Order of operations with decimals

 

Reviewing the convention

To avoid confusion with numerical calculations, mathematicians decided on a strict order of operations when calculations are are done involving any or all of the basic operations.

To understand the confusion, think about this calculation:

$6+5\times2+4$6+5×2+4

Working from left to right, which is not how mathematicians work, we might add $5$5 to $6$6 first to get $11$11, then times the result by $2$2 to get $22$22, and then add $4$4 to that, giving a final answer of $26$26.

Working from right to left, as some languages are written, we might add $2$2 to the $4$4 to get $6$6 then times that result by $5$5 to get $30$30, and then finally add $6$6 to give a final answer of $36$36.

Neither of these answers would be correct, according to a mathematician.

The mathematician would understand a global convention, where,

1. In the first instance one should deal with the inside of anything bracketed, and if there are other parentheses around the brackets, deal with those next.
2. After that, one should deal with all multiplications and divisions, starting from left and going right
3. Finally, one should add and subtract the clumps of things that remain, from left to right again.

 

So where does that leave us with the calculation $6+5\times2+4$6+5×2+4?

Well, there are no brackets or parentheses, so we can move straight to multiplication and division. But there's only a single multiplication, so we'll do that first:

$6+10+4$6+10+4

Finally, the addition becomes $20$20, and so $20$20 is the mathematical answer.  

 

Examples with decimals

You might like to think about what we just discusses about order of operations in these examples involving decimals:

Example 1

Evaluate $3\times\left(4.5+3.75\times2\right)+12\div6$3×(4.5+3.75×2)+12÷​6

Look carefully at what happens when we simplify the calculation line by line:

$3\times\left(4.5+3.75\times2\right)+12\div6$3×(4.5+3.75×2)+12÷​6	$=$=	$3\times\left(4.5+7.5\right)+12\div6$3×(4.5+7.5)+12÷​6
	$=$=	$3\times12+12\div6$3×12+12÷​6
	$=$=	$36+12\div6$36+12÷​6
	$=$=	$36+2$36+2
$Answer$Answer	$=$=	$38$38

Notice how the bracketed things are dealt with first, and within those brackets the multiplication is done before the addition.

After the brackets have been completely dealt with, the multiplication left and the division are done separately before we add what remains.

 

Example 2

Evaluate $12.8\div6.4\div2$12.8÷​6.4÷​2

This calculation only involves two divisions only.

The wrong way to proceed is to deal with the second division before the first. The only time we would do that is if the last division was bracketed. Then $12.8\div\left(6.4\div2\right)$12.8÷​(6.4÷​2) becomes $12.8\div3.2=4$12.8÷​3.2=4. This turns out to be four times the actual answer!

If we proceed correctly, however, we would calculate it as...

$12.8\div6.4\div2$12.8÷​6.4÷​2	$=$=	$2\div2$2÷​2
	$=$=	$1$1

Example 3

Evaluate $9+912.15\div9$9+912.15÷​9

If we reach for a calculator to evaluate this, we had better hope that the calculator is a scientific calculator, because a simple desk top calculator can give you the wrong answer.

I used my simple desk top calculator and got an answer of $102.35$102.35

I used a scientific calculator and got an answer of $110.35$110.35

The desk top calculator calculated from left to right. The scientific calculator completed the division first.

 

Example 4

Evaluate $3.71+23.22\div3-9.45$3.71+23.22÷​3−9.45 

My desk top calculator got it wrong again. Some desk top calculators follow the conventions, and you should test it yourself before relying on mathematical answers.

The correct answer is $2$2 because the division must be done first.

 

Example 5

Evaluate $\left(34.56\div16\right)\div\left(1.8\times1.2\right)$(34.56÷​16)÷​(1.8×1.2)

The left hand bracket is dealt with first, the the right hand bracket is dealt with, and then finally the division between the two brackets happens. In steps..

$\left(34.56\div16\right)\div\left(1.8\times1.2\right)$(34.56÷​16)÷​(1.8×1.2)	$=$=	$2.16\div\left(1.8\times1.2\right)$2.16÷​(1.8×1.2)
	$=$=	$2.16\div2.16$2.16÷​2.16
	$=$=	$1$1

 That's exactly what my scientific calculator shows me! 

Worked Examples

Question 1

Question 2

Question 3

Question 4