What will come in the place of ‘?’ in the following equation, if ‘+’ and ‘ב are interchanged and also ‘−’ and ‘÷’ are interchanged? 4 + 8 ÷ 4 × 2 − 2 = ?
29
Evaluating Mathematical Expressions with Interchanged Operators
The question asks us to find the value of a given mathematical expression after interchanging certain operators. We are given the equation: \(4 + 8 \div 4 \times 2 - 2 = ?\)
The rules for interchanging operators are:
- '+' and '×' are interchanged.
- '−' and '÷' are interchanged.
Let's apply these changes to the original equation.
Applying the Operator Changes
The original equation is: \(4 + 8 \div 4 \times 2 - 2\)
Applying the changes:
- '+' becomes '×'
- '÷' becomes '−'
- '×' becomes '+'
- '−' becomes '÷'
So, the new equation becomes: \(4 \times 8 - 4 + 2 \div 2\)
Solving the New Equation using BODMAS/PEMDAS
Now we need to evaluate the new equation \(4 \times 8 - 4 + 2 \div 2\) using the order of operations (BODMAS/PEMDAS).
The order of operations is:
- Brackets / Parentheses
- Orders / Exponents
- Division and Multiplication (from left to right)
- Addition and Subtraction (from left to right)
Step-by-Step Calculation
- First, perform the Division: \(2 \div 2 = 1\)
The equation becomes: \(4 \times 8 - 4 + 1\) - Next, perform the Multiplication: \(4 \times 8 = 32\)
The equation becomes: \(32 - 4 + 1\) - Now, perform Addition and Subtraction from left to right.
Subtraction: \(32 - 4 = 28\)
The equation becomes: \(28 + 1\) - Finally, perform the Addition: \(28 + 1 = 29\)
So, the value of the expression after interchanging the operators is 29.
Summary of Steps
| Original Equation | Operator Changes Applied | New Equation |
|---|---|---|
| \(4 + 8 \div 4 \times 2 - 2\) | \(+ \leftrightarrow \times\), \( - \leftrightarrow \div\) | \(4 \times 8 - 4 + 2 \div 2\) |
| Step | Operation | Calculation | Resulting Expression |
|---|---|---|---|
| 1 | Division | \(2 \div 2 = 1\) | \(4 \times 8 - 4 + 1\) |
| 2 | Multiplication | \(4 \times 8 = 32\) | \(32 - 4 + 1\) |
| 3 | Subtraction | \(32 - 4 = 28\) | \(28 + 1\) |
| 4 | Addition | \(28 + 1 = 29\) | \(29\) |
The final result of the expression after the operator interchange is 29.
Revision Table: Operator Interchanges
| Original Operator | Interchanged With |
|---|---|
| + | × |
| × | + |
| − | ÷ |
| ÷ | − |
Additional Information: Understanding BODMAS/PEMDAS in Mathematical Operations
The BODMAS or PEMDAS rule is a mnemonic used to remember the correct order for evaluating mathematical expressions. It ensures that everyone gets the same answer when solving a multi-step calculation.
- B/P: Brackets/Parentheses - Operations inside brackets are performed first.
- O/E: Orders/Exponents - Powers and roots are calculated next.
- DM: Division and Multiplication - These are performed from left to right.
- AS: Addition and Subtraction - These are performed from left to right.
When solving problems with interchanged operators, it is crucial to first rewrite the expression with the new operators and then apply the BODMAS/PEMDAS rule correctly to the new expression.
Important Questions from Logical Puzzle
If ‘A’ denotes ‘addition’, ‘B’ denotes ‘subtraction’, ‘C’ denotes ‘multiplication’ and ‘D’ denotes ‘division’, then what will be the value of the following expression?
6 C (57 B 8) D 7 B 32 A 9
If, 5 + 7 = 47, 3 + 8 = 35, 9 + 2 = 29 then 7 + 4 = ?
Which two signs need to be interchanged to make the following equation correct?
63 ÷ 21 – 7 + 28 × 3 = 144
If ‘–’ stands for ‘÷’, ‘+’ stands for ‘×’, ‘÷’ stands for ‘–’ and ‘×’ stands for ‘+’, then 80 – 20 + 10 ÷ 8 × 10 is equal to:
Which two signs need to be interchanged to make the given equation correct?
3 ÷ 2 - 4 × 5 + 5 = 1
