Introduction To BODMAS

The acronym $BODMAS$ stands for:

$B$ - Bracket

$O$ - Of

$D$ - Division

$M$ - Multiplication

$A$ - Addition

$S$ - Subtraction

Given an expression containing two or more of the above operations, the rule tells us in which order we need to evaluate the expression. We will understand this rule, by taking a few examples:

How much is:

$3 + 4 \times 5$

If you perform the addition first you will get $3 + 4 = 12$ and then perform the multiplication, to get $12 \times 5 = 60$, but this is $incorrect$.

Take a look at the BODMAS rule, multiplication appears before addition. So we should perform the multiplication first, then the addition, so we get $4 \times 5 = 20$ and then $3 + 20 = 23$.

Our correct answer is $23$.

How, much is $20 - 12 \div 4$.

Here if we perform the subtraction first, then we get $20 - 12 = 8$ and then $8 \div 4 = 2$, but again, this is $incorrect$.

As per the BODMAS rule, division should be done before subtraction, therefore, we should get $12 \div 4 = 3$ and then,

$20 - 3 = 17$. Our correct answer is $17$.

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Now, let us take a look at some larger problems. Let us evaluate the following expression:

$70 \div 10 - 5 \times {3 - (4 \div \overline{7-5})}$

In this problem, you get to see a new symbol, which is the line above the numbers $7$ and $5$ which appears as $\overline{7-5}$. This symbol is called the $\unicode{0x201C}vinculum\unicode{0x201D}$. This sign means everything under the line, needs to be computed before any other operation outside the line. Let us get back to our problem:

$70 \div 10 - 5 \times {3 - (4 \div \overline{7-5})}$, we will perform the vinculum first, that is $7 - 5 = 2$.

$= 70 \div 10 - 5 \times {3 - (4 \div 2)}$

Now, we perform the operation inside the bracket. Remember, you will need to perform the innermost bracket first. Although, some textbooks mention $(,\ \{,\ [$ as first, second and third bracket respectively, it is always easy to go from the innermost bracket to the outermost bracket. So, now we perform the operation inside the innermost bracket and get $4 \div 2 = 2$

$= 70 \div 10 - 5 \times {3 - 2}$, now we will perform the operation inside the next bracket, which is $3 - 2 = 1$

$= 70 \div 10 - 5 \times 1$, now we will perform the division operation, which is $70 \div 10 = 7$

$= 7 - 5 \times 1$, now we will perform the multiplication, which is $5 \times 1 = 5$.

$= 7 - 5$

$= 2$

We have our answer as $2$.

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$\underline{Important\ Things\ To\ Remember}$

If your expression has two or more consecutive division $(\div)$ operations like the one below:

$24 \div 6 \div 2 + 8$ you must perform the division operations left to right. If you perform the second division first you will get an incorrect result, like:

$24 \div 6 \div 2 + 8$

$= 24 \div 3 + 8$

$= 8 + 8$

$= 16$, which is an incorrect answer.

The correct method would be to use the following steps:

$24 \div 6 \div 2 + 8$

$= 4 \div 2 + 8$

$= 2 + 8$

$= 10$, which is the correct answer.

Another thing you need to be very careful about. When your expression is left with no other operation but addition and subtraction, rearrange the expression with all the additions first then the subtractions, then perform the operations from left to right. We will see how:

Suppose we were to simplify $20 - 12 + 8$

Here if you perform the addition of $8$ with $12$ first you will get an incorrect result, like this:

$20 - 12 + 8$

$= 20 - 20$

$= 0$. But this is incorrect.

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If you rearrange the expression, and then go from left to right, you will get:

$20 - 12 + 8$

$= 20 + 8 - 12$

$= 28 - 12$

$= 16$ This is the correct answer.

We can take one more example of this:

Let us say you are simplifying:

$22 - 11 - 17 + 14$

You should not add $14$ to $17$, that will give you an incorrect result, rather we should first rearrange the expression as:

$22 + 14 - 11 - 17$

Now we should do the operations from left to right, like below:

$22 + 14 - 11 - 17$

$= 36 - 11 - 17$

$= 25 - 17$

$= 8$