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
Which two numbers should be interchanged to make the given equation correct?
9 + 7 × 5 – 18 ÷ 2 = 3 × 4 – 10 + 45 ÷ 5Select the correct combination of mathematical signs that can sequentially replace the * signs and balance the equation.
60 * 2 * 3 * 6 * 5 * 43
Which of the following interchange of numbers and mathematical signs would make the given equation correct?
30 ÷ 6 × 4 + 15 - 35 = 25
Which two signs need to be interchanged to make the following equation correct?
23 + 84 ÷ 14 × 8 − 3 = 5
Select the correct combination of mathematical signs that can sequentially replace the * signs and make the equation correct.
68 * 138* 23 * 54 * 20
