By Interchanging the given two numbers which of the following equation will be correct? 5 and 8
Solving Equations by Interchanging Numbers
The question asks us to find which equation becomes correct after interchanging the numbers 5 and 8 in the given expressions. We need to examine each option, swap the positions of 5 and 8, and then evaluate the resulting equation using the order of operations (BODMAS/PEMDAS).
Evaluating Option 1 after Interchanging 5 and 8
The original equation in Option 1 is 5 × 2 - 8 + 9 ÷ 1 = 20.
After interchanging 5 and 8, the equation becomes:
\(8 \times 2 - 5 + 9 \div 1 = ?\)
Let's evaluate this expression following BODMAS (Brackets, Orders/Exponents, Division and Multiplication (from left to right), Addition and Subtraction (from left to right)):
- First, perform the division: \(9 \div 1 = 9\). The equation is now \(8 \times 2 - 5 + 9 = ?\)
- Next, perform the multiplication: \(8 \times 2 = 16\). The equation is now \(16 - 5 + 9 = ?\)
- Now, perform addition and subtraction from left to right:
- \(16 - 5 = 11\). The equation is now \(11 + 9 = ?\)
- \(11 + 9 = 20\).
So, the result of the left side is 20. The equation becomes \(20 = 20\), which is a correct statement.
Thus, interchanging 5 and 8 makes the equation in Option 1 correct.
Evaluating Other Options (for completeness)
Let's also check the other options to confirm they are incorrect after interchanging 5 and 8.
Option 2 Analysis
Original equation: \(8 + 5 \times 4 - 9 \div 3 = 30\)
After interchanging 5 and 8: \(5 + 8 \times 4 - 9 \div 3 = ?\)
Evaluating:
- Division: \(9 \div 3 = 3\) → \(5 + 8 \times 4 - 3 = ?\)
- Multiplication: \(8 \times 4 = 32\) → \(5 + 32 - 3 = ?\)
- Addition/Subtraction: \(5 + 32 = 37\) → \(37 - 3 = ?\)
- \(37 - 3 = 34\)
Result: \(34 \neq 30\). Incorrect.
Option 3 Analysis
Original equation: \(5 \times 7 + 8 - 3 \div 1 = 55\)
After interchanging 5 and 8: \(8 \times 7 + 5 - 3 \div 1 = ?\)
Evaluating:
- Division: \(3 \div 1 = 3\) → \(8 \times 7 + 5 - 3 = ?\)
- Multiplication: \(8 \times 7 = 56\) → \(56 + 5 - 3 = ?\)
- Addition/Subtraction: \(56 + 5 = 61\) → \(61 - 3 = ?\)
- \(61 - 3 = 58\)
Result: \(58 \neq 55\). Incorrect.
Option 4 Analysis
Original equation: \(8 - 5 \times 2 + 6 \div 2 = -9\)
After interchanging 5 and 8: \(5 - 8 \times 2 + 6 \div 2 = ?\)
Evaluating:
- Division: \(6 \div 2 = 3\) → \(5 - 8 \times 2 + 3 = ?\)
- Multiplication: \(8 \times 2 = 16\) → \(5 - 16 + 3 = ?\)
- Addition/Subtraction: \(5 - 16 = -11\) → \(-11 + 3 = ?\)
- \(-11 + 3 = -8\)
Result: \(-8 \neq -9\). Incorrect.
Based on our evaluation, only the equation in Option 1 becomes correct after interchanging the numbers 5 and 8.
| Option | Original Equation | Equation after Interchanging 5 and 8 | Calculation Result | Correct? |
|---|---|---|---|---|
| 1 | \(5 \times 2 - 8 + 9 \div 1 = 20\) | \(8 \times 2 - 5 + 9 \div 1 = ?\) | 20 | Yes (20 = 20) |
| 2 | \(8 + 5 \times 4 - 9 \div 3 = 30\) | \(5 + 8 \times 4 - 9 \div 3 = ?\) | 34 | No (34 ≠ 30) |
| 3 | \(5 \times 7 + 8 - 3 \div 1 = 55\) | \(8 \times 7 + 5 - 3 \div 1 = ?\) | 58 | No (58 ≠ 55) |
| 4 | \(8 - 5 \times 2 + 6 \div 2 = -9\) | \(5 - 8 \times 2 + 6 \div 2 = ?\) | -8 | No (-8 ≠ -9) |
Revision Table: Checking Equation Interchanges
| Concept | Description | Application in Problem |
|---|---|---|
| Interchanging Numbers | Swapping the positions of two specific numbers within an expression or equation. | Swapping 5 and 8 in each given equation. |
| Order of Operations (BODMAS/PEMDAS) | Rules governing the sequence in which mathematical operations should be performed to ensure a unique result (Brackets, Orders, Division/Multiplication, Addition/Subtraction). | Used to evaluate the modified equations correctly after interchanging 5 and 8. |
| Equation Verification | Checking if the left side of an equation equals the right side after performing all calculations. | Comparing the calculated result of the left side to the given value on the right side for each modified equation. |
Additional Information: The Importance of BODMAS/PEMDAS
When solving mathematical expressions or equations, the order of operations is crucial. Without a standard order, the same expression could yield different results depending on which operation is performed first.
- Brackets (or Parentheses)
- Orders (or Exponents/Indices)
- Division and Multiplication (These have equal priority and are performed from left to right)
- Addition and Subtraction (These have equal priority and are performed from left to right)
In this problem, applying BODMAS correctly after interchanging the numbers is essential to determine if the resulting equation holds true.
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