Solving Math Equation by Interchanging Numbers

The problem asks us to find which pair of numbers, when swapped in the given equation, makes the equation true. We need to test each option provided and calculate the result of the equation after the interchange.

The original equation is:

\(18 + (12 \times 6) + (63 \div 7) \times 4 + {(27)^{\frac{1}{3}}} - 11 = 112\)

First, let's calculate the value of the original equation following the order of operations (BODMAS/PEMDAS):

• Brackets first: \(12 \times 6 = 72\) and \(63 \div 7 = 9\).
• Powers (cube root): \((27)^{\frac{1}{3}} = 3\) (since \(3 \times 3 \times 3 = 27\)).

The equation becomes:

\(18 + 72 + 9 \times 4 + 3 - 11\)

• Multiplication: \(9 \times 4 = 36\).

The equation is now:

\(18 + 72 + 36 + 3 - 11\)

• Addition and Subtraction from left to right:
• \(18 + 72 = 90\)
• \(90 + 36 = 126\)
• \(126 + 3 = 129\)
• \(129 - 11 = 118\)

The value of the original equation is 118, which is not equal to 112. So, we must interchange numbers as per the options.

Testing Option 1: Interchange 18 and 11

Swap 18 and 11 in the equation:

\(11 + (12 \times 6) + (63 \div 7) \times 4 + {(27)^{\frac{1}{3}}} - 18\)

Using the intermediate calculations from the original equation:

\(11 + 72 + 9 \times 4 + 3 - 18\)

After multiplication:

\(11 + 72 + 36 + 3 - 18\)

Add and subtract from left to right:

• \(11 + 72 = 83\)
• \(83 + 36 = 119\)
• \(119 + 3 = 122\)
• \(122 - 18 = 104\)

The result is 104, which is not 112. Option 1 is incorrect.

Testing Option 2: Interchange 12 and 4

Swap 12 and 4 in the equation:

\(18 + (4 \times 6) + (63 \div 7) \times 12 + {(27)^{\frac{1}{3}}} - 11\)

Following BODMAS:

• Brackets: \(4 \times 6 = 24\) and \(63 \div 7 = 9\).
• Power: \((27)^{\frac{1}{3}} = 3\).

The equation becomes:

\(18 + 24 + 9 \times 12 + 3 - 11\)

• Multiplication: \(9 \times 12 = 108\).

The equation is now:

\(18 + 24 + 108 + 3 - 11\)

Add and subtract from left to right:

• \(18 + 24 = 42\)
• \(42 + 108 = 150\)
• \(150 + 3 = 153\)
• \(153 - 11 = 142\)

The result is 142, which is not 112. Option 2 is incorrect.

Testing Option 3: Interchange 6 and 4

Swap 6 and 4 in the equation:

\(18 + (12 \times 4) + (63 \div 7) \times 6 + {(27)^{\frac{1}{3}}} - 11\)

Following BODMAS:

• Brackets: \(12 \times 4 = 48\) and \(63 \div 7 = 9\).
• Power: \((27)^{\frac{1}{3}} = 3\).

The equation becomes:

\(18 + 48 + 9 \times 6 + 3 - 11\)

• Multiplication: \(9 \times 6 = 54\).

The equation is now:

\(18 + 48 + 54 + 3 - 11\)

Add and subtract from left to right:

• \(18 + 48 = 66\)
• \(66 + 54 = 120\)
• \(120 + 3 = 123\)
• \(123 - 11 = 112\)

The result is 112, which matches the target value. Option 3 is correct.

Testing Option 4: Interchange 18 and 12

Swap 18 and 12 in the equation:

\(12 + (18 \times 6) + (63 \div 7) \times 4 + {(27)^{\frac{1}{3}}} - 11\)

Following BODMAS:

• Brackets: \(18 \times 6 = 108\) and \(63 \div 7 = 9\).
• Power: \((27)^{\frac{1}{3}} = 3\).

The equation becomes:

\(12 + 108 + 9 \times 4 + 3 - 11\)

After multiplication:

\(12 + 108 + 36 + 3 - 11\)

Add and subtract from left to right:

• \(12 + 108 = 120\)
• \(120 + 36 = 156\)
• \(156 + 3 = 159\)
• \(159 - 11 = 148\)

The result is 148, which is not 112. Option 4 is incorrect.

Based on the calculations, interchanging the numbers 6 and 4 makes the equation correct.

Revision Table: Equation Interchange Results

Additional Information: Understanding Order of Operations and Cube Roots

Solving mathematical equations requires following a specific order of operations to ensure consistency and accuracy. A commonly used mnemonic is BODMAS or PEMDAS:

• B/P: Brackets / Parentheses (solve operations inside first)
• O/E: Orders / Exponents (powers, roots, etc.)
• DM: Division and Multiplication (from left to right)
• AS: Addition and Subtraction (from left to right)

In this problem, we also encountered a cube root: \((27)^{\frac{1}{3}}\). The cube root of a number is the value that, when multiplied by itself three times, gives the original number. For example, the cube root of 27 is 3 because \(3 \times 3 \times 3 = 27\).

Applying these rules systematically is crucial when solving problems involving multiple operations and brackets, especially when testing interchanges.