Solving Equations by Interchanging Numbers and Signs

This question requires us to find the correct interchange of numbers and mathematical signs that makes the given equation true. The given equation is:

\(30 \div 6 \times 4 + 15 - 35 = 25\)

We need to evaluate each option by applying the suggested interchanges to the equation and checking if the resulting equation is correct.

Evaluating Option 1: Interchange 25 and 30, × and -

Apply the interchange of numbers (25 and 30) and signs (× and -) to the given equation:

\(30 \div 6 \times 4 + 15 - 35 = 25\)

Interchanging 25 and 30 means replacing 30 with 25 and 25 with 30 wherever they appear in the equation.

Interchanging × and - means replacing × with - and - with × wherever they appear in the equation.

Applying these changes:

• Replace 30 with 25
• Keep ÷
• Keep 6
• Replace × with -
• Keep 4
• Keep +
• Keep 15
• Replace - with ×
• Keep 35
• Keep =
• Replace 25 with 30

The new equation becomes:

\(25 \div 6 - 4 + 15 \times 35 = 30\)

Now, let's evaluate the Left Hand Side (LHS) using the BODMAS/PEMDAS rule (Brackets, Orders, Division/Multiplication, Addition/Subtraction):

\(LHS = 25 \div 6 - 4 + 15 \times 35\)

First, perform division and multiplication from left to right:

\(25 \div 6 \approx 4.17\)

\(15 \times 35 = 525\)

So, \(LHS \approx 4.17 - 4 + 525\)

Next, perform addition and subtraction from left to right:

\(4.17 - 4 = 0.17\)

\(0.17 + 525 = 525.17\)

The LHS evaluates to approximately 525.17. The RHS is 30.

\(525.17 \neq 30\)

So, Option 1 does not make the equation correct.

Evaluating Option 2: Interchange 4 and 6, + and -

Apply the interchange of numbers (4 and 6) and signs (+ and -) to the given equation:

\(30 \div 6 \times 4 + 15 - 35 = 25\)

Interchanging 4 and 6 means replacing 4 with 6 and 6 with 4.

Interchanging + and - means replacing + with - and - with +.

Applying these changes:

• Keep 30
• Keep ÷
• Replace 6 with 4
• Keep ×
• Replace 4 with 6
• Replace + with -
• Keep 15
• Replace - with +
• Keep 35
• Keep =
• Keep 25

The new equation becomes:

\(30 \div 4 \times 6 - 15 + 35 = 25\)

Now, let's evaluate the Left Hand Side (LHS) using BODMAS/PEMDAS:

\(LHS = 30 \div 4 \times 6 - 15 + 35\)

First, perform division and multiplication from left to right:

\(30 \div 4 = 7.5\)

\(7.5 \times 6 = 45\)

So, \(LHS = 45 - 15 + 35\)

Next, perform addition and subtraction from left to right:

\(45 - 15 = 30\)

\(30 + 35 = 65\)

The LHS evaluates to 65. The RHS is 25.

\(65 \neq 25\)

So, Option 2 does not make the equation correct.

Evaluating Option 4: Interchange 30 and 35, - and ÷

Apply the interchange of numbers (30 and 35) and signs (- and ÷) to the given equation:

\(30 \div 6 \times 4 + 15 - 35 = 25\)

Interchanging 30 and 35 means replacing 30 with 35 and 35 with 30.

Interchanging - and ÷ means replacing - with ÷ and ÷ with -.

Applying these changes:

• Replace 30 with 35
• Replace ÷ with -
• Keep 6
• Keep ×
• Keep 4
• Keep +
• Keep 15
• Replace - with ÷
• Replace 35 with 30
• Keep =
• Keep 25

The new equation becomes:

\(35 - 6 \times 4 + 15 \div 30 = 25\)

Now, let's evaluate the Left Hand Side (LHS) using BODMAS/PEMDAS:

\(LHS = 35 - 6 \times 4 + 15 \div 30\)

First, perform multiplication and division from left to right:

\(6 \times 4 = 24\)

\(15 \div 30 = 0.5\)

So, \(LHS = 35 - 24 + 0.5\)

Next, perform addition and subtraction from left to right:

\(35 - 24 = 11\)

\(11 + 0.5 = 11.5\)

The LHS evaluates to 11.5. The RHS is 25.

\(11.5 \neq 25\)

So, Option 4 does not make the equation correct.

Evaluating Option 3: Interchange 25 and 35, × and ÷

Apply the interchange of numbers (25 and 35) and signs (× and ÷) to the given equation:

\(30 \div 6 \times 4 + 15 - 35 = 25\)

Interchanging 25 and 35 means replacing 25 with 35 and 35 with 25 wherever they appear.

Interchanging × and ÷ means replacing × with ÷ and ÷ with × wherever they appear.

Applying these changes:

• Keep 30
• Replace ÷ with ×
• Keep 6
• Replace × with ÷
• Keep 4
• Keep +
• Keep 15
• Keep -
• Replace 35 with 25
• Keep =
• Replace 25 with 35

The new equation becomes:

\(30 \times 6 \div 4 + 15 - 25 = 35\)

Now, let's evaluate the Left Hand Side (LHS) using BODMAS/PEMDAS:

\(LHS = 30 \times 6 \div 4 + 15 - 25\)

First, perform multiplication and division from left to right:

\(30 \times 6 = 180\)

\(180 \div 4 = 45\)

So, \(LHS = 45 + 15 - 25\)

Next, perform addition and subtraction from left to right:

\(45 + 15 = 60\)

\(60 - 25 = 35\)

The LHS evaluates to 35. The RHS is 35.

\(35 = 35\)

Since LHS = RHS, the new equation is correct. Therefore, interchanging 25 and 35, and × and ÷ makes the equation a true statement.

Summary of Evaluation

The interchange in option 3 makes the modified equation correct.

Revision Table: Mathematical Operations and Interchange Problems

Additional Information: Solving Mathematical Reasoning Questions

Solving mathematical reasoning problems involving interchange of signs and numbers requires careful application of the given rules and strict adherence to the order of operations. Here are some tips:

• Understand the Scope: Clarify if the interchange applies only to the Left Hand Side (LHS) expression or to the entire equation (including the Right Hand Side - RHS). In this problem, the interchange seems to apply throughout the equation to make the statement true.
• Apply Changes Systematically: Go through the equation from left to right, replacing numbers and signs as specified by the option.
• Follow BODMAS/PEMDAS: Always evaluate the modified expression using the correct order of operations to avoid errors.
• Check Each Option: Although one option is correct, practicing with all options helps reinforce the method and ensures accuracy under timed conditions.
• Use Parentheses: When interchanged operations involve potential ambiguity (e.g., division and multiplication), using parentheses based on the left-to-right rule within the same priority level can help organize calculation steps.

These types of questions test your ability to follow instructions precisely and perform calculations accurately. Practice with different variations helps build speed and confidence.