Solving Mathematical Equation by Symbol Interchange

The problem asks us to identify which pair of mathematical symbols, when interchanged in the given equation, makes the equation correct. The equation is:

\((35 \div 7 \times 50) \div 11 \times 1 + 3 = 8\)

We will test each option by swapping the indicated symbols and then evaluating the expression using the order of operations (BODMAS/PEMDAS: Brackets, Orders, Division/Multiplication, Addition/Subtraction).

Checking Option 1: Interchange + and −

If we interchange the symbols '+' and '\(-\)', the equation becomes:

\((35 \div 7 \times 50) \div 11 \times 1 - 3 = 8\)

Let's evaluate the left side:

• First, calculate the expression inside the brackets: \((35 \div 7 \times 50)\)
• \(35 \div 7 = 5\)
• \(5 \times 50 = 250\)
• So, the equation becomes: \(250 \div 11 \times 1 - 3 = 8\)
• Next, perform division and multiplication from left to right:
• \(250 \div 11 \approx 22.727\)
• \(22.727 \times 1 = 22.727\)
• The equation is now: \(22.727 - 3 = 8\)
• Finally, perform subtraction:
• \(22.727 - 3 = 19.727\)

So, \(19.727 = 8\), which is false. Interchanging + and − does not make the equation correct.

Checking Option 2: Interchange ÷ and +

If we interchange the symbols '\(\div\)' and '+', the equation becomes:

\((35 + 7 \times 50) + 11 \times 1 \div 3 = 8\)

Let's evaluate the left side:

• First, calculate inside the brackets: \((35 + 7 \times 50)\)
• Inside brackets, perform multiplication first: \(7 \times 50 = 350\)
• Then, perform addition: \(35 + 350 = 385\)
• So, the equation becomes: \(385 + 11 \times 1 \div 3 = 8\)
• Next, perform multiplication and division from left to right:
• \(11 \times 1 = 11\)
• \(11 \div 3 \approx 3.667\)
• The equation is now: \(385 + 3.667 = 8\)
• Finally, perform addition:
• \(385 + 3.667 = 388.667\)

So, \(388.667 = 8\), which is false. Interchanging ÷ and + does not make the equation correct.

Checking Option 3: Interchange × and ÷

If we interchange the symbols '\(\times\)' and '\(\div\)', the equation becomes:

\((35 \times 7 \div 50) \times 11 \div 1 + 3 = 8\)

Let's evaluate the left side:

• First, calculate inside the brackets: \((35 \times 7 \div 50)\)
• Inside brackets, perform multiplication and division from left to right:
• \(35 \times 7 = 245\)
• \(245 \div 50 = 4.9\)
• So, the equation becomes: \(4.9 \times 11 \div 1 + 3 = 8\)
• Next, perform multiplication and division from left to right:
• \(4.9 \times 11 = 53.9\)
• \(53.9 \div 1 = 53.9\)
• The equation is now: \(53.9 + 3 = 8\)
• Finally, perform addition:
• \(53.9 + 3 = 56.9\)

So, \(56.9 = 8\), which is false. Interchanging × and ÷ does not make the equation correct.

Checking Option 4: Interchange + and ×

If we interchange the symbols '+' and '\(\times\)', the equation becomes:

\((35 \div 7 + 50) \div 11 + 1 \times 3 = 8\)

Let's evaluate the left side:

• First, calculate inside the brackets: \((35 \div 7 + 50)\)
• Inside brackets, perform division first: \(35 \div 7 = 5\)
• Then, perform addition: \(5 + 50 = 55\)
• So, the equation becomes: \(55 \div 11 + 1 \times 3 = 8\)
• Next, perform division and multiplication from left to right:
• \(55 \div 11 = 5\)
• \(1 \times 3 = 3\)
• The equation is now: \(5 + 3 = 8\)
• Finally, perform addition:
• \(5 + 3 = 8\)

So, \(8 = 8\), which is true. Interchanging + and \(\times\) makes the equation correct.

Therefore, interchanging the symbols '+' and '\(\times\)' makes the given equation correct.

Revision Table: Symbol Interchange Results

Additional Information on Order of Operations

The order of operations is crucial when evaluating mathematical expressions. The widely used acronyms BODMAS or PEMDAS help remember the correct sequence:

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

In the given problem, we applied this order diligently within brackets first, then performed division and multiplication, and finally addition and subtraction to correctly evaluate the modified equations after each symbol interchange.