Interchange two signs and two numbers in the equation 8 - 9 x 4 / 6 + 7 = 5 to make it correct.

To solve this problem, we need to find which two signs and which two numbers, when interchanged, will make the given equation correct.

The original equation is:

$$8 - 9 \times 4 \div 6 + 7 = 5$$

First, let's evaluate the left side of the original equation using the order of operations (BODMAS/PEMDAS):

1. Multiplication: $9 \times 4 = 36$
2. Division: $36 \div 6 = 6$ Substitute these values back into the expression:

$$8 - 6 + 7$$

3. Subtraction: $8 - 6 = 2$
4. Addition: $2 + 7 = 9$ So, the original equation evaluates to $9 = 5$, which is incorrect. We need to find an interchange that results in 5.

Now, let's test each option:

• Option 1: Interchange $+$ and $-$, and 9 and 6 The equation becomes:

  $$8 + 6 \times 4 \div 9 - 7 = 5$$

  Evaluate the left side:

  1. Multiplication: $6 \times 4 = 24$
  2. Division: $24 \div 9 = \frac{24}{9} = \frac{8}{3}$ Substitute these values:
  $$8 + \frac{8}{3} - 7$$ $$1 + \frac{8}{3} = \frac{3}{3} + \frac{8}{3} = \frac{11}{3}$$

  Since $\frac{11}{3} \neq 5$, this option is incorrect.

• Option 2: Interchange $+$ and $-$, and 8 and 7 The equation becomes:

  $$7 - 9 \times 4 \div 6 + 8 = 5$$

  Evaluate the left side:

  1. Multiplication: $9 \times 4 = 36$
  2. Division: $36 \div 6 = 6$ Substitute these values:
  $$7 - 6 + 8$$
  3. Subtraction: $7 - 6 = 1$
  4. Addition: $1 + 8 = 9$ Wait, I made a mistake in my scratchpad. Let's re-evaluate this option carefully. Original: $8 - 9 \times 4 \div 6 + 7 = 5$ Interchange $+$ and $-$ : $8 + 9 \times 4 \div 6 - 7 = 5$ Interchange 8 and 7: $7 + 9 \times 4 \div 6 - 8 = 5$

  Evaluate the left side of the modified equation:

  $$7 + 9 \times 4 \div 6 - 8$$
  1. Multiplication: $9 \times 4 = 36$
  2. Division: $36 \div 6 = 6$ Substitute these values:
  $$7 + 6 - 8$$
  3. Addition: $7 + 6 = 13$
  4. Subtraction: $13 - 8 = 5$ Since $5 = 5$, this option makes the equation correct.

• Option 3: Interchange $\times$ and $\div$, and 6 and 7 The equation becomes:

  $$8 - 9 \div 4 \times 7 + 6 = 5$$

  Evaluate the left side:

  1. Division: $9 \div 4 = \frac{9}{4}$
  2. Multiplication: $\frac{9}{4} \times 7 = \frac{63}{4}$ Substitute these values:
  $$8 - \frac{63}{4} + 6$$ $$14 - \frac{63}{4} = \frac{56}{4} - \frac{63}{4} = -\frac{7}{4}$$

  Since $-\frac{7}{4} \neq 5$, this option is incorrect.

• Option 4: Interchange $\div$ and $+$, and 8 and 7 The equation becomes:

  $$7 - 9 \times 4 + 6 \div 8 = 5$$

  Evaluate the left side:

  1. Multiplication: $9 \times 4 = 36$
  2. Division: $6 \div 8 = \frac{6}{8} = \frac{3}{4}$ Substitute these values:
  $$7 - 36 + \frac{3}{4}$$ $$-29 + \frac{3}{4} = -\frac{116}{4} + \frac{3}{4} = -\frac{113}{4}$$

  Since $-\frac{113}{4} \neq 5$, this option is incorrect.

Therefore, interchanging the $+$ and $-$ signs, and the numbers 8 and 7 makes the equation correct.

The final answer is $\text{2}$.