When using the word order of operations subtraction is performed before division?

Reducing ambiguity by agreement

In general, nobody wants to be misunderstood. In mathematics, it is so important that readers understand expressions exactly the way the writer intended that mathematics establishes conventions, agreed-upon rules, for interpreting mathematical expressions.

Does 10 − 5 − 3 mean that we start with 10, subtract 5, and then subtract 3 more leaving 2? Or does it mean that we are subtracting 5 − 3 from 10?
Does 2 + 3 × 10 equal 50 because 2 + 3 is 5 and then we multiply by 10, or does the writer intend that we add 2 to the result of 3 × 10?

To avoid these and other possible ambiguities, mathematics has established conventions (agreements) for the way we interpret mathematical expressions. One of these conventions states that when all of the operations are the same, we proceed left to right, so 10 − 5 − 3 = 2, so a writer who wanted the other interpretation would have to write the expression differently: 10 − (5 − 2). When the operations are not the same, as in 2 + 3 × 10, some may be given preference over others. In particular, multiplication is performed before addition regardless of which appears first when reading left to right. For example, in 2 + 3 × 10, the multiplication must be performed first, even though it appears to the right of the addition, and the expression means 2 + 30.
See full rules for order of operations below.

Conventions for reading and writing mathematical expressions

The basic principle: “more powerful” operations have priority over “less powerful” ones.

Using a number as an exponent (e.g., 58 = 390625) has, in general, the “most powerful” effect; using the same number as a multiplier (e.g., 5 ×8 = 40) has a weaker effect; addition has, in general, the “weakest” effect (e.g., 5 + 8 = 13). Although these terms (powerful, weak) are not used in mathematics, the sense is preserved in the language of “raising 5 to the 8th power.” Exponentiation is “powerful” and so it comes first! Addition/subtraction are “weak,” so they come last. Multiplication/division come in between.

When it is important to specify a different order, as it sometimes is, we use parentheses to package the numbers and a weaker operation as if they represented a single number.

For example, while 2 + 3 × 8 means the same as 2 + 24 (because the multiplication takes priority and is done first), (2 + 3) × 8 means 5 × 8, because the (2 + 3) is a package deal, a quantity that must be figured out before using it. In fact (2 + 3) × 8 is often pronounced “two plus three, the quantity, times eight” (or “the quantity two plus three all times eight”).

Summary of the rules:

• Parentheses first. Referring to these as “packages” often helps children remember their purpose and role.
• Exponents next.
• Multiplication and division next. (Neither takes priority, and when there is a consecutive string of them, they are performed left to right.)
• Addition and subtraction last. (Again, neither takes priority and a consecutive string of them are performed left to right.)

Common Misconceptions

Many students learn the order of operations using PEMDAS (Parentheses, Exponents, Multiplication, Division…) as a memory aid. This very often leads to the misconception that multiplication comes before division and that addition comes before subtraction. Understanding the principle is probably the best memory aid.

Order of Operations: BODMAS

Order of Operations

In mathematics, an operation is an action such as addition, subtraction, multiplication and division.

In a given mathematical expression, the order in which we carry out a calculation is important. The wrong order of operations will often lead to the wrong answer.

For example, consider the expression \[4\div5+6\times2.\]

• If we first divide $4$ by $5$ to get $0.8$, then multiply $6$ by $2$ to get $12$, and then add $0.8$ to $12$ we get $12.8$.
• If instead we first add $5$ and $6$ to get $11$, then multiply $11$ by $2$ to get $22$, and then divide the $4$ by $22$ we get $0.182$ (to $3$ d.p.).

We can see that the result is very different when we perform the operations in a different order.

BODMAS

BODMAS is an acronym which tells us the correct order in which we should carry out mathematical operations:

Division and multiplication, and addition and subtraction, have the same priority - the convention is to work from left to right when the order of operations would be unclear.

Note: An alternative form of this mnemonic is BIDMAS, where the I stands for indices. PEMDAS (“Parentheses, exponents, ...”) and BEDMAS are also used in the USA and Australia.

Returning to the above example, the correct answer would be the first answer as it follows the rules of BODMAS: division can be done before multiplication and must be done before addition, and multiplication comes before addition. So the answer is $12.8$.

We will now look at more examples to practice using BODMAS.

Example 1

Evaluate the following expression: $20\times(100+1)$.

Solution

Applying the BODMAS rule, we know that we must first consider everything inside the brackets. Since the only operation inside the brackets is a single sum, we first add $1$ to $100$ to get $101$. The expression then becomes: \[20\times101\].

All that is left is to multiply the two numbers together. This gives $2020$ and so we have: \[20\times(100+1)=2020\].

Example 2

Evaluate the following expression: $(-4)\times102$.

Solution

First notice that although this expression contains a pair of brackets, the brackets are only there to indicate that $-4$ is a negative number.

Applying the BODMAS rule, we must first evaluate the exponent: $10^2=100$.

Finally, we multiply $100$ by $(-4)$ to get $-400$. We thus have

\[(-4)\times10^2=-400\]

Example 3

Evaluate $2+4\times3-1$

Solution

Applying BODMAS, we do the multiplication first. $3\times4=12$ so we have: \[2+12-1\] Then addition and subtraction have the same priority, so we can do either next. Performing the addition first we have: \[14-1=13.\] Check for yourself that doing the subtraction before the addition gives the same answer.

Example 4

Evaluate $3+2^2$

Solution

Applying BODMAS, we evaluate the power first, then the addition. $2^2=4$ so we have: \[3+4=7.\]

Test Yourself

Try our Numbas test: Arithmetic operations

External Resources

• BODMAS at Maths is Fun
• BODMAS by the University of East Anglia.

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What comes first in order of operations?

What comes first in order of operations addition or subtraction?

Which mathematical operation should be done first?