Evaluating Mathematical Expressions with Symbol Substitution

This problem requires us to substitute specific mathematical operations for given letters in an expression and then evaluate the resulting numerical expression following the standard order of operations.

The given mappings are:

• 'A' denotes 'addition' ($+$)
• 'B' denotes 'multiplication' ($\times$)
• 'C' denotes 'subtraction' ($-$)
• 'D' denotes 'division' ($\div$)

The expression to evaluate is:

50 D (2 B 5) A 3 B (15 A 12) C 2 B (18 C 15)

Step 1: Substitute the Symbols with Operations

Replace each letter with its corresponding mathematical operation:

50 $\div$ (2 $\times$ 5) $+$ 3 $\times$ (15 $+$ 12) $-$ 2 $\times$ (18 $-$ 15)

Step 2: Evaluate the Expression using Order of Operations (BODMAS/PEMDAS)

We follow the order of operations: Brackets first, then Orders (powers and roots), then Division and Multiplication (from left to right), and finally Addition and Subtraction (from left to right).

Evaluate Expressions inside Brackets:

• (2 $\times$ 5) = 10
• (15 $+$ 12) = 27
• (18 $-$ 15) = 3

Substituting these values back into the expression, we get:

50 $\div$ 10 $+$ 3 $\times$ 27 $-$ 2 $\times$ 3

Perform Division and Multiplication (from left to right):

• 50 $\div$ 10 = 5
• 3 $\times$ 27 = 81
• 2 $\times$ 3 = 6

The expression now becomes:

5 $+$ 81 $-$ 6

Perform Addition and Subtraction (from left to right):

• 5 $+$ 81 = 86
• 86 $-$ 6 = 80

The final value of the expression is 80.

Result of the Expression Evaluation

After substituting the operations and following the order of operations, the value of the expression 50 D (2 B 5) A 3 B(15 A 12) C 2 B(18 C 15) is 80.

Revision Table: Symbol Mapping

Additional Information: Order of Operations

The order of operations is a rule used in mathematics to clarify which procedures should be performed first in a given mathematical expression. Common mnemonics include BODMAS and PEMDAS.

• BODMAS: Brackets, Orders (powers, square roots, etc.), Division, Multiplication, Addition, Subtraction.
• PEMDAS: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.

Division and Multiplication have the same priority and are performed from left to right. Similarly, Addition and Subtraction have the same priority and are performed from left to right.

In this problem, correctly applying the order of operations after substituting the symbols was crucial to arrive at the correct result.