Analyzing Equation by Interchanging Signs and Numbers

The problem asks us to find which pair of numbers and which pair of signs, when interchanged in the given equation, will make the equation mathematically correct. The given equation is:

$$784 \div 6 \times 9 - 30 + 14 = 528$$

We need to test each option provided to see which interchange results in a true statement when we evaluate the left side of the equation using the BODMAS rule (Brackets, Orders, Division and Multiplication, Addition and Subtraction).

Testing Option 1: Interchange 14 and 6, + and -

Let's apply the interchange proposed in Option 1. We swap the numbers 14 and 6, and the signs + and -.

• Original numbers: 6, 14
• Interchanged numbers: 14, 6
• Original signs: -, +
• Interchanged signs: +, -

The new equation after interchanging 14 with 6 and + with - becomes:

$$784 \div 14 \times 9 + 30 - 6$$

Now, let's evaluate this expression using BODMAS:

1. Division: $$784 \div 14 = 56$$
2. Multiplication: $$56 \times 9 = 504$$
3. Addition: $$504 + 30 = 534$$
4. Subtraction: $$534 - 6 = 528$$

The result of the left side after this interchange is 528. This matches the right side of the original equation (which is also 528). Therefore, interchanging 14 and 6, and + and - makes the equation correct.

Testing Other Options

Although we found the correct interchange, let's quickly check the other options to demonstrate why they are incorrect.

Testing Option 2: Interchange 784 and 30, + and x

Interchanging 784 with 30 and + with x in the original equation $784 \div 6 \times 9 - 30 + 14$ gives:

$$30 \div 6 + 9 - 784 \times 14$$

Evaluate using BODMAS:

1. Division: $$30 \div 6 = 5$$
2. Multiplication: $$784 \times 14 = 10976$$
3. Addition/Subtraction (left to right): $$5 + 9 - 10976 = 14 - 10976 = -10962$$

$-10962 \neq 528$. So, Option 2 is incorrect.

Testing Option 3: Interchange 784 and 6, × and -

Interchanging 784 with 6 and × with - in the original equation $784 \div 6 \times 9 - 30 + 14$ gives:

$$6 \div 784 - 9 \times 30 + 14$$

Evaluate using BODMAS:

1. Division: $$6 \div 784 \approx 0.00765$$
2. Multiplication: $$9 \times 30 = 270$$
3. Addition/Subtraction (left to right): $$0.00765 - 270 + 14 \approx -255.99$$

$-255.99 \neq 528$. So, Option 3 is incorrect.

Testing Option 4: Interchange 9 and 6, ÷ and +

Interchanging 9 with 6 and ÷ with + in the original equation $784 \div 6 \times 9 - 30 + 14$ gives:

$$784 + 9 \div 6 - 30 \times 14$$

Evaluate using BODMAS:

1. Division: $$9 \div 6 = 1.5$$
2. Multiplication: $$30 \times 14 = 420$$
3. Addition/Subtraction (left to right): $$784 + 1.5 - 420 = 785.5 - 420 = 365.5$$

$$365.5 \neq 528$$. So, Option 4 is incorrect.

Only the interchange proposed in Option 1 results in the correct equation.

Revision Table: BODMAS Rule

Remembering the correct order of operations is crucial for solving these types of problems accurately.

Additional Information: Solving Equation Interchange Problems

Problems involving interchanging signs and numbers require careful application of the order of operations after performing the specified swaps. It's essentially a trial-and-error method where you test each given option.

• Always write down the new equation clearly after the interchange.
• Apply the BODMAS/PEMDAS rule step-by-step.
• Perform division and multiplication from left to right as they appear.
• Perform addition and subtraction from left to right as they appear.
• Compare the final result on the left side with the number on the right side of the original equation.

This systematic approach helps avoid errors and efficiently finds the correct combination of interchanges.