Let's break down this problem step by step. We are given a rule for replacing letters with mathematical operators and then asked to evaluate an expression using these rules.

Understanding the Symbol Substitution Rules

The question provides the following rules for substituting letters with operators:

• A denotes '+' (addition)
• B denotes '×' (multiplication)
• C denotes '−' (subtraction)
• D denotes '÷' (division)

Applying Symbol Substitution to the Expression

The given expression is: 10 B 19 A 10 D 10 C 7

Now, we will replace each letter with its corresponding operator according to the rules:

10 B 19 A 10 D 10 C 7 becomes:

$$10 \times 19 + 10 \div 10 - 7$$

Now we have a standard mathematical expression to evaluate.

Evaluating the Expression using Order of Operations (BODMAS/PEDMAS)

To correctly evaluate the expression $$10 \times 19 + 10 \div 10 - 7$$, we must follow the order of operations, commonly remembered by acronyms like BODMAS or PEDMAS.

• Brackets (or Parentheses)
• Orders (or Exponents)
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

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

Step 1: Division

First, perform the division operation: $$10 \div 10 = 1$$

The expression now becomes: $$10 \times 19 + 1 - 7$$

Step 2: Multiplication

Next, perform the multiplication operation: $$10 \times 19 = 190$$

The expression now becomes: $$190 + 1 - 7$$

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

Now, perform the addition: $$190 + 1 = 191$$

The expression becomes: $$191 - 7$$

Finally, perform the subtraction: $$191 - 7 = 184$$

The value of the expression is 184.

Final Value of the Expression

After performing the symbol substitution and evaluating the expression using the order of operations, the final value is 184.

Revision Table: Symbol Substitution and Order of Operations

Here's a quick summary of the steps involved in solving this type of problem.

Additional Information: BODMAS/PEDMAS Explained

The order of operations is a crucial concept in mathematics to ensure everyone gets the same answer when evaluating an expression. Here's a little more detail:

• B/P: Brackets or Parentheses - Operations inside brackets are always done first.
• O/E: Orders or Exponents - Powers and square roots are evaluated next.
• D/M: Division and Multiplication - These are done next, working from left to right in the expression. They have equal priority.
• A/S: Addition and Subtraction - These are done last, working from left to right in the expression. They also have equal priority.

In our problem, we had only Multiplication, Addition, Division, and Subtraction. We first handled the division and multiplication from left to right, and then the addition and subtraction from left to right.