Solve the following expression. 50 - [20 + (30 - (25 - 5)]
20
Understanding Mathematical Expressions and the Order of Operations
Solving mathematical expressions correctly requires following a specific order of operations. This order ensures that everyone gets the same answer when solving the same problem. A common way to remember this order is using mnemonics like BODMAS or PEMDAS.
- BODMAS stands for: Brackets, Orders (powers and square roots), Division and Multiplication (from left to right), Addition and Subtraction (from left to right).
- PEMDAS stands for: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
In the given expression, $50 - [20 + (30 - (25 - 5))]$, we have different types of brackets (parentheses and square brackets). We must start with the innermost bracket and work our way outwards.
Step-by-Step Solution
Let's break down the expression $50 - [20 + (30 - (25 - 5))]$ using the BODMAS/PEMDAS rule.
- Innermost Brackets (Parentheses): First, solve the expression inside the innermost parentheses.
Expression: $25 - 5$
Calculation: $25 - 5 = 20$
The expression now becomes: $50 - [20 + (30 - 20)]$
- Next Level Brackets (Parentheses): Now, solve the expression inside the next set of parentheses.
Expression: $30 - 20$
Calculation: $30 - 20 = 10$
The expression now becomes: $50 - [20 + 10]$
- Outer Brackets (Square Brackets): Next, solve the expression inside the square brackets.
Expression: $20 + 10$
Calculation: $20 + 10 = 30$
The expression now becomes: $50 - 30$
- Final Operation (Subtraction): Finally, perform the remaining subtraction.
Expression: $50 - 30$
Calculation: $50 - 30 = 20$
So, the value of the expression $50 - [20 + (30 - (25 - 5))]$ is 20.
Summary of Calculation Steps
| Step | Expression Part | Calculation | Result |
|---|---|---|---|
| 1 | $25 - 5$ | $25 - 5 = 20$ | $50 - [20 + (30 - 20)]$ |
| 2 | $30 - 20$ | $30 - 20 = 10$ | $50 - [20 + 10]$ |
| 3 | $20 + 10$ | $20 + 10 = 30$ | $50 - 30$ |
| 4 | $50 - 30$ | $50 - 30 = 20$ | $20$ |
The final result is 20.
Revision Table: Order of Operations (BODMAS/PEMDAS)
| Order | Operation Type | Examples |
|---|---|---|
| 1 | Brackets/Parentheses | $( ), [ ], \{ \}$ |
| 2 | Orders/Exponents | $x^2, \sqrt{x}$ |
| 3 | Division and Multiplication | $\div, \times$ (Left to Right) |
| 4 | Addition and Subtraction | $+, -$ (Left to Right) |
Additional Information: Importance of Order of Operations
Understanding and applying the correct order of operations is fundamental in mathematics. It ensures consistency and accuracy in calculations. Without a standard order, expressions could have multiple possible answers, leading to confusion. This rule is essential not only for basic arithmetic but also for algebra, calculus, and all other areas of mathematics. When dealing with nested brackets, always start from the innermost set and move outwards, simplifying each part according to the BODMAS/PEMDAS rules.
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