If + means ÷ , – means + , x means – and ÷ means x, then what will come in place of ?in the given equation? 6 ÷ 4 × 10 + 2 – 3 = ?
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Solving Operator Substitution Math Puzzles
This type of question requires us to substitute the given mathematical operators with new ones and then evaluate the expression according to the standard order of operations, commonly known as BODMAS or PEMDAS.
Understanding the Operator Substitution Rules
The question provides the following rules for changing the operators:
- The symbol '+' means '÷'.
- The symbol '–' means '+'.
- The symbol 'x' means '–'.
- The symbol '÷' means 'x'.
Applying Substitution to the Equation
The given equation is: 6 ÷ 4 × 10 + 2 – 3 = ?
Let's apply the substitution rules step-by-step to rewrite the equation:
- Replace '÷' with 'x': The first part 6 ÷ 4 becomes 6 × 4.
- Replace '×' with '–': The next part 4 × 10 (after substituting the ÷) becomes 4 – 10.
- Replace '+' with '÷': The part 10 + 2 becomes 10 ÷ 2.
- Replace '–' with '+': The part 2 – 3 becomes 2 + 3.
So, the equation after substitution becomes:
\(6 \times 4 - 10 \div 2 + 3 = ?\)
Evaluating the Substituted Expression using BODMAS
Now we need to solve the new expression using the BODMAS (Brackets, Orders, Division/Multiplication, Addition/Subtraction) rule.
The BODMAS rule dictates the order of performing operations:
- Brackets
- Orders (powers, roots)
- Division and Multiplication (from left to right)
- Addition and Subtraction (from left to right)
Our expression is: \(6 \times 4 - 10 \div 2 + 3\)
Step-by-step Calculation:
- Division: Perform the division first.
\(10 \div 2 = 5\)
The expression is now: \(6 \times 4 - 5 + 3\)
- Multiplication: Perform the multiplication next.
\(6 \times 4 = 24\)
The expression is now: \(24 - 5 + 3\)
- Addition and Subtraction: Perform addition and subtraction from left to right.
\(24 - 5 = 19\)
The expression is now: \(19 + 3\)
\(19 + 3 = 22\)
The final value of the expression is 22.
Summary of Steps
| Original Expression | Substitution Rule | Substituted Expression |
|---|---|---|
| \(6 \div 4 \times 10 + 2 - 3\) | \(+ \to \div\) , \(-\to +\) , \( \times \to - \) , \( \div \to \times\) | \(6 \times 4 - 10 \div 2 + 3\) |
| Substituted Expression | BODMAS Step | Calculation | Resulting Expression |
|---|---|---|---|
| \(6 \times 4 - 10 \div 2 + 3\) | Division | \(10 \div 2 = 5\) | \(6 \times 4 - 5 + 3\) |
| \(6 \times 4 - 5 + 3\) | Multiplication | \(6 \times 4 = 24\) | \(24 - 5 + 3\) |
| \(24 - 5 + 3\) | Subtraction | \(24 - 5 = 19\) | \(19 + 3\) |
| \(19 + 3\) | Addition | \(19 + 3 = 22\) | \(22\) |
The value that comes in place of the '?' is 22.
Revision Table: Key Concepts for Operator Problems
| Concept | Description | Importance in Solving |
|---|---|---|
| Operator Substitution | Replacing standard mathematical symbols (+, -, ×, ÷) with different ones based on given rules. | First step to translate the problem into a solvable mathematical expression. |
| Order of Operations (BODMAS/PEMDAS) | Standard rule governing the sequence of calculations (Brackets, Orders, Division/Multiplication, Addition/Subtraction). | Crucial for correctly evaluating the substituted expression to arrive at the unique correct answer. |
| Accuracy in Calculation | Performing basic arithmetic operations (addition, subtraction, multiplication, division) without errors. | Essential to get the correct final result after applying substitution and BODMAS. |
Additional Information: Types of Operator Questions
Operator-based reasoning questions can appear in various forms in competitive exams. Beyond simple substitution and evaluation, they might involve:
- Finding the correct set of operators to satisfy a given equation.
- Determining the correct sequence of numbers to satisfy an equation with given operators.
- Questions where symbols represent relationships or conditions instead of mathematical operations.
- Problems combining operator substitution with other logical reasoning patterns.
Mastering the basic substitution and BODMAS application is fundamental to tackling more complex variations of these operator problems.
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