What will come in the place of ‘?’ in the following equation, if ‘+’ and ‘ב are interchanged and also ‘−’ and ‘÷’ are interchanged? 4 + 8 ÷ 4 × 2 − 2 = ?
29
Evaluating Mathematical Expressions with Interchanged Operators
The question asks us to find the value of a given mathematical expression after interchanging certain operators. We are given the equation: \(4 + 8 \div 4 \times 2 - 2 = ?\)
The rules for interchanging operators are:
- '+' and '×' are interchanged.
- '−' and '÷' are interchanged.
Let's apply these changes to the original equation.
Applying the Operator Changes
The original equation is: \(4 + 8 \div 4 \times 2 - 2\)
Applying the changes:
- '+' becomes '×'
- '÷' becomes '−'
- '×' becomes '+'
- '−' becomes '÷'
So, the new equation becomes: \(4 \times 8 - 4 + 2 \div 2\)
Solving the New Equation using BODMAS/PEMDAS
Now we need to evaluate the new equation \(4 \times 8 - 4 + 2 \div 2\) using the order of operations (BODMAS/PEMDAS).
The order of operations is:
- Brackets / Parentheses
- Orders / Exponents
- Division and Multiplication (from left to right)
- Addition and Subtraction (from left to right)
Step-by-Step Calculation
- First, perform the Division: \(2 \div 2 = 1\)
The equation becomes: \(4 \times 8 - 4 + 1\) - Next, perform the Multiplication: \(4 \times 8 = 32\)
The equation becomes: \(32 - 4 + 1\) - Now, perform Addition and Subtraction from left to right.
Subtraction: \(32 - 4 = 28\)
The equation becomes: \(28 + 1\) - Finally, perform the Addition: \(28 + 1 = 29\)
So, the value of the expression after interchanging the operators is 29.
Summary of Steps
| Original Equation | Operator Changes Applied | New Equation |
|---|---|---|
| \(4 + 8 \div 4 \times 2 - 2\) | \(+ \leftrightarrow \times\), \( - \leftrightarrow \div\) | \(4 \times 8 - 4 + 2 \div 2\) |
| Step | Operation | Calculation | Resulting Expression |
|---|---|---|---|
| 1 | Division | \(2 \div 2 = 1\) | \(4 \times 8 - 4 + 1\) |
| 2 | Multiplication | \(4 \times 8 = 32\) | \(32 - 4 + 1\) |
| 3 | Subtraction | \(32 - 4 = 28\) | \(28 + 1\) |
| 4 | Addition | \(28 + 1 = 29\) | \(29\) |
The final result of the expression after the operator interchange is 29.
Revision Table: Operator Interchanges
| Original Operator | Interchanged With |
|---|---|
| + | × |
| × | + |
| − | ÷ |
| ÷ | − |
Additional Information: Understanding BODMAS/PEMDAS in Mathematical Operations
The BODMAS or PEMDAS rule is a mnemonic used to remember the correct order for evaluating mathematical expressions. It ensures that everyone gets the same answer when solving a multi-step calculation.
- B/P: Brackets/Parentheses - Operations inside brackets are performed first.
- O/E: Orders/Exponents - Powers and roots are calculated next.
- DM: Division and Multiplication - These are performed from left to right.
- AS: Addition and Subtraction - These are performed from left to right.
When solving problems with interchanged operators, it is crucial to first rewrite the expression with the new operators and then apply the BODMAS/PEMDAS rule correctly to the new expression.
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?
