Solving Mathematical Equation by Interchanging Signs

The problem asks us to find which two mathematical signs in the given equation should be swapped to make the equation true. The original equation is:

\( 25 + 14 \times 63 - 870 \div 29 = 383 \)

Let's first calculate the value of the original expression following the standard order of operations (BODMAS/PEMDAS):

• Division: \( 870 \div 29 = 30 \)
• Multiplication: \( 14 \times 63 = 882 \)
• Now substitute these back: \( 25 + 882 - 30 \)
• Addition: \( 25 + 882 = 907 \)
• Subtraction: \( 907 - 30 = 877 \)

So, the original equation is \( 877 = 383 \), which is false. We need to test each given option by interchanging the specified signs.

Testing Options for Sign Interchange

Option 1: Interchange + and -

If we interchange the '+' and '-' signs, the equation becomes:

\( 25 - 14 \times 63 + 870 \div 29 \)

Let's calculate the value:

• Division: \( 870 \div 29 = 30 \)
• Multiplication: \( 14 \times 63 = 882 \)
• Substitute back: \( 25 - 882 + 30 \)
• Subtraction/Addition (from left to right): \( 25 - 882 = -857 \)
• \( -857 + 30 = -827 \)

So, the equation is \( -827 = 383 \), which is false.

Option 2: Interchange × and -

If we interchange the '\(\times\)' and '-' signs, the equation becomes:

\( 25 + 14 - 63 \times 870 \div 29 \)

Let's calculate the value following BODMAS/PEMDAS:

• Multiplication/Division (from left to right): \( 63 \times 870 = 54810 \)
• \( 54810 \div 29 = 1890 \)
• Substitute back: \( 25 + 14 - 1890 \)
• Addition/Subtraction (from left to right): \( 25 + 14 = 39 \)
• \( 39 - 1890 = -1851 \)

So, the equation is \( -1851 = 383 \), which is false.

Option 3: Interchange - and ÷

If we interchange the '-' and '\(\div\)' signs, the equation becomes:

\( 25 + 14 \times 63 \div 870 - 29 \)

Let's calculate the value following BODMAS/PEMDAS:

• Multiplication/Division (from left to right): \( 14 \times 63 = 882 \)
• \( 882 \div 870 = \frac{882}{870} = \frac{147}{145} \)
• Substitute back: \( 25 + \frac{147}{145} - 29 \)
• Addition/Subtraction (from left to right): \( 25 - 29 = -4 \)
• \( -4 + \frac{147}{145} = \frac{-4 \times 145 + 147}{145} = \frac{-580 + 147}{145} = \frac{-433}{145} \)

So, the equation is \( \frac{-433}{145} = 383 \) (or approximately \( -2.986 = 383 \)), which is false.

Option 4: Interchange × and +

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

\( 25 \times 14 + 63 - 870 \div 29 \)

Let's calculate the value following BODMAS/PEMDAS:

• Multiplication: \( 25 \times 14 = 350 \)
• Division: \( 870 \div 29 = 30 \)
• Substitute back: \( 350 + 63 - 30 \)
• Addition/Subtraction (from left to right): \( 350 + 63 = 413 \)
• \( 413 - 30 = 383 \)

So, the equation is \( 383 = 383 \), which is true.

Interchanging the '\(\times\)' and '+' signs makes the equation correct.

Revision Table: Equation Sign Swap

The analysis confirms that swapping the multiplication (\(\times\)) and addition (+) signs results in a correct mathematical statement.

Additional Information: Order of Operations (BODMAS/PEMDAS)

When solving mathematical expressions with multiple operations, we follow a specific order to ensure consistency. This order is commonly remembered by the acronyms BODMAS or PEMDAS.

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

Understanding and applying this order is crucial for accurately solving mathematical expressions and equations.