Which two numbers (not digits) given in the options must be interchanged to balance the given equation?
77 ÷ 7 − 2 × 3 + 3 = 4
Evaluating the Original Equation
The initial equation provided is:
\( 77 \div 7 - 2 \times 3 + 3 = 4 \)
To determine if the equation is balanced, we need to evaluate the left side using the standard order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Division and Multiplication (from left to right), Addition and Subtraction (from left to right).
- Division: First, calculate \( 77 \div 7 \). The result is \( 11 \).
- Multiplication: Next, calculate \( 2 \times 3 \). The result is \( 6 \).
- Substitute Results: Replace the division and multiplication parts in the equation: \( 11 - 6 + 3 \).
- Subtraction: Perform the subtraction: \( 11 - 6 = 5 \).
- Addition: Finally, perform the addition: \( 5 + 3 = 8 \).
The left side of the equation evaluates to 8. The original equation thus becomes \( 8 = 4 \), which is clearly false. The equation is unbalanced.
Interchanging Numbers to Balance the Equation
The problem requires us to interchange two specific numbers from the options provided to make the equation true. Let's test the option suggesting the interchange of the numbers 2 and 4.
The original equation structure is: \( 77 \div 7 - \underline{2} \times 3 + 3 = 4 \)
By interchanging 2 and 4, we replace every instance of '2' with '4' and '4' with '2'. The equation becomes:
\( 77 \div 7 - 4 \times 3 + 3 = 2 \)
Now, let's evaluate this modified equation using the order of operations:
- Division: \( 77 \div 7 = 11 \)
- Multiplication: \( 4 \times 3 = 12 \)
- Substitute Results: The equation becomes \( 11 - 12 + 3 \).
- Subtraction: \( 11 - 12 = -1 \)
- Addition: \( -1 + 3 = 2 \)
The left side now evaluates to 2. The equation is \( 2 = 2 \), which is a correct statement. This confirms that interchanging the numbers 2 and 4 successfully balances the equation.
Conclusion
Interchanging the numbers 2 and 4 in the equation \( 77 \div 7 - 2 \times 3 + 3 = 4 \) transforms it into \( 77 \div 7 - 4 \times 3 + 3 = 2 \), which is a valid mathematical statement.
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.)
