Analyzing the Equation and Goal

The question asks us to find which two mathematical signs in the given equation need to be swapped to make the equation correct. The equation provided is: \(625 \div 25 + 7 \times 318 - 112 = 381\).

First, let's evaluate the original equation using the order of operations, commonly known as BODMAS or PEMDAS, to see if it is already correct.

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

Let's calculate the left side of the original equation:

\(625 \div 25 + 7 \times 318 - 112\)

Perform Division and Multiplication first (from left to right):

• \(625 \div 25 = 25\)
• \(7 \times 318 = 2226\)

Substitute these values back into the equation:

\(25 + 2226 - 112\)

Now, perform Addition and Subtraction (from left to right):

• \(25 + 2226 = 2251\)
• \(2251 - 112 = 2139\)

So, the original equation evaluates to \(2139\). The equation states \(2139 = 381\), which is incorrect.

Now we will test each option by swapping the indicated signs and re-evaluating the equation to see if it becomes correct (equal to 381).

Testing Sign Interchange Options

Option 1: Interchange - and +

Original equation: \(625 \div 25 + 7 \times 318 - 112 = 381\)

Swap + and - signs:

\(625 \div 25 - 7 \times 318 + 112\)

Evaluate using BODMAS:

• Division: \(625 \div 25 = 25\)
• Multiplication: \(7 \times 318 = 2226\)

Substitute and calculate:

\(25 - 2226 + 112\)

• Subtraction: \(25 - 2226 = -2201\)
• Addition: \(-2201 + 112 = -2089\)

The result is \(-2089\). \(-2089 = 381\) is incorrect.

Option 2: Interchange ÷ and +

Original equation: \(625 \div 25 + 7 \times 318 - 112 = 381\)

Swap \(\div\) and + signs:

\(625 + 25 \div 7 \times 318 - 112\)

Evaluate using BODMAS:

• Division (left to right): \(25 \div 7\) (This division does not result in a whole number).
• Multiplication: \((25 \div 7) \times 318 = (25/7) \times 318 = 7950/7 \approx 1135.71\)

Substitute and calculate:

\(625 + (7950/7) - 112\)

Calculating further with fractions or decimals shows this will not equal 381.

\(625 + 1135.71 - 112 \approx 1648.71\). \(1648.71 = 381\) is incorrect.

Option 3: Interchange ÷ and -

Original equation: \(625 \div 25 + 7 \times 318 - 112 = 381\)

Swap \(\div\) and - signs:

\(625 - 25 + 7 \times 318 \div 112\)

Evaluate using BODMAS:

• Multiplication (left to right): \(7 \times 318 = 2226\)
• Division: \(2226 \div 112\) (This division does not result in a whole number). \(2226/112 = 1113/56 \approx 19.875\)

Substitute and calculate:

\(625 - 25 + (1113/56)\)

• Subtraction: \(625 - 25 = 600\)
• Addition: \(600 + (1113/56) \approx 600 + 19.875 = 619.875\)

The result is approximately \(619.875\). \(619.875 = 381\) is incorrect.

Option 4: Interchange + and x

Original equation: \(625 \div 25 + 7 \times 318 - 112 = 381\)

Swap + and \(\times\) signs:

\(625 \div 25 \times 7 + 318 - 112\)

Evaluate using BODMAS:

• Division and Multiplication (from left to right):
• \(625 \div 25 = 25\)
• \(25 \times 7 = 175\)

Substitute these values back into the equation:

\(175 + 318 - 112\)

Now, perform Addition and Subtraction (from left to right):

• \(175 + 318 = 493\)
• \(493 - 112 = 381\)

The result is \(381\). \(381 = 381\) is correct!

Conclusion on Swapping Signs

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

Revision Table: Equation Balancing Concepts

Additional Information: Operator Precedence

Operator precedence rules dictate the order in which operations are performed in a mathematical expression. BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) and PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) are two common acronyms for remembering this order. Division and Multiplication have the same level of precedence, and they are performed from left to right as they appear in the expression. Similarly, Addition and Subtraction have the same level of precedence and are performed from left to right.

In the solved problem, correctly applying the left-to-right rule for division and multiplication in the modified equation \(625 \div 25 \times 7 + 318 - 112\) was crucial. It was performed as \((625 \div 25) \times 7\), not \(625 \div (25 \times 7)\).

Understanding operator precedence is fundamental to solving algebraic and arithmetic problems accurately, especially those involving multiple operations.