Solving Equations by Interchanging Signs

This question asks us to find which pair of arithmetic signs, when interchanged in the given equation, will make the equation mathematically correct. The given equation is:

13 + 15 × 8 ÷ 96 − 4 = 109

To solve this type of problem, we need to test each given option by swapping the specified signs and then evaluating the resulting expression according to the order of operations (BODMAS or PEMDAS).

Let's examine each option:

Checking Option 1: Interchanging + and ÷

If we interchange the '+' and '÷' signs, the equation becomes:

13 ÷ 15 × 8 + 96 − 4

Let's calculate the value using the order of operations:

• First, Division: $\frac{13}{15}$
• Next, Multiplication: $\frac{13}{15} \times 8 = \frac{104}{15}$
• Then, Addition/Subtraction from left to right:
• $\frac{104}{15} + 96 - 4$
• $= \frac{104}{15} + 92$
• $= 6.933... + 92 = 98.933...$

Since $98.933... \neq 109$, interchanging '+' and '÷' is not the correct solution.

Checking Option 2: Interchanging − and +

If we interchange the '−' and '+' signs, the equation becomes:

13 − 15 × 8 ÷ 96 + 4

Let's calculate the value using the order of operations:

• First, Division: $8 \div 96 = \frac{8}{96} = \frac{1}{12}$
• Next, Multiplication: $15 \times \frac{1}{12} = \frac{15}{12} = \frac{5}{4} = 1.25$
• Then, Addition/Subtraction from left to right:
• $13 - 1.25 + 4$
• $= 11.75 + 4 = 15.75$

Since $15.75 \neq 109$, interchanging '−' and '+' is not the correct solution.

Checking Option 3: Interchanging × and −

If we interchange the '×' and '−' signs, the equation becomes:

13 + 15 − 8 ÷ 96 × 4

Let's calculate the value using the order of operations:

• First, Division: $8 \div 96 = \frac{8}{96} = \frac{1}{12}$
• Next, Multiplication: $\frac{1}{12} \times 4 = \frac{4}{12} = \frac{1}{3}$
• Then, Addition/Subtraction from left to right:
• $13 + 15 - \frac{1}{3}$
• $= 28 - \frac{1}{3} = 27.666...$

Since $27.666... \neq 109$, interchanging '×' and '−' is not the correct solution.

Checking Option 4: Interchanging ÷ and −

If we interchange the '÷' and '−' signs, the equation becomes:

13 + 15 × 8 − 96 ÷ 4

Let's calculate the value using the order of operations (BODMAS/PEMDAS - Brackets, Orders/Exponents, Division/Multiplication, Addition/Subtraction):

• First, Division: $96 \div 4 = 24$
• Next, Multiplication: $15 \times 8 = 120$
• Then, Addition and Subtraction from left to right:
• $13 + 120 - 24$
• $= 133 - 24 = 109$

Since the result is $109$, which matches the right side of the original equation, interchanging '÷' and '−' makes the equation correct.

Summary of Sign Interchange Results

Based on the calculations, interchanging the '÷' and '−' signs makes the equation correct.

Revision Table: Understanding Order of Operations

Both BODMAS and PEMDAS represent the same hierarchy. Division and Multiplication have equal priority and are done from left to right. Similarly, Addition and Subtraction have equal priority and are done from left to right.

Additional Information: Types of Mathematical Reasoning Problems

Problems involving interchanging signs in equations fall under logical reasoning or mathematical reasoning. These problems often test a student's understanding of:

• Order of Operations (BODMAS/PEMDAS).
• Basic arithmetic calculations (addition, subtraction, multiplication, division).
• Systematic trial and error approach to test options.

Practicing such problems helps improve calculation speed and logical thinking for competitive exams.