Understanding Symbol Substitution in Mathematical Expressions

The question asks us to substitute specific letters with mathematical symbols and then evaluate the resulting expression to find which one is correct. We are given the following mapping:

• A stands for Multiplication ($\times$)
• B stands for Division ($\div$)
• C stands for Addition ($+$)
• D stands for Subtraction ($-$)
• E stands for Equal to ($=$)
• F stands for Greater than ($>$)
• G stands for Less than ($<$)

We need to apply these substitutions to the given options and check which mathematical statement holds true.

Evaluating the Options

Let's convert each option into a mathematical expression using the given symbols and evaluate it.

Option 1: 512B4D16C21A5F217

Substituting the letters with symbols:

512 $\div$ 4 - 16 + 21 $\times$ 5 > 217

Now, let's evaluate the left side following the order of operations (Division and Multiplication first, then Addition and Subtraction from left to right):

1. Division: $512 \div 4 = 128$
2. Multiplication: $21 \times 5 = 105$
3. Expression becomes: $128 - 16 + 105$
4. Subtraction: $128 - 16 = 112$
5. Addition: $112 + 105 = 217$

So, the inequality is $217 > 217$. This statement is false, as 217 is equal to 217, not greater than 217.

Option 2: 512B4D16C21A5G217

Substituting the letters with symbols:

512 $\div$ 4 - 16 + 21 $\times$ 5 < 217

From our calculation in Option 1, we know the left side evaluates to 217.

So, the inequality is $217 < 217$. This statement is false, as 217 is equal to 217, not less than 217.

Option 3: 512B4D16C21A5E217

Substituting the letters with symbols:

512 $\div$ 4 - 16 + 21 $\times$ 5 = 217

Again, we evaluate the left side following the order of operations:

1. Division: $512 \div 4 = 128$
2. Multiplication: $21 \times 5 = 105$
3. Expression becomes: $128 - 16 + 105$
4. Subtraction: $128 - 16 = 112$
5. Addition: $112 + 105 = 217$

So, the equality is $217 = 217$. This statement is true.

Option 4: 512B4D16A21A5E217

Substituting the letters with symbols:

512 $\div$ 4 - 16 $\times$ 21 $\times$ 5 = 217

Let's evaluate the left side following the order of operations:

1. Division: $512 \div 4 = 128$
2. Multiplication: $16 \times 21 \times 5$ (evaluate from left to right or in any order) $16 \times 21 = 336$ $336 \times 5 = 1680$
3. Expression becomes: $128 - 1680$
4. Subtraction: $128 - 1680 = -1552$

So, the equality is $-1552 = 217$. This statement is false.

Conclusion

Based on our evaluation, only Option 3 results in a true mathematical statement ($217 = 217$). Therefore, Option 3 is the correct one according to the symbols given in the directions.

Revision Table: Symbol Substitution and Evaluation

When solving problems like this, it's crucial to correctly substitute symbols and follow the order of mathematical operations (BODMAS/PEMDAS).

• BODMAS/PEMDAS: Brackets/Parentheses, Orders/Exponents, Division and Multiplication (from left to right), Addition and Subtraction (from left to right).
• Symbol Mapping: Keep a clear list of which letter corresponds to which symbol.
• Step-by-step Calculation: Evaluate the expression on one side of the equality or inequality step by step to avoid errors.
• Verification: Compare the calculated value with the value on the other side of the equality/inequality to determine if the statement is true or false.

Additional Information: Order of Operations

The order of operations is a set of rules that dictates the sequence in which operations in a mathematical expression should be performed. This ensures consistency and a unique result for any given expression. The widely used acronyms are BODMAS or PEMDAS.

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

Note that Division and Multiplication have the same priority and should be performed from left to right. Similarly, Addition and Subtraction have the same priority and should be performed from left to right.

For example, in the expression $10 - 3 \times 2 + 6 \div 2$:

1. Multiplication/Division (left to right): $3 \times 2 = 6$ and $6 \div 2 = 3$. The expression becomes $10 - 6 + 3$.
2. Addition/Subtraction (left to right): $10 - 6 = 4$. The expression becomes $4 + 3$.
3. $4 + 3 = 7$.

Following the order of operations is essential for correctly evaluating mathematical expressions resulting from symbol substitution.