Which two signs should be interchanged to make the below equation mathematically correct? $132 \div 6 - 24 \times 5 + 9 = 43$
Equation Balancing: Signs Interchange Problem
The main goal is to determine which pair of arithmetic operation signs, when swapped in the given equation, results in a mathematically accurate statement. The equation provided is: $$132 \div 6 - 24 \times 5 + 9 = 43$$
To solve this, we must first evaluate the original equation using the standard order of operations, commonly remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right) or BODMAS.
Original Equation Evaluation Steps
Let's apply the order of operations to the original equation:
- Division first (leftmost): $132 \div 6 = 22$
- Next, multiplication: $24 \times 5 = 120$
- Substitute these values back into the equation: $22 - 120 + 9$
- Perform subtraction: $22 - 120 = -98$
- Finally, perform addition: $-98 + 9 = -89$
The original equation evaluates to $-89$, not $43$. Therefore, the signs need to be interchanged.
Sign Interchange Testing and Verification
We will now examine each option to see which sign swap corrects the equation:
- Option 1: Interchange $ \div $ and $ \times $
The equation becomes: $132 \times 6 - 24 \div 5 + 9 = 43$
Evaluation: $792 - 4.8 + 9 = 796.2$. This is not equal to $43$.
- Option 2: Interchange $ + $ and $ \div $
The equation becomes: $132 + 6 - 24 \times 5 \div 9 = 43$
Evaluation: $132 + 6 - (24 \times 5) \div 9 = 138 - 120 \div 9 = 138 - 13.33... = 124.67...$. This is not equal to $43$.
- Option 3: Interchange $ - $ and $ + $
The equation becomes: $132 \div 6 + 24 - 5 \times 9 = 43$
Evaluation: $22 + 24 - 45 = 46 - 45 = 1$. This is not equal to $43$.
- Option 4: Interchange $ + $ and $ \times $
The equation becomes: $132 \div 6 - 24 + 5 \times 9 = 43$
Applying order of operations (PEMDAS/BODMAS):
First, Division: $132 \div 6 = 22$
Next, Multiplication: $5 \times 9 = 45$
The equation transforms to: $22 - 24 + 45$
Performing subtraction: $22 - 24 = -2$
Performing addition: $-2 + 45 = 43$
This result ($43$) matches the right side of the original equation ($43$). Thus, $43 = 43$. This option is correct.
Final Verification of Correct Equation
Interchanging the addition ($ + $) sign and the multiplication ($ \times $) sign successfully corrects the equation. The verified equation is:
$$132 \div 6 - 24 + 5 \times 9 = 43$$The step-by-step calculation confirming this is:
- Perform Division: $132 \div 6 = 22$
- Perform Multiplication: $5 \times 9 = 45$
- The equation is now: $22 - 24 + 45$
- Perform Subtraction: $22 - 24 = -2$
- Perform Addition: $-2 + 45 = 43$
Therefore, swapping the signs $ + $ and $ \times $ is the correct solution.
Important Questions from Mathematical Operations
What will come in the place of ‘?’ in the following equation if ‘+’ and ‘−’ are interchanged and ‘×’ and ‘÷’ are interchanged?
18 ÷ 9 + 24 × 3 − 16 = ?If ‘A’ stands for ‘÷’, ‘B’ stands for ‘×’, ‘C’ stands for ‘+’, and ‘D’ stands for ‘−’, what will come in place of the question mark (?) in the following equation?
88 B 3 D 12 A 4 C 6 = ?Select the set in which the numbers are related in the same way as are the numbers of the given sets.
(NOTE: Operations should be performed on whole numbers without breaking down the numbers into their constituent digits. E.g., 13 – Operations on 13 such as adding/subtracting/multiplying, etc., to 13 can be performed. Breaking down 13 into 1 and 3 and then performing mathematical operations on 1 and 3 is not allowed.)
Given sets:
(9, 21, 193)
(12, 15, 184)What will come in the place of the question mark (?) in the following equation, if ‘÷’ and ‘+’ are interchanged and ‘−’ and ‘×’ are interchanged?
44 ÷ 78 + 6 − 3 × 19 = ?Two sets of numbers are given below. In each set of numbers, certain mathematical operation(s) on the first number result(s) in the second number. Similarly, certain mathematical operation(s) on the second number result(s) in the third number and so on. Which of the given options follows the same set of operations as in the given sets?
(NOTE: Operations should be performed on the whole numbers, without breaking down the numbers into their constituent digits. E.g. 13 – Operations on 13 such as adding/subtracting/multiplying to 13 can be performed. Breaking down 13 into 1 and 3 and then performing mathematical operations on 1 and 3 is not allowed.)
