If A denotes '+', B denotes '×', C denotes '-', and D denotes '÷', then what will come in place of '?' in the following equation? (51 D 17) A (21 B 9) C 78 A (16 D 4) = ?
118
Understanding the Operator Substitution Problem
This problem requires us to substitute specific mathematical operators for given letters and then evaluate the resulting expression following the standard order of operations (BODMAS/PEMDAS).
Operator Substitutions
The question provides the following substitutions:
- A denotes '+' (Addition)
- B denotes '×' (Multiplication)
- C denotes '-' (Subtraction)
- D denotes '÷' (Division)
The Given Expression
The expression we need to evaluate is:
(51 D 17) A (21 B 9) C 78 A (16 D 4) = ?
Substituting the Operators
Now, let's replace the letters with their corresponding mathematical operators:
$(51 \div 17) + (21 \times 9) - 78 + (16 \div 4) = ?$
Step-by-Step Evaluation using BODMAS/PEMDAS
We follow the BODMAS/PEMDAS rule which dictates the order of operations:
- B/P: Brackets/Parentheses
- O/E: Orders/Exponents
- D/M: Division/Multiplication (from left to right)
- A/S: Addition/Subtraction (from left to right)
Step 1: Evaluate expressions within Parentheses
First, calculate the values inside the parentheses:
- $(51 \div 17) = 3$
- $(21 \times 9) = 189$
- $(16 \div 4) = 4$
The expression now becomes:
$3 + 189 - 78 + 4 = ?$
Step 2: Perform Addition and Subtraction from Left to Right
Now, we perform addition and subtraction operations as they appear from left to right:
- $3 + 189 = 192$
- $192 - 78 = 114$
- $114 + 4 = 118$
So, the final value of the expression is 118.
Summary of Evaluation
Here is a summary of the steps:
| Step | Expression | Operation | Result |
|---|---|---|---|
| Initial Expression | (51 D 17) A (21 B 9) C 78 A (16 D 4) | Substitution | $(51 \div 17) + (21 \times 9) - 78 + (16 \div 4)$ |
| Evaluate Parentheses 1 | $(51 \div 17)$ | Division | 3 |
| Evaluate Parentheses 2 | $(21 \times 9)$ | Multiplication | 189 |
| Evaluate Parentheses 3 | $(16 \div 4)$ | Division | 4 |
| Expression after Parentheses | $3 + 189 - 78 + 4$ | Simplification | $3 + 189 - 78 + 4$ |
| Addition 1 | $3 + 189$ | Addition | 192 |
| Subtraction | $192 - 78$ | Subtraction | 114 |
| Addition 2 | $114 + 4$ | Addition | 118 |
| Final Result | $?$ | Evaluation Complete | 118 |
The value that comes in place of '?' is 118.
Revision Table: Key Concepts in Operator Substitution
| Concept | Description |
|---|---|
| Operator Substitution | Replacing given symbols or letters with standard mathematical operators (+, -, ×, ÷). |
| Mathematical Expression | A combination of numbers, variables, and mathematical operators. |
| Order of Operations (BODMAS/PEMDAS) | Rules defining the sequence for evaluating a mathematical expression to ensure a unique result. Brackets/Parentheses, Orders/Exponents, Division/Multiplication, Addition/Subtraction. |
Additional Information: Solving Mathematical Expressions
Solving mathematical expressions involving multiple operations requires careful application of the order of operations. Failing to follow the correct order can lead to an incorrect result. Parentheses are always evaluated first. Then, exponents or orders are calculated. After that, multiplication and division are performed from left to right. Finally, addition and subtraction are performed from left to right. Practice with different types of expressions and substitutions is key to mastering these problems.
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
