Interchanging Signs to Correct the Equation

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

$$152 + 8 \times 16 \div 9 - 4 = 309$$

To solve this, we need to test each option by swapping the specified signs and then evaluating the equation using the standard order of operations (BODMAS/PEMDAS): Brackets, Orders, Division and Multiplication (from left to right), Addition and Subtraction (from left to right).

Evaluating the Original Equation

Let's first check the original equation:

$$152 + 8 \times 16 \div 9 - 4$$

• Multiplication first: $8 \times 16 = 128$
• Equation becomes: $152 + 128 \div 9 - 4$
• Division next: $128 \div 9 \approx 14.22$
• Equation becomes: $152 + 14.22 - 4$
• Addition: $152 + 14.22 = 166.22$
• Subtraction: $166.22 - 4 = 162.22$

Since $162.22 \neq 309$, the original equation is incorrect.

Testing the Options

Now, let's test each option:

Option 1: Interchange $\div$ and -

The equation becomes:

$$152 + 8 \times 16 - 9 \div 4 = 309$$

• Multiplication: $8 \times 16 = 128$
• Equation: $152 + 128 - 9 \div 4$
• Division: $9 \div 4 = 2.25$
• Equation: $152 + 128 - 2.25$
• Addition: $152 + 128 = 280$
• Subtraction: $280 - 2.25 = 277.75$

Since $277.75 \neq 309$, this option is incorrect.

Option 2: Interchange $\div$ and +

The equation becomes:

$$152 \div 8 \times 16 + 9 - 4 = 309$$

• Division first: $152 \div 8 = 19$
• Equation becomes: $19 \times 16 + 9 - 4$
• Multiplication next: $19 \times 16 = 304$
• Equation becomes: $304 + 9 - 4$
• Addition: $304 + 9 = 313$
• Subtraction: $313 - 4 = 309$

Since $309 = 309$, this option makes the equation correct.

Option 3: Interchange - and +

The equation becomes:

$$152 - 8 \times 16 \div 9 + 4 = 309$$

• Multiplication: $8 \times 16 = 128$
• Equation: $152 - 128 \div 9 + 4$
• Division: $128 \div 9 \approx 14.22$
• Equation: $152 - 14.22 + 4$
• Subtraction: $152 - 14.22 = 137.78$
• Addition: $137.78 + 4 = 141.78$

Since $141.78 \neq 309$, this option is incorrect.

Option 4: Interchange + and x

The equation becomes:

$$152 \times 8 + 16 \div 9 - 4 = 309$$

• Multiplication first: $152 \times 8 = 1216$
• Equation becomes: $1216 + 16 \div 9 - 4$
• Division next: $16 \div 9 \approx 1.78$
• Equation becomes: $1216 + 1.78 - 4$
• Addition: $1216 + 1.78 = 1217.78$
• Subtraction: $1217.78 - 4 = 1213.78$

Since $1213.78 \neq 309$, this option is incorrect.

Conclusion

By testing all the options, we found that interchanging the division ($\div$) and addition (+) signs results in a correct equation. The steps for Option 2 are:

1. Swap '+' and '$\div$': $152 \div 8 \times 16 + 9 - 4 = 309$
2. Perform division: $19 \times 16 + 9 - 4 = 309$
3. Perform multiplication: $304 + 9 - 4 = 309$
4. Perform addition: $313 - 4 = 309$
5. Perform subtraction: $309 = 309$

Therefore, the correct signs to interchange are $\div$ and +.