Solving Equation by Interchanging Numbers

The problem asks us to identify which pair of numbers from the given options, when swapped in the equation, makes the equation true.

The original equation is:

\((357 \div 2 + 3) \div 11 - 4 = (28 \times 3 + 7) \div 9\)

Let's first evaluate both sides of the original equation to see if it's initially true (it usually isn't in these types of problems):

• Left Hand Side (LHS):
• \(357 \div 2 = 178.5\)
• \(178.5 + 3 = 181.5\)
• \(181.5 \div 11 = 16.5\)
• \(16.5 - 4 = 12.5\)
• So, LHS = 12.5
• Right Hand Side (RHS):
• \(28 \times 3 = 84\)
• \(84 + 7 = 91\)
• \(91 \div 9 \approx 10.11\)
• So, RHS \(\approx 10.11\)

Since \(12.5 \ne 10.11\), the original equation is false.

Now, we will test each option by interchanging the specified numbers in the equation and checking if the LHS equals the RHS.

Testing Option 1: Interchange 11 and 28

The new equation after swapping 11 and 28 is:

\((357 \div 2 + 3) \div 28 - 4 = (11 \times 3 + 7) \div 9\)

• LHS:
• \(357 \div 2 = 178.5\)
• \(178.5 + 3 = 181.5\)
• \(181.5 \div 28 \approx 6.48\)
• \(6.48 - 4 \approx 2.48\)
• So, LHS \(\approx 2.48\)
• RHS:
• \(11 \times 3 = 33\)
• \(33 + 7 = 40\)
• \(40 \div 9 \approx 4.44\)
• So, RHS \(\approx 4.44\)

Since \(2.48 \ne 4.44\), swapping 11 and 28 does not make the equation correct.

Testing Option 2: Interchange 2 and 3

The new equation after swapping 2 and 3 is:

\((357 \div 3 + 2) \div 11 - 4 = (28 \times 2 + 7) \div 9\)

• LHS:
• \(357 \div 3 = 119\)
• \(119 + 2 = 121\)
• \(121 \div 11 = 11\)
• \(11 - 4 = 7\)
• So, LHS = 7
• RHS:
• \(28 \times 2 = 56\)
• \(56 + 7 = 63\)
• \(63 \div 9 = 7\)
• So, RHS = 7

Since \(7 = 7\), swapping 2 and 3 makes the equation correct.

Testing Option 3: Interchange 7 and 9

The new equation after swapping 7 and 9 is:

\((357 \div 2 + 3) \div 11 - 4 = (28 \times 3 + 9) \div 7\)

• LHS: (Same as original LHS, as 2, 3, 11, 4 are not swapped)
• LHS = 12.5
• RHS:
• \(28 \times 3 = 84\)
• \(84 + 9 = 93\)
• \(93 \div 7 \approx 13.28\)
• So, RHS \(\approx 13.28\)

Since \(12.5 \ne 13.28\), swapping 7 and 9 does not make the equation correct.

Testing Option 4: Interchange 2 and 9

The new equation after swapping 2 and 9 is:

\((357 \div 9 + 3) \div 11 - 4 = (28 \times 3 + 7) \div 2\)

• LHS:
• \(357 \div 9 \approx 39.67\)
• \(39.67 + 3 = 42.67\)
• \(42.67 \div 11 \approx 3.88\)
• \(3.88 - 4 \approx -0.12\)
• So, LHS \(\approx -0.12\)
• RHS:
• \(28 \times 3 = 84\)
• \(84 + 7 = 91\)
• \(91 \div 2 = 45.5\)
• So, RHS = 45.5

Since \(-0.12 \ne 45.5\), swapping 2 and 9 does not make the equation correct.

Based on the calculations, interchanging the numbers 2 and 3 makes the given equation correct.

Here is a summary of the results:

Therefore, interchanging 2 and 3 makes the equation correct.

Revision Table: Key Concepts

Additional Information: Mathematical Operations

When solving equations or evaluating expressions, it's crucial to follow the correct order of mathematical operations. The acronyms BODMAS or PEMDAS help remember this order:

• B/P: Brackets / Parentheses - Solve the operations inside brackets first.
• O/E: Orders / Exponents - Solve powers, square roots, etc.
• DM: Division and Multiplication - Perform division and multiplication from left to right.
• AS: Addition and Subtraction - Perform addition and subtraction from left to right.

In this problem, we first evaluated expressions within parentheses, then performed division and multiplication, and finally addition and subtraction to simplify each side of the equation after interchanging the numbers.