Evaluating Numerical Expressions with Order of Operations

Welcome. This opening lesson establishes a rule you will use across the whole May/June preparation plan: when one calculation contains several operations, every student, calculator, spreadsheet, and examiner must interpret it in the same way. It matters immediately in mathematics and later in formula work for Chemistry and Physics.

Today you will evaluate expressions containing integers, fractions, and decimals by applying the correct order of operations. You will also learn the two priority ties that commonly cause lost marks: multiplication with division, and addition with subtraction.

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One expression, one agreed meaning

Consider the expression

$$4+2\times 3$$

If calculations were always done from left to right, the answer would be $18$. But multiplication has higher priority than addition, so the intended calculation is

$$4+(2\times 3)=4+6=10$$

Order of operations is the shared grammar of mathematics: it tells us which parts of a calculation belong together without having to write every grouping symbol.

You may see several memory aids:

• BODMAS: Brackets, Orders, Division, Multiplication, Addition, Subtraction
• BIDMAS: Brackets, Indices, Division, Multiplication, Addition, Subtraction
• PEMDAS/PEDMAS: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction

They describe the same underlying structure. For Cambridge-style questions, BODMAS or BIDMAS is common, but the letters can be misleading if treated as six completely separate levels.

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The four real priority levels

It is more accurate to remember four levels, not six:

1. Brackets: work from the innermost brackets outward.
2. Orders / indices: powers such as $3^2$, and roots when they occur.
3. Multiplication and division: equal priority, so work left to right.
4. Addition and subtraction: equal priority, so work left to right.

The key point is that the letters $D$ and $M$ do not mean “always divide before multiplying.” Similarly, $A$ and $S$ do not mean “always add before subtracting.”

For example:

$$18÷9\times 2$$

Division and multiplication have equal priority. Read from the left:

$$18÷9\times 2=2\times 2=4$$

It would be wrong to multiply $9$ and $2$ first. Doing that would silently change the original expression into a different one.

The same applies at the final level:

$$10-3+5$$

Work left to right:

$$10-3+5=7+5=12$$

Do not add $3$ and $5$ first. The subtraction sign belongs to the $3$, so that method changes the calculation.

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A dependable written method

In a non-calculator paper, neat lines are not just presentation: they protect you from changing an expression accidentally.

Use this routine:

1. Scan for brackets, including fraction bars.
2. Complete any powers.
3. Complete multiplication and division, moving from left to right where they occur in one uninterrupted calculation.
4. Complete addition and subtraction, again from left to right.
5. On each line, change only the part you have calculated. Copy all remaining terms exactly.

Here is a mixed example:

$$20-12÷3+4\times 2$$

There are no brackets or powers. Division and multiplication come before addition and subtraction:

$$20-12÷3+4\times 2=20-4+4\times 2=20-4+8$$

Now only subtraction and addition remain, so read from the left:

$$20-4+8=16+8=24$$

The answer is $24$.

Notice that writing

$$20-4+8=20-12$$

would be incorrect: it combines $4$ and $8$ by addition before the subtraction to their left has been handled.

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Brackets and simple powers

Brackets tell you to temporarily focus on the enclosed calculation. Within those brackets, you still use the same order of operations.

Consider:

$$18÷(2+4)+3^2-1.5\times 4$$

First deal with the bracket:

$$18÷6+3^2-1.5\times 4$$

Next calculate the power:

$$18÷6+9-1.5\times 4$$

Now complete division and multiplication:

$$3+9-6$$

Finally, calculate from left to right:

$$3+9-6=12-6=6$$

So the expression has value $6$.

A power applies only to the number or bracket immediately before it. Parentheses can therefore be important with negative numbers:

$$(-3{)}^2=9$$

but

$$-3^2=-9$$

In the second expression, the square applies to $3$, and the negative sign is applied afterwards. You will study indices in more depth later; for now, recognise that powers come before multiplication, division, addition, and subtraction.

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Fractions are built-in grouping symbols

A fraction bar means that the whole numerator is grouped together and the whole denominator is grouped together. Treat it much like brackets.

For example:

$$\frac{5+7}{8-2}+0.25\times 8$$

Calculate the numerator and denominator first:

$$\frac{12}{6}+0.25\times 8$$

Now simplify the fraction and multiply the decimal:

$$2+2=4$$

The answer is $4$.

A common mistake is to calculate only one piece of a fraction, perhaps changing

$$\frac{5+7}{8-2}$$

into

$$\frac{12}{8}-2$$

That is not allowed: the denominator is $8-2$ as a complete grouped expression. It must become $6$ before the division represented by the fraction bar is done.

Fractions do not create a new order-of-operations rule. They simply require careful fraction skills within the normal order. If multiplication or division involving fractions appears, complete it before any surrounding addition or subtraction.

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Decimals and negative integers follow exactly the same rule

The type of number does not alter the priority. A decimal is still a number; a negative integer is still a number.

For example:

$$7.8-0.6\times (3.5+1.5)$$

Start with the bracket:

$$7.8-0.6\times 5$$

Then multiply:

$$7.8-3=4.8$$

The answer is $4.8$.

With negative values, calculate the operation first and then handle the sign carefully:

$$-24÷6-3\times (-2)$$

The division gives $-4$, and the product of a positive and negative number is $-6$:

$$-4-(-6)$$

Subtracting a negative is equivalent to adding its positive value:

$$-4+6=2$$

Brackets around a negative number, such as $(-2)$, make the intended sign clear. This is especially useful in exam working.

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Final-answer checks for O Level questions

Before committing to an answer, use a short audit:

• Priority check: Did you complete brackets, powers, multiplication/division, then addition/subtraction?
• Tie check: Where multiplication and division, or addition and subtraction, appeared together, did you work left to right?
• Fraction check: Did you simplify both the numerator and denominator as complete groups?
• Sign check: Did any negative signs remain attached to the correct number?
• Reasonableness check: Does your answer have roughly the expected size and sign?

For calculator work, enter brackets exactly as they appear in the expression. A calculator can check arithmetic, but it cannot rescue an expression that has been entered with missing brackets or an incorrectly placed negative sign. In a written solution, show the key intermediate lines so that even a minor arithmetic slip does not hide otherwise correct method marks.

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Key takeaways

Order of operations has four practical levels: brackets, powers, multiplication/division, and addition/subtraction. The final two pairs are tied: multiplication and division are performed left to right, as are addition and subtraction.

Fractions act as grouping symbols, and integers, decimals, and negative numbers all obey the same structure. The safest exam method is to simplify one valid part per line while copying the rest of the expression unchanged.

Next, you will build on accurate numerical calculation by estimating values and writing error intervals to a stated degree of accuracy.