Which two numbers and two signs should be interchanged to make the given equation correct? 5 + 44 - 8 × 25 = 203
8, 5 and -, ×
Solving Equations by Interchanging Numbers and Signs
The problem asks us to find which specific interchange of two numbers and two signs will make the given mathematical equation correct. The original equation is:
\(5 + 44 - 8 \times 25 = 203\)
Currently, let's evaluate the left side of the original equation using the BODMAS/PEMDAS rule (Multiplication before Addition and Subtraction):
\(5 + 44 - 8 \times 25 = 5 + 44 - 200\)
\(= 49 - 200\)
\(= -151\)
Since \(-151 \neq 203\), the original equation is incorrect. We need to test the given options by swapping the numbers and signs as specified and then re-evaluate the equation.
Testing Option 1: Interchange 8 and 5, and - and +
Original equation: \(5 + 44 - 8 \times 25 = 203\)
First, swap the numbers 8 and 5:
\(8 + 44 - 5 \times 25\)
Next, swap the signs - and + where they appear in the equation. The sign + is before 44, and - is before 8 (now 5):
\(8 - 44 + 5 \times 25\)
Now, evaluate the new equation following BODMAS:
\(8 - 44 + 5 \times 25 = 8 - 44 + 125\)
\(= -36 + 125\)
\(= 89\)
The result \(89\) is not equal to \(203\). So, this interchange is incorrect.
Testing Option 2: Interchange 8 and 44, and - and +
Original equation: \(5 + 44 - 8 \times 25 = 203\)
First, swap the numbers 8 and 44:
\(5 + 8 - 44 \times 25\)
Next, swap the signs - and +:
\(5 - 8 + 44 \times 25\)
Now, evaluate the new equation:
\(5 - 8 + 44 \times 25 = 5 - 8 + 1100\)
\(= -3 + 1100\)
\(= 1097\)
The result \(1097\) is not equal to \(203\). So, this interchange is incorrect.
Testing Option 3: Interchange 8 and 5, and - and ×
Original equation: \(5 + 44 - 8 \times 25 = 203\)
First, swap the numbers 8 and 5:
\(8 + 44 - 5 \times 25\)
Next, swap the signs - and ×:
\(8 + 44 \times 5 - 25\)
Now, evaluate the new equation:
\(8 + 44 \times 5 - 25 = 8 + 220 - 25\)
\(= 228 - 25\)
\(= 203\)
The result \(203\) is equal to \(203\). So, this interchange makes the equation correct.
Testing Option 4: Interchange 8 and 44, and - and ×
Original equation: \(5 + 44 - 8 \times 25 = 203\)
First, swap the numbers 8 and 44:
\(5 + 8 - 44 \times 25\)
Next, swap the signs - and ×:
\(5 + 8 \times 44 - 25\)
Now, evaluate the new equation:
\(5 + 8 \times 44 - 25 = 5 + 352 - 25\)
\(= 357 - 25\)
\(= 332\)
The result \(332\) is not equal to \(203\). So, this interchange is incorrect.
Conclusion: Correct Interchange for Equation
After testing all the options, the interchange of the numbers 8 and 5 and the signs - and × results in the correct equation:
\(8 + 44 \times 5 - 25 = 203\)
Revision Table: Equation Interchange Results
| Interchange (Numbers, Signs) | New Equation | Evaluation | Result | Correct? |
|---|---|---|---|---|
| (8, 5), (-, +) | \(8 - 44 + 5 \times 25\) | \(8 - 44 + 125 = -36 + 125\) | \(89\) | No |
| (8, 44), (-, +) | \(5 - 8 + 44 \times 25\) | \(5 - 8 + 1100 = -3 + 1100\) | \(1097\) | No |
| (8, 5), (-, ×) | \(8 + 44 \times 5 - 25\) | \(8 + 220 - 25 = 228 - 25\) | \(203\) | Yes |
| (8, 44), (-, ×) | \(5 + 8 \times 44 - 25\) | \(5 + 352 - 25 = 357 - 25\) | \(332\) | No |
Additional Information: Order of Mathematical Operations
The order in which mathematical operations are performed significantly affects the result of an expression. Following the standard order ensures consistent and correct calculations.
- BODMAS / PEMDAS: This rule dictates the sequence:
- B/P: Brackets / Parentheses - Evaluate expressions inside brackets first.
- O/E: Orders / Exponents - Evaluate powers and roots next.
- DM: Division and Multiplication - Perform these from left to right.
- AS: Addition and Subtraction - Perform these from left to right.
- In the given problem, we only have addition, subtraction, and multiplication, so the order is Multiplication first, then Addition and Subtraction from left to right.
- Interchanging numbers or signs changes the structure of the equation, requiring re-evaluation according to these rules.
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