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
Select the set, from the given options, in which the numbers are related in the same way as are the numbers of the following sets.
(NOTE: Operations should be performed on the whole numbers, without breaking down the numbers into its 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.)
(6, 13, 78)
(5, 22, 110)
If + means -, - means ×, × means ÷, ÷ means +, then what will come in place of the question mark (?) in the following equation?
28 - 6 ÷ 4 + 14 × 14 = ?
In the following number-pairs, the second number is obtained by applying certain mathematical operations to the first number. Which numbers should replace X and Y so that the pattern followed by the two numbers on the left side of :: is the same as that on the right side of :: ?
(NOTE: Operations should be performed on the whole numbers, without breaking down the numbers into its 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.)
X : 649 :: 956 : Y
Which two numbers should be interchanged to make the given equation correct?
75 ÷ 5 + 15 × 4 - 17 × 3 + (60 ÷ 6) × 10 = 100
(Note: Interchange should be done of the entire number and not individual digits of a given number.)
If '+' means minus, '-' means multiplication, '÷' means plus, and 'x' means division, then solve the equation:
15 - 3 + 10 x 5 ÷ 5
