Analyzing Equation Interchange Problems

The problem asks us to find the correct interchange of numbers and mathematical signs that makes the given equation true. The original equation is:

\(24 \times 9 + 6 \div 24 - 18 = 22\)

We need to test each given option by performing the suggested swaps and then evaluating the resulting equation using the BODMAS rule (Brackets, Orders, Division and Multiplication, Addition and Subtraction) to see if the Left Hand Side (LHS) equals the Right Hand Side (RHS), which is 22.

Step-by-Step Testing of Interchange Options

Option 1: Interchange 18 and 22, + and -

Original equation: \(24 \times 9 + 6 \div 24 - 18 = 22\)

As per Option 1, we swap the numbers 18 and 22, and the signs + and -.

The number 18 appears on the LHS, and 22 is the RHS value. We interchange these numbers and also interchange the + and - signs within the expression.

The modified equation becomes:

\(24 \times 9 - 6 \div 24 + 22 = 18\)

Now, let's evaluate the LHS using BODMAS:

• First, perform Multiplication and Division from left to right:
• \(24 \times 9 = 216\)
• \(6 \div 24 = 0.25\)
• The equation becomes: \(216 - 0.25 + 22\)
• Next, perform Addition and Subtraction from left to right:
• \(216 - 0.25 = 215.75\)
• \(215.75 + 22 = 237.75\)

The LHS is 237.75, and the RHS is 18. Since \(237.75 \neq 18\), this interchange does not make the equation correct.

Option 2: Interchange 24 and 6, × and +

Original equation: \(24 \times 9 + 6 \div 24 - 18 = 22\)

As per Option 2, we swap the numbers 24 and 6, and the signs × and +.

• Swap 24 with 6 and 6 with 24 in the equation: \(6 \times 9 + 24 \div 6 - 18 = 22\)
• Swap × with + and + with × in the equation: \(6 + 9 \times 24 \div 6 - 18 = 22\)

The modified equation is:

\(6 + 9 \times 24 \div 6 - 18 = 22\)

Now, let's evaluate the LHS using BODMAS:

• First, perform Multiplication and Division from left to right:
• \(24 \div 6 = 4\)
• The equation becomes: \(6 + 9 \times 4 - 18\)
• \(9 \times 4 = 36\)
• The equation becomes: \(6 + 36 - 18\)
• Next, perform Addition and Subtraction from left to right:
• \(6 + 36 = 42\)
• \(42 - 18 = 24\)

The LHS is 24, and the RHS is 22. Since \(24 \neq 22\), this interchange does not make the equation correct.

Option 3: Interchange 6 and 9, ÷ and +

Original equation: \(24 \times 9 + 6 \div 24 - 18 = 22\)

As per Option 3, we swap the numbers 6 and 9, and the signs ÷ and +.

• Swap 6 with 9 and 9 with 6 in the equation: \(24 \times 6 + 9 \div 24 - 18 = 22\)
• Swap ÷ with + and + with ÷ in the equation: \(24 \times 6 \div 9 + 24 - 18 = 22\)

The modified equation is:

\(24 \times 6 \div 9 + 24 - 18 = 22\)

Now, let's evaluate the LHS using BODMAS:

• First, perform Multiplication and Division from left to right:
• \(24 \times 6 = 144\)
• The equation becomes: \(144 \div 9 + 24 - 18\)
• \(144 \div 9 = 16\)
• The equation becomes: \(16 + 24 - 18\)
• Next, perform Addition and Subtraction from left to right:
• \(16 + 24 = 40\)
• \(40 - 18 = 22\)

The LHS is 22, and the RHS is 22. Since \(22 = 22\), this interchange makes the equation correct.

Option 4: Interchange 6 and 18, ÷ and ×

Original equation: \(24 \times 9 + 6 \div 24 - 18 = 22\)

As per Option 4, we swap the numbers 6 and 18, and the signs ÷ and ×.

• Swap 6 with 18 and 18 with 6 in the equation: \(24 \times 9 + 18 \div 24 - 6 = 22\)
• Swap ÷ with × and × with ÷ in the equation: \(24 \div 9 + 18 \times 24 - 6 = 22\)

The modified equation is:

\(24 \div 9 + 18 \times 24 - 6 = 22\)

Now, let's evaluate the LHS using BODMAS:

• First, perform Multiplication and Division from left to right:
• \(24 \div 9 = \frac{24}{9} = \frac{8}{3}\)
• \(18 \times 24 = 432\)
• The equation becomes: \(\frac{8}{3} + 432 - 6\)
• Next, perform Addition and Subtraction from left to right:
• \(\frac{8}{3} + 432 - 6 = \frac{8}{3} + 426\)
• \(\frac{8}{3} + 426 = \frac{8 + 426 \times 3}{3} = \frac{8 + 1278}{3} = \frac{1286}{3}\)

The LHS is \(\frac{1286}{3}\), and the RHS is 22. Since \(\frac{1286}{3} \neq 22\), this interchange does not make the equation correct.

Conclusion

After testing all the given options, we found that interchanging the numbers 6 and 9, and the signs ÷ and + makes the equation \(24 \times 9 + 6 \div 24 - 18 = 22\) correct. The equation becomes \(24 \times 6 \div 9 + 24 - 18 = 22\), which simplifies to \(22 = 22\).

Revision Table: Checking Options

Additional Information: BODMAS Rule and Equation Solving

Problems involving interchanging signs and numbers require careful application of the order of mathematical operations, commonly remembered by the acronym BODMAS or PEMDAS.

• BODMAS:
  • Brackets
  • Orders (powers, roots)
  • Division and Multiplication (from left to right)
  • Addition and Subtraction (from left to right)

When solving such problems, always remember to first apply the required interchanges correctly to the original equation. Then, evaluate the new equation step-by-step following the BODMAS rule. Compare the final result of the LHS evaluation with the RHS value to verify if the equation holds true. This systematic approach helps avoid errors in calculations and ensures the correct identification of the valid interchange.