An arithmetic expression involving two numbers is easy to solve. On the other hand, expressions involving multiple operations such as addition, subtraction, multiplication and division are not easy to solve. We should take special attention when the expression contains brackets.
Order of operations (or operator precedence) is the order in which certain operations must be completed. For example, we do addition before subtraction and division before multiplication.
Brackets are of four types: ( ), { }, [ ] and ¯. The last one (¯) is known as vinculum (or bar). A vinculum is a horizontal line used in mathematical notation for a specific purpose. It is placed over a mathematical expression to indicate that the expression is to be considered as a group. If an expression contains vinculum, we must evaluate it first.
To solve these type of problems, we should use BODMAS, which means Bracket, Of, Division, Multiplication, Addition and Subtraction.
While using BODMAS, if an expression contains brackets like ( ), { } or [ ], we have first simplify the bracket followed by of (roughly means multiplication), then division, multiplication, addition and then subtraction from left to right. Solving the problem in the wrong order will result in a wrong answer.
Remember that addition, subtraction, multiplication, division are binary operations. In other words, using two numbers we can get one (with one exception that we cannot divide by zero). Moreover, the precedence of division and multiplication is more than that of addition and subtraction.
Following examples given in the table show how to evaluate the expressions.
| Expression | Wrong Way | Correct Way |
| 1 – 2 + 3 | = 1 – 5 = – 4 | = 1 + 3 – 2 = 4 – 2 = 2 |
| 5 × (4 + 3) | = 5 × 4 + 3 = 20 + 3 = 23 | = 5 × (7) = 5 × 7 = 35 |
| 3 + 5 × 6 | = 8 × 6 = 48 | = 3 + 30 = 33 |
| 60 ÷ 6 × 5 | = 60 ÷ 30 = 2 | = 10 × 5 = 50 |
Following steps are important while solving the problems of BODMAS.
- Step 1. Vinculum(s) must be evaluated FIRST. Then the other brackets ( ), { } and [ ] will be evaluated.
- Step 2. Next we will calculate of, if it is there in the expression.
- Step 3. Solve any multiplication or division problems, going from left to right.
- Step 4. Solve any addition or subtraction problems, going from left to right.
Note. If an expression is a mixture of positive and negative terms, then add all the positive terms and all the negative terms separately, then subtract.
Example 1. Find the value of the expression 10 + 8 × 90 ÷ 9 – 4.
Solution. To find the value of the given expression, we will use BODMAS. Following table will help us to understand the solution stepwise. Here, given expression is 10 + 8 × 90 ÷ 9 – 4.
| Evaluate | Final Expression | |
| Step 1 | 90 ÷ 9 = 10 | = 10 + 8 × 10 – 4 |
| Step 2 | 8 × 10 = 80 | = 10 + 80 – 4 |
| Step 3 | 10 + 80 = 90 | = 90 – 4 |
| Step 4 | 90 – 4 = 86 | = 86 |
Therefore, required value of the expression is 86.
Example 2. Find the value of the expression 14 – 8 + 3 + 8 × (24 ÷ 8).
Solution. To find the value of the given expression, we will use BODMAS. Following table will help us to understand the solution stepwise. Here, given expression is 14 – 8 + 3 + 8 × (24 ÷ 8).
| Evaluate | Final Expression | |
| Step 1 | (24 ÷ 8) = 3 | = 14 – 8 + 3 + 8 × 3 |
| Step 2 | 8 × 3 = 24 | = 14 – 8 + 3 + 24 |
| Step 3 | Rearrange | = 14 + 3 + 24 – 8 |
| Step 4 | 14 + 3 = 17 | = 17 + 24 – 8 |
| Step 5 | 17 + 24 = 41 | = 41 – 8 |
| Step 6 | 41 – 8 = 33 | = 33 |
Therefore, required value of the expression is 33.
Example 3. Find the value of the expression 4 × 5 + (14 + 8) – 36 ÷ 9.
Solution. To find the value of the given expression, we will use BODMAS. Following table will help us to understand the solution stepwise. Here, given expression is 4 × 5 + (14 + 8) – 36 ÷ 9.
| Evaluate | Final Expression | |
| Step 1 | (14 + 8) = 22 | = 4 × 5 + 22 – 36 ÷ 9 |
| Step 2 | 4 × 5 = 20 | = 20 + 22 – 36 ÷ 9 |
| Step 3 | 36 ÷ 9 = 4 | = 20 + 22 – 4 |
| Step 4 | 20 + 22 = 42 | = 42 – 4 |
| Step 5 | 42 – 4 = 38 | = 38 |
Therefore, required value of the expression is 38.
Example 4. Find the value of the expression 18 ÷ 6 × (4 − 3) + 6.
Solution. To find the value of the given expression, we will use BODMAS. Following table will help us to understand the solution stepwise. Here, given expression is 18 ÷ 6 × (4 − 3) + 6.
| Evaluate | Final Expression | |
| Step 1 | (4 − 3) = 1 | = 18 ÷ 6 × 1 + 6 |
| Step 2 | 18 ÷ 6 = 3 | = 3 × 1 + 6 |
| Step 3 | 3 × 1 = 3 | = 3 + 6 |
| Step 4 | 3 + 6 = 9 | = 9 |
Therefore, required value of the expression is 9.
Example 5. Find the value of the expression (28 ÷ 4) + 3 + (10 − 8) × 5.
Solution. To find the value of the given expression, we will use BODMAS. Here, given expression is (28 ÷ 4) + 3 + (10 − 8) × 5.
There are two brackets here. We can solve them simultaneously or we can do them one by one. Following table will help us to understand the solution stepwise.
| Evaluate | Final Expression | |
| Step 1 | (28 ÷ 4) = 7 | = 7 + 3 + (10 − 8) × 5 |
| Step 2 | (10 − 8) = 2 | = 7 + 3 + 2 × 5 |
| Step 3 | 2 × 5 = 10 | = 7 + 3 + 10 |
| Step 4 | 7 + 3 = 10 | = 10 + 10 |
| Step 5 | 10 + 10 = 20 | = 20 |
Therefore, required value of the expression is 20.
Example 6. Find the value of the expression (12 ÷ 3) + 3 + (16 − 7) × 4.
Solution. To find the value of the given expression, we will use BODMAS. Here, given expression is (12 ÷ 3) + 3 + (16 − 7) × 4.
There are two brackets here. We can solve them simultaneously or we can do them one by one. Following table will help us to understand the solution stepwise.
| Evaluate | Final Expression | |
| Step 1 | (12 ÷ 3) = 4 | = 4 + 3 + (16 − 7) × 4 |
| Step 2 | (16 − 7) = 9 | = 4 + 3 + 9 × 4 |
| Step 3 | 9 × 4 = 36 | = 4 + 3 + 36 |
| Step 4 | 4 + 3 = 7 | = 7 + 36 |
| Step 5 | 7 + 36 = 43 | = 43 |
Therefore, required value of the expression is 43.
Example 7. Find the value of the expression 3.5 ÷ 0.1 of 0.7 + 0.5 × 0.3 – 0.1.
Solution. To find the value of the given expression, we will use BODMAS. Here, given expression is 3.5 ÷ 0.1 of 0.7 + 0.5 × 0.3 – 0.1.
| Evaluate | Final Expression | |
| Step 1 | 0.1 of 0.7 = (0.1 × 0.7) = 0.07 | = 3.5 ÷ 0.07 + 0.5 × 0.3 – 0.1 |
| Step 2 | 3.5 ÷ 0.07 = 50 | = 50 + 0.5 × 0.3 – 0.1 |
| Step 3 | 0.5 × 0.3 = 0.15 | = 50 + 0.15 – 0.1 |
| Step 4 | 50 + 0.15 = 50.15 | = 50.15 – 0.1 |
| Step 5 | 50.15 – 0.1 = 50.05 | = 50.05 |
Therefore, required value of the expression is 50.05.
Example 8. Find the value of the expression 0.01 of 0.3 + 0.4 × 0.5 – 0.1 × 0.12.
Solution. To find the value of the given expression, we will use BODMAS. Here, given expression is 0.01 of 0.3 + 0.4 × 0.5 – 0.1 × 0.12.
| Evaluate | Final Expression | |
| Step 1 | 0.01 of 0.3 = (0.01 × 0.3) = 0.003 | = 0.003 + 0.4 × 0.5 – 0.1 × 0.12 |
| Step 2 | 0.4 × 0.5 = 0.02 | = 0.003 + 0.02 – 0.1 × 0.12 |
| Step 3 | 0.1 × 0.12 = 0.012 | = 0.003 + 0.02 – 0.012 |
| Step 4 | 0.003 + 0.02 = 0.023 | = 0.023 – 0.012 |
| Step 5 | 0.023 – 0.012 = 0.011 | = 0.011 |
Therefore, required value of the expression is 0.011.
Example 9. Simplify the expression 27 – [5 + {28 – (29 – 7)}].
Solution. To find the value of the given expression, we will use BODMAS. Here, given expression is 27 – [5 + {28 – (29 – 7)}].
Clearly, we need calculate ( ) bracket first. Then we will calculate { } and finally [ ] bracket.
| Evaluate | Final Expression | |
| Step 1 | (29 – 7) = 22 | = 27 – [5 + {28 – 22}] |
| Step 2 | {28 – 22} = 6 | = 27 – [5 + 6] |
| Step 3 | [5 + 6] = 11 | = 27 – 11 |
| Step 4 | 27 – 11 = 16 | = 16 |
Therefore, required value of the expression is 16.
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