Solving the Equation by Swapping Numbers

The problem asks us to find which pair of numbers, when interchanged from the given options, will make the following equation correct:

\( 168 \div 4 + 216 \div ( 78 \times 1 - 8) = 14 \)

We need to test each option by swapping the specified numbers and evaluating the left-hand side (LHS) of the equation to see if it equals the right-hand side (RHS) after the swap.

Testing Option 1: Swapping 78 and 216

If we interchange 78 and 216, the equation becomes:

\( 168 \div 4 + 78 \div ( 216 \times 1 - 8) = 14 \)

Let's calculate the LHS:

• Calculate the first term: \( 168 \div 4 = 42 \)
• Calculate the expression in the parenthesis: \( 216 \times 1 - 8 = 216 - 8 = 208 \)
• Calculate the second term: \( 78 \div 208 \)

So, LHS = \( 42 + 78 \div 208 \approx 42 + 0.375 = 42.375 \)

This does not equal the RHS (14). So, option 1 is incorrect.

Testing Option 2: Swapping 14 and 8

If we interchange 14 and 8, the equation becomes:

\( 168 \div 4 + 216 \div ( 78 \times 1 - 14) = 8 \)

Let's calculate the LHS:

• Calculate the first term: \( 168 \div 4 = 42 \)
• Calculate the expression in the parenthesis: \( 78 \times 1 - 14 = 78 - 14 = 64 \)
• Calculate the second term: \( 216 \div 64 \)

So, LHS = \( 42 + 216 \div 64 = 42 + 3.375 = 45.375 \)

This does not equal the RHS (8). So, option 2 is incorrect.

Testing Option 3: Swapping 78 and 14

If we interchange 78 and 14, the equation becomes:

\( 168 \div 4 + 216 \div ( 14 \times 1 - 8) = 78 \)

Let's calculate the LHS step-by-step following the order of operations (BODMAS/PEMDAS):

• Expression within parenthesis: \( 14 \times 1 - 8 \)
• Multiplication first: \( 14 \times 1 = 14 \)
• Then subtraction: \( 14 - 8 = 6 \)

Now the equation is:

\( 168 \div 4 + 216 \div 6 = 78 \)

• Perform divisions:
• First term: \( 168 \div 4 = 42 \)
• Second term: \( 216 \div 6 = 36 \)

Now the equation is:

\( 42 + 36 = 78 \)

• Perform addition: \( 42 + 36 = 78 \)

The LHS is 78, which is equal to the RHS (78). This swap makes the equation correct.

Testing Option 4: Swapping 216 and 168

If we interchange 216 and 168, the equation becomes:

\( 216 \div 4 + 168 \div ( 78 \times 1 - 8) = 14 \)

Let's calculate the LHS:

• Calculate the first term: \( 216 \div 4 = 54 \)
• Calculate the expression in the parenthesis: \( 78 \times 1 - 8 = 78 - 8 = 70 \)
• Calculate the second term: \( 168 \div 70 \)

So, LHS = \( 54 + 168 \div 70 = 54 + 2.4 = 56.4 \)

This does not equal the RHS (14). So, option 4 is incorrect.

Conclusion

Based on the evaluation of each option, interchanging the numbers 78 and 14 makes the given equation correct.

Revision Table: Checking Equation Swaps

Additional Information: Order of Operations

When solving mathematical equations or expressions, it is crucial to follow a specific order of operations. This order ensures that everyone gets the same answer for the same problem. A common acronym used to remember this order is BODMAS or PEMDAS.

• B/P: Brackets (Parentheses) - Operations inside brackets are performed first.
• O/E: Orders (Exponents) - Powers and square roots are calculated next.
• D/MD: Division and Multiplication - These are performed from left to right.
• A/AS: Addition and Subtraction - These are performed from left to right.

In the equation swapping problem, we consistently applied this order to correctly evaluate the expressions after swapping the numbers.