If ‘A’ denotes ‘addition’, ‘B’ denotes ‘multiplication’, ‘C’ denotes ‘subtraction’, and ‘D’ denotes ‘division’, then what will be the value of the following expression ? 100 D (1 B 4) A 2 B (13 A 11) C 3 B (17 C 12)
58
Understanding the Symbolic Expression
The question asks us to evaluate a mathematical expression where standard arithmetic operations are represented by letters:
- 'A' denotes 'addition' (+)
- 'B' denotes 'multiplication' (*)
- 'C' denotes 'subtraction' (-)
- 'D' denotes 'division' (/)
The given expression is: 100 D (1 B 4) A 2 B (13 A 11) C 3 B (17 C 12)
First, we need to replace the letters with their corresponding mathematical operators to convert the symbolic expression into a standard arithmetic expression.
Converting to a Mathematical Expression
Let's substitute the operators:
Dbecomes/Bbecomes*Abecomes+Cbecomes-
The expression becomes:
\(100 \div (1 \times 4) + 2 \times (13 + 11) - 3 \times (17 - 12)\)
Evaluating the Mathematical Expression using BODMAS/PEMDAS
To find the value of this expression, we need to follow the order of operations (BODMAS or PEMDAS):
- B/P: Brackets/Parentheses first.
- O/E: Orders/Exponents (none in this expression).
- DM: Division and Multiplication (from left to right).
- AS: Addition and Subtraction (from left to right).
Step-by-Step Calculation:
Let's evaluate the expression step by step:
- Evaluate the expressions inside the parentheses:
- First set of parentheses: \(1 \times 4 = 4\)
- Second set of parentheses: \(13 + 11 = 24\)
- Third set of parentheses: \(17 - 12 = 5\)
The expression now looks like: \(100 \div 4 + 2 \times 24 - 3 \times 5\)
- Perform Division and Multiplication from left to right:
- Division: \(100 \div 4 = 25\)
- First Multiplication: \(2 \times 24 = 48\)
- Second Multiplication: \(3 \times 5 = 15\)
The expression now looks like: \(25 + 48 - 15\)
- Perform Addition and Subtraction from left to right:
- Addition: \(25 + 48 = 73\)
- Subtraction: \(73 - 15 = 58\)
The final value of the expression is 58.
Final Result of the Expression Value
After converting the symbolic expression to a standard mathematical expression and applying the order of operations (BODMAS/PEMDAS), the value of the expression 100 D (1 B 4) A 2 B (13 A 11) C 3 B (17 C 12) is found to be 58.
| Step | Expression | Calculation | Result |
|---|---|---|---|
| Original | 100 D (1 B 4) A 2 B (13 A 11) C 3 B (17 C 12) |
Symbolic expression | - |
| Convert | \(100 \div (1 \times 4) + 2 \times (13 + 11) - 3 \times (17 - 12)\) | Replace symbols with operators | - |
| Parentheses 1 | \(100 \div (4) + 2 \times (13 + 11) - 3 \times (17 - 12)\) | \(1 \times 4\) | 4 |
| Parentheses 2 | \(100 \div 4 + 2 \times (24) - 3 \times (17 - 12)\) | \(13 + 11\) | 24 |
| Parentheses 3 | \(100 \div 4 + 2 \times 24 - 3 \times (5)\) | \(17 - 12\) | 5 |
| Division | \(25 + 2 \times 24 - 3 \times 5\) | \(100 \div 4\) | 25 |
| Multiplication 1 | \(25 + 48 - 3 \times 5\) | \(2 \times 24\) | 48 |
| Multiplication 2 | \(25 + 48 - 15\) | \(3 \times 5\) | 15 |
| Addition | \(73 - 15\) | \(25 + 48\) | 73 |
| Subtraction | \(58\) | \(73 - 15\) | 58 |
Revision Table: Key Concepts Revisited
| Concept | Description | Importance in this problem |
|---|---|---|
| Symbolic Representation | Using symbols (like letters) to represent mathematical operations. | Decoding the original question into a solvable mathematical expression. |
| Arithmetic Operations | Addition, Subtraction, Multiplication, Division. | The core operations involved in the calculation. |
| Order of Operations | Rules like BODMAS/PEMDAS dictate the sequence of operations (Brackets/Parentheses, Orders/Exponents, Division/Multiplication, Addition/Subtraction). | Essential for evaluating complex expressions correctly to arrive at the unique correct answer. |
| Expression Evaluation | Calculating the value of a mathematical expression by performing the operations in the correct order. | The ultimate goal of the problem. |
Additional Information: BODMAS/PEMDAS Explained
The order of operations is a crucial concept in mathematics to ensure consistency in evaluating expressions. Let's break down BODMAS and PEMDAS:
- BODMAS:
- B - Brackets
- O - Orders (powers, square roots)
- D - Division
- M - Multiplication
- A - Addition
- S - Subtraction
- PEMDAS:
- P - Parentheses
- E - Exponents (powers, square roots)
- M - Multiplication
- D - Division
- A - Addition
- S - Subtraction
Both acronyms represent the same order. Division and Multiplication have equal priority and are performed from left to right as they appear in the expression. Similarly, Addition and Subtraction have equal priority and are performed from left to right.
In this problem, strictly following the BODMAS/PEMDAS rule ensures that the operations inside parentheses are done first, followed by multiplications and divisions, and finally additions and subtractions, leading to the correct value of the expression.
Important Questions from Logical Puzzle
Which two numbers should be interchanged to make the given equation correct?
9 + 7 × 5 – 18 ÷ 2 = 3 × 4 – 10 + 45 ÷ 5Select the correct combination of mathematical signs that can sequentially replace the * signs and balance the equation.
60 * 2 * 3 * 6 * 5 * 43
Which of the following interchange of numbers and mathematical signs would make the given equation correct?
30 ÷ 6 × 4 + 15 - 35 = 25
Which two signs need to be interchanged to make the following equation correct?
23 + 84 ÷ 14 × 8 − 3 = 5
Select the correct combination of mathematical signs that can sequentially replace the * signs and make the equation correct.
68 * 138* 23 * 54 * 20
