Finding the Incorrect Equation by Interchanging Signs and Numbers

The question asks us to identify which of the given equations becomes incorrect after applying specific interchanges: swapping the addition sign (+) with the subtraction sign (–) and swapping the number 3 with the number 5.

Let's break down the rule:

• Replace every '+' with '–'.
• Replace every '–' with '+'.
• Replace every '3' with '5'.
• Replace every '5' with '3'.

We will apply these rules to each equation and then evaluate if the resulting equation holds true.

Analyzing Option 1

Original equation: 5 × 4 + 3 – 6 ÷ 2 = 10

Applying the interchange rule:

• '5' becomes '3'
• '+' becomes '–'
• '3' becomes '5'
• '–' becomes '+'

The new equation is: 3 × 4 – 5 + 6 ÷ 2

Now, let's evaluate this expression using the BODMAS/PEMDAS rule (Brackets, Orders, Division/Multiplication, Addition/Subtraction):

• First, Multiplication and Division from left to right:
• 3 × 4 = 12
• 6 ÷ 2 = 3
• The expression becomes: 12 – 5 + 3
• Next, Addition and Subtraction from left to right:
• 12 – 5 = 7
• 7 + 3 = 10

So, the equation after interchange becomes 10 = 10. This equation is correct.

Analyzing Option 2

Original equation: 7 × 4 ÷ 2 – 5 + 3 = 12

Applying the interchange rule:

• '–' becomes '+'
• '5' becomes '3'
• '+' becomes '–'
• '3' becomes '5'

The new equation is: 7 × 4 ÷ 2 + 3 – 5

Now, let's evaluate this expression using BODMAS/PEMDAS:

• First, Multiplication and Division from left to right:
• 7 × 4 = 28
• 28 ÷ 2 = 14
• The expression becomes: 14 + 3 – 5
• Next, Addition and Subtraction from left to right:
• 14 + 3 = 17
• 17 – 5 = 12

So, the equation after interchange becomes 12 = 12. This equation is correct.

Analyzing Option 3

Original equation: 5 × 4 – 3 + 4 ÷ 2 = 15

Applying the interchange rule:

• '5' becomes '3'
• '–' becomes '+'
• '3' becomes '5'
• '+' becomes '–'

The new equation is: 3 × 4 + 5 – 4 ÷ 2

Now, let's evaluate this expression using BODMAS/PEMDAS:

• First, Multiplication and Division from left to right:
• 3 × 4 = 12
• 4 ÷ 2 = 2
• The expression becomes: 12 + 5 – 2
• Next, Addition and Subtraction from left to right:
• 12 + 5 = 17
• 17 – 2 = 15

So, the equation after interchange becomes 15 = 15. This equation is correct.

Analyzing Option 4

Original equation: 5 – 3 + 8 × 4 ÷ 2 = 10

Applying the interchange rule:

• '5' becomes '3'
• '–' becomes '+'
• '3' becomes '5'
• '+' becomes '–'

The new equation is: 3 + 5 – 8 × 4 ÷ 2

Now, let's evaluate this expression using BODMAS/PEMDAS:

• First, Multiplication and Division from left to right:
• 8 × 4 = 32
• 32 ÷ 2 = 16
• The expression becomes: 3 + 5 – 16
• Next, Addition and Subtraction from left to right:
• 3 + 5 = 8
• 8 – 16 = -8

So, the equation after interchange becomes -8 = 10. This equation is not correct.

Conclusion on Incorrect Equation

After interchanging the signs '+' and '–', and the numbers '3' and '5', three out of the four given equations resulted in a correct mathematical statement. Only Option 4 resulted in an incorrect statement (-8 = 10).

Revision Table: Summary of Results After Interchange

Additional Information: Understanding BODMAS/PEMDAS in Equation Solving

When solving mathematical expressions like the ones above, it's crucial to follow the correct order of operations. This order is often remembered using acronyms like BODMAS or PEMDAS.

• BODMAS: Brackets, Orders (powers/roots), Division, Multiplication, Addition, Subtraction.
• PEMDAS: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.

The order is essentially the same: perform operations within brackets/parentheses first, then powers/exponents, followed by multiplication and division (from left to right as they appear), and finally addition and subtraction (from left to right as they appear).

In these problems involving interchange of signs and numbers, applying the BODMAS/PEMDAS rule correctly after performing the substitutions is key to finding the correct result of the expression.