Solving the Symbol Substitution Expression

This question requires us to evaluate a mathematical expression where standard arithmetic operators (+, -, ×, ÷) are represented by letters (A, B, C, D). We need to substitute the letters with their corresponding operators and then solve the resulting expression following the correct order of operations.

Understanding the Symbol Substitutions

The question provides the following rules for substitution:

• ‘A’ means ‘+’ (Addition)
• ‘B’ means ‘–’ (Subtraction)
• ‘C’ means ‘×’ (Multiplication)
• ‘D’ means ‘÷’ (Division)

The given expression is:

50 A 40 B 10 C 12 D 2

Rewriting the Expression

Let's replace the letters with their corresponding mathematical operators:

$50 \text{ A } 40 \text{ B } 10 \text{ C } 12 \text{ D } 2 \implies 50 + 40 - 10 \times 12 \div 2$

Applying the Order of Operations (BODMAS/PEMDAS)

To correctly evaluate the expression $50 + 40 - 10 \times 12 \div 2$, we must follow the order of operations. A common acronym for this order is BODMAS or PEMDAS:

• Brackets / Parentheses
• Orders (powers, square roots, etc.) / Exponents
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

In our expression, we have Addition, Subtraction, Multiplication, and Division. According to BODMAS/PEMDAS, we perform Division and Multiplication before Addition and Subtraction.

Step-by-Step Calculation

Let's solve the expression step-by-step:

Expression: $50 + 40 - 10 \times 12 \div 2$

Step 1: Perform Division (from left to right)

We have $12 \div 2$.

$12 \div 2 = 6$

The expression becomes:

$50 + 40 - 10 \times 6$

Step 2: Perform Multiplication (from left to right)

We have $10 \times 6$.

$10 \times 6 = 60$

The expression becomes:

$50 + 40 - 60$

Step 3: Perform Addition and Subtraction (from left to right)

First, perform Addition: $50 + 40$.

$50 + 40 = 90$

The expression becomes:

$90 - 60$

Next, perform Subtraction: $90 - 60$.

$90 - 60 = 30$

The final value of the expression is 30.

Final Value of the Expression

After substituting the symbols and applying the order of operations, the value of the expression 50 A 40 B 10 C 12 D 2 is 30.

Revision Table: Symbol Substitution and Calculation

Additional Information on Order of Operations

The order of operations is a crucial concept in mathematics to ensure consistency in evaluating expressions. Without a standard order, the same expression could yield different results depending on which operation is performed first.

• BODMAS: Brackets, Orders, Division, Multiplication, Addition, Subtraction.
• PEMDAS: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.

Both acronyms represent the same hierarchy. Multiplication and Division are at the same level, and they should be performed from left to right as they appear in the expression. Similarly, Addition and Subtraction are at the same level and are performed from left to right.

In the given problem, the division ($12 \div 2$) comes before the multiplication ($10 \times 6$) when reading from left to right within that priority level. However, in the expression $10 \times 12 \div 2$, multiplication and division are adjacent. The rule 'from left to right' dictates the order when operators of the same priority level appear consecutively or in a group. In this case, we calculate $12 \div 2$ first, then multiply the result by 10. If it were $10 \div 2 \times 12$, we would calculate $10 \div 2$ first, then multiply by 12.