Solving Equations by Interchanging Numbers

This problem requires us to find which pair of numbers, when swapped in the given equation, makes the equation mathematically correct. The given equation is:

\(7 + 56 \div 8 \times 2 – 13 = 11\)

To solve this, we will check each option by interchanging the specified numbers and then evaluating the resulting equation using the order of operations (BODMAS/PEMDAS).

Understanding BODMAS/PEMDAS

BODMAS or PEMDAS is a rule that dictates the sequence in which operations should be performed in a mathematical expression. It stands for:

• Brackets / Parentheses
• Orders (powers, roots) / Exponents
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

Testing the Options by Interchanging Numbers

Option 1: Interchange 7 and 2

If we interchange 7 and 2 in the original equation, the equation becomes:

\(2 + 56 \div 8 \times 7 – 13\)

Now, let's evaluate this expression using BODMAS:

• Division: \(56 \div 8 = 7\)
• Multiplication: \(7 \times 7 = 49\)
• Addition and Subtraction (from left to right): \(2 + 49 – 13 = 51 – 13 = 38\)

The result is 38. Since \(38 \neq 11\), interchanging 7 and 2 does not make the equation correct.

Option 2: Interchange 8 and 2

If we interchange 8 and 2 in the original equation, the equation becomes:

\(7 + 56 \div 2 \times 8 – 13\)

Now, let's evaluate this expression using BODMAS:

• Division: \(56 \div 2 = 28\)
• Multiplication: \(28 \times 8 = 224\)
• Addition and Subtraction (from left to right): \(7 + 224 – 13 = 231 – 13 = 218\)

The result is 218. Since \(218 \neq 11\), interchanging 8 and 2 does not make the equation correct.

Option 3: Interchange 7 and 13

If we interchange 7 and 13 in the original equation, the equation becomes:

\(13 + 56 \div 8 \times 2 – 7\)

Now, let's evaluate this expression using BODMAS:

• Division: \(56 \div 8 = 7\)
• Multiplication: \(7 \times 2 = 14\)
• Addition and Subtraction (from left to right): \(13 + 14 – 7 = 27 – 7 = 20\)

The result is 20. Since \(20 \neq 11\), interchanging 7 and 13 does not make the equation correct.

Option 4: Interchange 7 and 8

If we interchange 7 and 8 in the original equation, the equation becomes:

\(8 + 56 \div 7 \times 2 – 13\)

Now, let's evaluate this expression using BODMAS:

• Division: \(56 \div 7 = 8\)
• Multiplication: \(8 \times 2 = 16\)
• Addition and Subtraction (from left to right): \(8 + 16 – 13 = 24 – 13 = 11\)

The result is 11. Since \(11 = 11\), interchanging 7 and 8 makes the equation correct.

Conclusion

By systematically testing each option and applying the BODMAS rule, we find that interchanging the numbers 7 and 8 makes the equation \(8 + 56 \div 7 \times 2 – 13\) evaluate to 11, which is the right-hand side of the original target equation.

Revision Table: Key Concepts

Additional Information: Equation Solving Strategy

Problems involving making an equation correct by interchanging numbers often appear in reasoning and quantitative aptitude tests. A common strategy is to systematically test the given options. For each option:

1. Identify the numbers to be interchanged.
2. Rewrite the original equation with the numbers swapped.
3. Evaluate the new equation following the order of operations (BODMAS/PEMDAS).
4. Compare the result with the required value (the number on the right side of the original equation).
5. The option that yields the correct result is the answer.

This systematic approach ensures that all possibilities are checked and helps in accurately determining which interchange achieves the desired result.