If ‘A’ is replaced by ‘+’; if ‘B’ is replaced by ‘-‘ ; ‘c’ is replaced by ‘÷’ and ‘D’ replaced by ‘×’ find the value of the following equation.
31
Evaluating Mathematical Expressions with Operator Replacements
The question asks us to find the value of a given expression where letters represent specific mathematical operators. We need to replace each letter with its corresponding operator and then evaluate the resulting mathematical expression following the standard order of operations.
Understanding the Operator Replacements
The question provides the following rules for replacement:
- ‘A’ is replaced by ‘+’ (Addition)
- ‘B’ is replaced by ‘-‘ (Subtraction)
- ‘C’ is replaced by ‘÷’ (Division)
- ‘D’ is replaced by ‘×’ (Multiplication)
Transforming the Expression
The given expression is: 17D22C11A5B8
Let's apply the replacements step-by-step:
- Replace 'D' with '×': $17 \times 22\text{C}11\text{A}5\text{B}8$
- Replace 'C' with '÷': $17 \times 22 \div 11\text{A}5\text{B}8$
- Replace 'A' with '+': $17 \times 22 \div 11 + 5\text{B}8$
- Replace 'B' with '-': $17 \times 22 \div 11 + 5 - 8$
The expression to evaluate is: $17 \times 22 \div 11 + 5 - 8$
Applying the Order of Operations (BODMAS/PEMDAS)
To correctly evaluate the expression $17 \times 22 \div 11 + 5 - 8$, we must follow the order of operations, commonly known as BODMAS or PEMDAS.
- Brackets (or Parentheses)
- Orders (or Exponents)
- Division and Multiplication (from left to right)
- Addition and Subtraction (from left to right)
Let's perform the operations in the correct order:
- Division and Multiplication (from left to right):
- First, calculate $22 \div 11$: $$22 \div 11 = 2$$ The expression becomes: $17 \times 2 + 5 - 8$
- Next, calculate $17 \times 2$: $$17 \times 2 = 34$$ The expression becomes: $34 + 5 - 8$
- Addition and Subtraction (from left to right):
- First, calculate $34 + 5$: $$34 + 5 = 39$$ The expression becomes: $39 - 8$
- Next, calculate $39 - 8$: $$39 - 8 = 31$$
The final value of the expression is 31.
Summary of Calculation Steps
| Step | Operation | Expression | Result |
|---|---|---|---|
| Initial | Replace letters | $17\text{D}22\text{C}11\text{A}5\text{B}8$ | $17 \times 22 \div 11 + 5 - 8$ |
| 1 | Division ($22 \div 11$) | $17 \times \underline{22 \div 11} + 5 - 8$ | $17 \times 2 + 5 - 8$ |
| 2 | Multiplication ($17 \times 2$) | $\underline{17 \times 2} + 5 - 8$ | $34 + 5 - 8$ |
| 3 | Addition ($34 + 5$) | $\underline{34 + 5} - 8$ | $39 - 8$ |
| 4 | Subtraction ($39 - 8$) | $\underline{39 - 8}$ | 31 |
Conclusion
By replacing the letters with the specified mathematical operators and following the BODMAS/PEMDAS rule for the order of operations, the value of the expression 17D22C11A5B8 is found to be 31.
Revision Table: Operator Replacement and Order of Operations
This type of question tests your ability to follow instructions for symbol replacement and apply the standard rules for evaluating mathematical expressions.
- Key Concept: Symbol/Operator replacement.
- Key Concept: Order of operations (BODMAS/PEMDAS).
- BODMAS/PEMDAS Sequence: Brackets > Orders > Division/Multiplication > Addition/Subtraction. Operations at the same level (like Division and Multiplication) are performed from left to right.
Additional Information: Understanding BODMAS/PEMDAS
BODMAS (or PEMDAS) is a mnemonic used to remember the correct sequence of operations when evaluating an expression. Performing operations in the wrong order will lead to an incorrect result. For example, in $17 \times 22 \div 11$, doing multiplication first ($17 \times 22 = 374$) and then division ($374 \div 11 = 34$) gives a different result than doing division first ($22 \div 11 = 2$) and then multiplication ($17 \times 2 = 34$). When multiplication and division appear together, or addition and subtraction appear together, you perform them in the order they appear from left to right.
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