Solving Equations by Interchanging Numbers and Signs

The question asks us to find the new values of two equations after applying specific interchanges to both numbers and mathematical signs. We need to interchange the numbers 3 and 9, and the signs × (multiplication) and + (addition).

Understanding the Interchange Rules

The rules for interchanging are:

• Wherever you see the number 3, replace it with the number 9.
• Wherever you see the number 9, replace it with the number 3.
• Wherever you see the sign ×, replace it with the sign +.
• Wherever you see the sign +, replace it with the sign ×.
• The other numbers (7, 8, 2, 4) and signs (–, ÷) remain unchanged.

Applying Interchanges and Solving Equation I

Let's apply the interchanges to the first equation:

Equation (I) original: $7 \times 9 \ndash 8 \div 2 + 3$

Applying the rules:

• Replace × with +
• Replace 9 with 3
• Replace + with ×
• Replace 3 with 9

Equation (I) after interchange: $7 + 3 \ndash 8 \div 2 \times 9$

Now, we evaluate the new equation using the BODMAS (or PEMDAS) rule, which dictates the order of operations:

• Brackets (or Parentheses)
• Orders (or Exponents)
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

Let's evaluate $7 + 3 \ndash 8 \div 2 \times 9$:

1. First, perform the division: $8 \div 2 = 4$. The equation becomes $7 + 3 \ndash 4 \times 9$.
2. Next, perform the multiplication: $4 \times 9 = 36$. The equation becomes $7 + 3 \ndash 36$.
3. Now, perform the addition: $7 + 3 = 10$. The equation becomes $10 \ndash 36$.
4. Finally, perform the subtraction: $10 \ndash 36 = \ndash 26$.

The value of equation (I) after interchanging is $\ndash 26$.

Applying Interchanges and Solving Equation II

Now, let's apply the interchanges to the second equation:

Equation (II) original: $4 \times 9 \ndash 3 + 8 \div 2$

Applying the rules:

• Replace × with +
• Replace 9 with 3
• Replace 3 with 9
• Replace + with ×

Equation (II) after interchange: $4 + 3 \ndash 9 \times 8 \div 2$

Let's evaluate $4 + 3 \ndash 9 \times 8 \div 2$ using the BODMAS rule:

1. First, perform the multiplication: $9 \times 8 = 72$. The equation becomes $4 + 3 \ndash 72 \div 2$.
2. Next, perform the division: $72 \div 2 = 36$. The equation becomes $4 + 3 \ndash 36$.
3. Now, perform the addition: $4 + 3 = 7$. The equation becomes $7 \ndash 36$.
4. Finally, perform the subtraction: $7 \ndash 36 = \ndash 29$.

The value of equation (II) after interchanging is $\ndash 29$.

Summary of Results

After interchanging numbers (3 and 9) and signs (× and +):

• The value of equation (I) is $\ndash 26$.
• The value of equation (II) is $\ndash 29$.

Therefore, the values of equation (I) and (II) respectively are $\ndash 26, \ndash 29$.

Revision Table: Key Concepts

Additional Information: Understanding BODMAS/PEMDAS

The BODMAS rule is crucial for solving mathematical expressions correctly. If operations are performed in the wrong order, the result will be incorrect. Let's break down the order:

• Brackets/Parentheses: Calculate anything inside brackets first. For example, $(2+3) \times 5$ requires you to calculate $2+3=5$ before multiplying by 5.
• Orders/Exponents: Next, calculate any powers or roots. For example, $5^2 + 10$ requires you to calculate $5^2 = 25$ before adding 10.
• Division and Multiplication: Perform division and multiplication from left to right as they appear in the expression. They have equal priority. For example, in $10 \div 2 \times 3$, you would first calculate $10 \div 2 = 5$, then $5 \times 3 = 15$.
• Addition and Subtraction: Finally, perform addition and subtraction from left to right as they appear. They also have equal priority. For example, in $5 + 8 \ndash 3$, you would calculate $5 + 8 = 13$, then $13 \ndash 3 = 10$.

Following this specific order ensures consistency and accuracy in evaluating mathematical expressions, especially in reasoning questions involving altered equations.