Solving the Equation by Interchanging Numbers

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

\( (63 \div 9 + 17) \div 11 = 15 - 195 \div 17 - 2 \)

To solve this, we will test each option by swapping the specified numbers and then evaluate both the Left Hand Side (LHS) and the Right Hand Side (RHS) of the modified equation to see if they are equal.

Testing the Options for Number Interchange

Testing Option 1: Swap 11 and 15

If we swap 11 and 15, the equation becomes:

\( (63 \div 9 + 17) \div \underline{15} = \underline{11} - 195 \div 17 - 2 \)

Let's evaluate the LHS:

\( \text{LHS} = (63 \div 9 + 17) \div 15 \)

\( = (7 + 17) \div 15 \)

\( = 24 \div 15 \)

\( = \frac{24}{15} = \frac{8}{5} = 1.6 \)

Now let's evaluate the RHS:

\( \text{RHS} = 11 - 195 \div 17 - 2 \)

\( = 11 - \frac{195}{17} - 2 \)

\( \frac{195}{17} \approx 11.47 \)

\( \text{RHS} \approx 11 - 11.47 - 2 = -0.47 - 2 = -2.47 \)

Since \( 1.6 \neq -2.47 \), swapping 11 and 15 does not make the equation correct.

Testing Option 2: Swap 17 and 15

If we swap 17 and 15, we replace every instance of 17 with 15 and every instance of 15 with 17. The equation becomes:

\( (63 \div 9 + \underline{15}) \div 11 = \underline{17} - 195 \div \underline{15} - 2 \)

Let's evaluate the LHS:

\( \text{LHS} = (63 \div 9 + 15) \div 11 \)

\( = (7 + 15) \div 11 \)

\( = 22 \div 11 \)

\( = 2 \)

Now let's evaluate the RHS:

\( \text{RHS} = 17 - 195 \div 15 - 2 \)

First, calculate \( 195 \div 15 \):

\( 195 \div 15 = 13 \)

Now substitute this value back into the RHS expression:

\( \text{RHS} = 17 - 13 - 2 \)

\( = 4 - 2 \)

\( = 2 \)

Since LHS = 2 and RHS = 2, LHS = RHS. Swapping 17 and 15 makes the equation correct.

Testing Option 3: Swap 195 and 63

If we swap 195 and 63, the equation becomes:

\( (\underline{195} \div 9 + 17) \div 11 = 15 - \underline{63} \div 17 - 2 \)

Let's evaluate the LHS:

\( \text{LHS} = (195 \div 9 + 17) \div 11 \)

\( 195 \div 9 = \frac{195}{9} = \frac{65}{3} \approx 21.67 \)

\( \text{LHS} = (\frac{65}{3} + 17) \div 11 \)

\( = (\frac{65 + 51}{3}) \div 11 = \frac{116}{3} \div 11 = \frac{116}{3 \times 11} = \frac{116}{33} \approx 3.52 \)

Now let's evaluate the RHS:

\( \text{RHS} = 15 - 63 \div 17 - 2 \)

\( 63 \div 17 = \frac{63}{17} \approx 3.71 \)

\( \text{RHS} = 15 - \frac{63}{17} - 2 \)

\( = 13 - \frac{63}{17} = \frac{13 \times 17 - 63}{17} = \frac{221 - 63}{17} = \frac{158}{17} \approx 9.29 \)

Since \( 3.52 \neq 9.29 \), swapping 195 and 63 does not make the equation correct.

Testing Option 4: Swap 2 and 15

If we swap 2 and 15, the equation becomes:

\( (63 \div 9 + 17) \div 11 = \underline{2} - 195 \div 17 - \underline{15} \)

Let's evaluate the LHS. The LHS remains the same as the original equation:

\( \text{LHS} = (63 \div 9 + 17) \div 11 \)

\( = (7 + 17) \div 11 \)

\( = 24 \div 11 \)

\( = \frac{24}{11} \approx 2.18 \)

Now let's evaluate the RHS:

\( \text{RHS} = 2 - 195 \div 17 - 15 \)

\( = 2 - \frac{195}{17} - 15 \)

\( \frac{195}{17} \approx 11.47 \)

\( \text{RHS} \approx 2 - 11.47 - 15 = -9.47 - 15 = -24.47 \)

Since \( 2.18 \neq -24.47 \), swapping 2 and 15 does not make the equation correct.

Conclusion

After testing all the options, we found that interchanging the numbers 17 and 15 makes the given equation correct.

Revision Table: Summary of Swaps

Additional Information on Solving Equations

When solving equations that require interchanging numbers or operators, it's crucial to carefully apply the BODMAS/PEMDAS rule (Brackets/Parentheses, Orders/Exponents, Division and Multiplication, Addition and Subtraction). This rule dictates the order of operations in a mathematical expression.

• Brackets/Parentheses: Solve expressions inside brackets first.
• Orders/Exponents: Calculate powers and roots next.
• Division and Multiplication: Perform division and multiplication from left to right.
• Addition and Subtraction: Perform addition and subtraction from left to right.

In this problem, after swapping the numbers, we used the BODMAS rule to evaluate both sides of the equation step-by-step, performing division before addition/subtraction, and working within parentheses first.