Which two signs need to be interchanged to make the following equation correct? 35 + 7 × 12 − 6 ÷ 2 = 6
- and ×
Solving the Equation by Interchanging Signs
The question asks us to find which pair of mathematical signs, when swapped in the given equation, makes the equation correct. The original equation is:
\(35 + 7 \times 12 - 6 \div 2 = 6\)
Let's evaluate the original equation using the order of operations (BODMAS/PEMDAS: Brackets, Orders/Exponents, Division and Multiplication (from left to right), Addition and Subtraction (from left to right)).
- Division: \(6 \div 2 = 3\)
- Multiplication: \(7 \times 12 = 84\)
- Addition/Subtraction (from left to right): \(35 + 84 - 3 = 119 - 3 = 116\)
So, \(35 + 7 \times 12 - 6 \div 2 = 116\). The equation \(116 = 6\) is false. We need to interchange signs to make it true.
Testing the Options for Sign Interchange
We will test each option by swapping the indicated signs and re-evaluating the equation.
Option 1: Interchange - and ×
Swap the minus sign (-) and the multiplication sign (×) in the original equation. The new equation becomes:
\(35 + 7 - 12 \times 6 \div 2\)
Now, let's evaluate this new equation using BODMAS/PEMDAS:
- Multiplication: \(12 \times 6 = 72\)
- Division: \(72 \div 2 = 36\)
- Addition/Subtraction (from left to right): \(35 + 7 - 36 = 42 - 36 = 6\)
So, \(35 + 7 - 12 \times 6 \div 2 = 6\). The equation \(6 = 6\) is true.
This option makes the equation correct.
Option 2: Interchange - and ÷
Swap the minus sign (-) and the division sign (÷) in the original equation. The new equation becomes:
\(35 + 7 \times 12 \div 6 - 2\)
Evaluate using BODMAS/PEMDAS:
- Multiplication: \(7 \times 12 = 84\)
- Division: \(84 \div 6 = 14\)
- Addition/Subtraction (from left to right): \(35 + 14 - 2 = 49 - 2 = 47\)
The equation becomes \(47 = 6\), which is false.
Option 3: Interchange + and ×
Swap the addition sign (+) and the multiplication sign (×) in the original equation. The new equation becomes:
\(35 \times 7 + 12 - 6 \div 2\)
Evaluate using BODMAS/PEMDAS:
- Division: \(6 \div 2 = 3\)
- Multiplication: \(35 \times 7 = 245\)
- Addition/Subtraction (from left to right): \(245 + 12 - 3 = 257 - 3 = 254\)
The equation becomes \(254 = 6\), which is false.
Option 4: Interchange × and ÷
Swap the multiplication sign (×) and the division sign (÷) in the original equation. The new equation becomes:
\(35 + 7 \div 12 - 6 \times 2\)
Evaluate using BODMAS/PEMDAS:
- Division: \(7 \div 12 = \frac{7}{12}\)
- Multiplication: \(6 \times 2 = 12\)
- Addition/Subtraction (from left to right): \(35 + \frac{7}{12} - 12 = 23 + \frac{7}{12}\)
The equation becomes \(23 + \frac{7}{12} = 6\), which is false.
Based on the testing, interchanging the minus (-) and multiplication (×) signs makes the equation correct.
Step-by-Step Solution for the Correct Sign Interchange
The correct option is to interchange the signs - and ×.
Original Equation: \(35 + 7 \times 12 - 6 \div 2 = 6\)
Step 1: Identify the signs to be interchanged, which are - and ×.
Step 2: Rewrite the equation with the signs interchanged.
The original equation has × between 7 and 12, and - between 12 and 6.
After interchanging, - will be between 7 and 12, and × will be between 12 and 6.
New equation: \(35 + 7 - 12 \times 6 \div 2\)
Step 3: Apply the order of operations (BODMAS/PEMDAS) to evaluate the new equation.
- First, evaluate Multiplication and Division from left to right.
- Multiplication: \(12 \times 6 = 72\). The equation becomes \(35 + 7 - 72 \div 2\).
- Division: \(72 \div 2 = 36\). The equation becomes \(35 + 7 - 36\).
- Next, evaluate Addition and Subtraction from left to right.
- Addition: \(35 + 7 = 42\). The equation becomes \(42 - 36\).
- Subtraction: \(42 - 36 = 6\).
Step 4: Compare the result with the right side of the original equation.
The result is 6, which matches the right side of the original equation \(... = 6\).
Thus, interchanging the signs - and × makes the equation \(35 + 7 - 12 \times 6 \div 2 = 6\) correct.
Revision Table: Understanding Operations Order
Understanding the correct order of mathematical operations is crucial for solving these types of problems correctly. The widely used acronyms are BODMAS or PEMDAS.
| Letter | BODMAS | PEMDAS | Operation | Notes |
| B / P | Brackets | Parentheses | Operations within ( ) { } [ ] | Solve first |
| O / E | Orders | Exponents | Powers, Square roots, etc. | Solve next |
| D / M | Division | Multiplication | ÷ and × | Solve from left to right |
| M / D | Multiplication | Division | × and ÷ | Solve from left to right |
| A / A | Addition | Addition | + and - | Solve from left to right |
| S / S | Subtraction | Subtraction | - and + | Solve from left to right |
Additional Information: Types of Mathematical Operations
Mathematical operations are fundamental actions performed on numbers. The basic arithmetic operations are:
- Addition (+): Combining two or more numbers to find their sum. Example: \(5 + 3 = 8\).
- Subtraction (-): Finding the difference between two numbers. Example: \(8 - 3 = 5\).
- Multiplication (× or *): Repeated addition or scaling numbers. Example: \(5 \times 3 = 15\).
- Division (÷ or /): Splitting a number into equal parts. Example: \(15 \div 3 = 5\).
Problems involving interchanging signs test your understanding of how these operations interact and the critical importance of following the correct order of operations to arrive at the accurate result.
Important Questions from Logical Puzzle
Below is given a figure, in which a horse is placed on one of the boxes in a square grid of size $5 \times 5$. The horse can move in either of the following ways in a single move:
(i) two boxes to either left or right side and one box to either up or down side
(ii) two boxes to either up or down side and one box to either left or right side
(iii) the horse cannot step on a grey box. The horse has to stay within the grid and any move to out of the grid is NOT legal.
How many white boxes are there which cannot be covered by horse within 3 moves?
- Six persons Raj, Rohit, Rachit, Rajat, Rishu, and Rinku are working in an office. Each of them has a favorite car viz. P, Q, R, S, T, and U, but not necessarily in the same order. Each of them got a different salary - 1000, 2000, 2500, 3000, 3500, and 4000, but not necessarily in the same order. Rajat gets a salary of 2500 and his favorite car is U. The one who likes car Q gets a salary of 3000. Rachit likes car P and gets a salary of 3500. The one who gets a salary of 1000 does not like car R or S. Rinku does not like car Q or S but gets a salary of 2000. Rishu gets a salary of 4000 and does not like a car R. Rohit does not get a salary of 1000. Who gets a salary of 3000?
- Six friends P, Q, R, S, T, and U are sitting at different positions in a row of the company, and each of them get a different salary viz – 500, 800, 1000, 1500, 1800 & 2000, but not necessary in the same order. The friend who gets a salary of 2000 does not sit at the extreme end of the row. Q sits immediately before R and gets a salary of 800. T sits between U and S. Q is second to the left of S. P sits at the left end of the row and gets a salary of 1000. T gets a salary of 1500. R gets a salary neither of 2000 nor of 1800. How much salary does U get?
- Seven men A, B, C, D, E, F and G are standing in a queue, one behind the other in that order. That is, A is first in queue while G brings up the rear. Each of them is wearing a hat of a different colour of the rainbow - Violet, Indigo, Blue, Green, Yellow, Orange, and Red, but not necessarily in that order. It is also known that:
• D can see in front of him green and blue hats, but not the violet hat.
• E can see violet and yellow hats, but not the red hat.
• G can see hats of all the colours except Orange.
• E is wearing the Indigo coloured hat.
What colour hat is worn by F? - Who amongst the females arrived third?
