paper but behave differently. In

5-(-2)

the first symbol is subtraction and the second is negation, and the result is

5+2=7

. Saying it out loud — "five minus negative two" — helps keep the two roles separate while you write each step.

Key terms

Order of operations.: The agreed sequence for evaluating an expression: grouping symbols, then exponents, then multiplication and division left to right, then addition and subtraction left to right.
Grouping symbol.: Any notation that bundles part of an expression to be evaluated first, including parentheses, brackets, absolute value bars, and the fraction bar.
Fraction bar.: A horizontal division bar that also acts as invisible parentheses around the entire numerator and the entire denominator.
Same-rank operations.: Operations of equal priority, such as multiplication with division or addition with subtraction; they are performed from left to right.
Base.: The number an exponent applies to. In $(-3)^2$ the base is $-3$ ; in $-3^2$ the base is $3$ .
Rational number.: Any number that can be written as a ratio of two integers, including negative fractions, terminating decimals, and repeating decimals.
Reciprocal.: The multiplicative inverse of a nonzero number; dividing by a number equals multiplying by its reciprocal, so $\div\frac{2}{3}$ is the same as $\cdot\frac{3}{2}$ .
Adding the opposite.: Rewriting $a-b$ as $a+(-b)$ so a string of additions and subtractions can be combined without order confusion.

Worked example

Evaluate

\dfrac{-2^2+10}{-3}-\dfrac{1}{2}\left(\dfrac{2}{3}-1.25\right)

Start by marking the groups. There is a fraction bar with numerator

-2^2+10

and denominator

-3

, and there is a set of parentheses containing

\frac{2}{3}-1.25

. Because

\frac{2}{3}

is present, work in fractions rather than decimals, and note that

1.25=\frac{5}{4}

.

Step 1, inside the numerator. The exponent applies only to

2

, so

-2^2=-4

. Then

-4+10=6

.

Step 2, finish the fraction. Now the bar means

6\div(-3)=-2

.

Step 3, inside the parentheses. Use a common denominator of 12:

\frac{2}{3}=\frac{8}{12}

and

\frac{5}{4}=\frac{15}{12}

, so

\frac{8}{12}-\frac{15}{12}=-\frac{7}{12}

.

The expression is now

-2-\frac{1}{2}\left(-\frac{7}{12}\right)

.

Step 4, rank 3 work. Multiply:

\frac{1}{2}\cdot\left(-\frac{7}{12}\right)=-\frac{7}{24}

. So the expression reads

-2-\left(-\frac{7}{24}\right)

.

Step 5, rank 4 work. Subtracting a negative is adding the opposite:

-2+\frac{7}{24}=-\frac{48}{24}+\frac{7}{24}=-\frac{41}{24}

.

Final answer:

-\frac{41}{24}

, or

-1\frac{17}{24}

.

Check with an estimate. Since

-\frac{7}{24}

is about

-0.29

, we expect

-2

minus about

-0.29

, which is roughly

-1.71

. And

-\frac{41}{24}\approx-1.708

, so the sign and size both make sense.

Practice questions

Evaluate

-3+4\cdot(-2)^2\div 8

$-5$
$-1$
$1$
$5$

Answer: $-1$

Exponent first:

(-2)^2=4

, because the parentheses put the negative sign inside the base. The expression becomes

-3+4\cdot 4\div 8

. Multiplication and division are the same rank, so go left to right:

4\cdot 4=16

, then

16\div 8=2

. Finally

-3+2=-1

. A common wrong answer is

-5

, which comes from reading

(-2)^2

-4

; another is

5

, from dividing

4\div 8

before multiplying.

Evaluate

-12\div 3\cdot 2

$-8$
$-2$
$2$
$8$

Answer: $-8$

Division and multiplication tie in rank, so position decides. Left to right:

-12\div 3=-4

, then

-4\cdot 2=-8

. Getting

-2

means the multiplication was done first, which happens when PEMDAS is read as putting M ahead of D. Rewriting the expression as

-12\cdot\frac{1}{3}\cdot 2

removes the ambiguity.

Evaluate

\dfrac{5-(-1)}{-4}\cdot 0.5-\left(-\dfrac{3}{4}\right)

and explain, in a sentence, why the fraction bar has to be handled before the multiplication by

0.5

Answer: The value is $0$ . The bar groups the whole numerator, so $5-(-1)=6$ must be finished before dividing by $-4$ , giving $-1.5$ ; then $-1.5\cdot 0.5=-0.75$ , and $-0.75+0.75=0$ .

First the numerator:

5-(-1)=5+1=6

. The bar then means

6\div(-4)=-\frac{3}{2}=-1.5

. Next, rank 3 left to right:

-1.5\cdot 0.5=-0.75

. Last, rank 4: subtracting

-\frac{3}{4}

means adding

0.75

, so

-0.75+0.75=0

. The fraction bar comes first because it is a grouping symbol, rank 1, while multiplication is rank 3; if you divided

5

-4

and ignored the

-(-1)

, you would be evaluating a different expression.

FAQ

Does PEMDAS mean I always multiply before I divide?: No. Multiplication and division share the same rank, and so do addition and subtraction, so there are really four steps, not six. Within a rank you work left to right. In $-12\div 3\cdot 2$ the division comes first only because it appears first, giving $-8$ .
Is the fraction bar really the same as parentheses?: For grouping purposes, yes. A bar means divide, and it silently encloses the whole numerator and the whole denominator. So $\frac{-8+2}{-3}$ must be read as $(-8+2)\div(-3)=2$ . This is why typing the same expression on one line requires you to add the parentheses yourself.
Why is $-4^2$ equal to $-16$ but $(-4)^2$ equal to $16$ ?: The exponent applies only to the base written directly under it. In $-4^2$ the base is $4$ , so you square to get $16$ and then take the opposite, $-16$ . The parentheses in $(-4)^2$ make $-4$ the base, so $(-4)(-4)=16$ .
Should I turn fractions into decimals or decimals into fractions?: Convert to whichever form stays exact. Terminating decimals like $0.5$ , $1.25$ , and $0.75$ become clean fractions, but $\frac{1}{3}$ and $\frac{2}{3}$ become repeating decimals, and rounding them mid-problem makes the final answer wrong. If any denominator is not a factor-friendly value like 2, 4, 5, 8, 10, or 25, work in fractions.

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