1.3: The Order of Operations

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Samar ElHitti, Marianna Bonanome, Holly Carley, Thomas Tradler, & Lin Zhou
CUNY New York City College of Technology & NYC College of Technology via New York City College of Technology at CUNY Academic Works

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What is the meaning of the expression '3 times 4 plus 5'. Some will answer 17 while others may answer 27. Why? To take the ambiguity out, we can write

\[(3 \times 4)+5=17 \nonumber\]

and

\[3 \cdot(4+5)=27, \nonumber\]

where we must first evaluate the quantity in parentheses. Since it can be somewhat cumbersome to write a lot of parentheses, there is an important convention or agreement that if we just write \(3 \times 4+5\) we mean \((3 \times 4)+5 .\) That is, in the absence of parentheses, we should multiply before we add. This is part of what is called The Order of Operations. This must be remembered.

Definition: 1.24: The Order of Operation

When evaluating an expression involving addition, subtraction, multiplication and division which has no parentheses or exponents, we first perform, from left to right, all of the multiplications and divisions. Then, from left to right, the additions and subtractions. If there are parts of the expression set off by parentheses, what is within the parentheses must be evaluated first.

Remark 1.25

Subtraction can be turned into addition and then addition can be done in any order, not necessarily from left to right. This explains why addition and subtraction come together in the order of operations. There will be a similar statement for multiplication and division but will be postponed until fractions are discussed.

’PE(MD)(AS)’ is an easy way to remember the order of operations. This means that the order is: Parentheses, Exponents (this will be incorporated later), Multiplication and Division (taken together from left to right), and finally, Addition and Subtraction (taken together from left to right).

Let us try a few problems.

Example 1.26

\(3+2(3+5)=3+2(8)=3+16=19\)
\(3-2(-4+7)=3-2(3)=3-6=-3\)
\(-3-4-2(-2 \cdot 6-5)=-3-4-2(-12-5)=-3-4-2(-17)=-3-4-(-34)=-3-4+34=27\)
\(-(3-(-6))-(1-4 \cdot(-5)+4)=-(3+6)-(1-(-20)+4)=-9-(1+20+4)=-9-25=-9+(-25)=-34\)
\(-2(-14 \div 7+7)=-2(-2+7)=-2(5)=-10\)
\(-3(-2 \cdot 7-(-5)(4) \div 2)=-3(-14-(-20) \div 2)=-3(-14-(-10))=-3(-4)=12\)
\(6 \div 2 \times 3=3 \times 3=9\) Note: \(6 \div 2 \times 36=6 \div 6=1\)
\(-2(3-1) 2-(8-22) \div 4=-2(2) 2-(8-4) \div 4=-2(4)-4 \div 4=-8-1=-9\)

Exit Problem

Evaluate: \(\left(3^{3}+5\right) \div 4-4(7-2)\)