Solving the Symbolic Expression Question

This question asks us to find the value of a symbolic mathematical expression, \( 14\$20\*8#4 \), based on information given in two statements. This is a typical Data Sufficiency problem where we need to determine which statement or combination of statements provides enough information to answer the question.

Analyzing the Question and Statements

The expression we need to evaluate is \( 14\$20\*8#4 \). To find its value, we must understand what the symbols $, *, and # represent in terms of standard mathematical operations like addition, subtraction, multiplication, or division.

We are given two statements:

1. Statement 1: \( \text{\$ implies -} \), \( \text{\* implies +} \), \( \text{\# implies \div} \)
2. Statement 2: \( 14\$20\*8 = 25 \)

We need to check if Statement 1 alone is sufficient, if Statement 2 alone is sufficient, if both are needed, or if neither is sufficient to determine the value of \( 14\$20\*8#4 \).

Evaluating Statement 1 Sufficiency for Symbolic Expression

Statement 1 explicitly defines what each symbol represents:

• \( \$ \) means subtraction \( (-) \)
• \( \* \) means addition \( (+) \)
• \( \# \) means division \( (\div) \)

With these definitions, we can replace the symbols in the expression \( 14\$20\*8#4 \) with the corresponding mathematical operations:

The expression becomes \( 14 - 20 + 8 \div 4 \).

Now, we can evaluate this standard mathematical expression following the order of operations (BODMAS/PEMDAS - Brackets, Orders/Exponents, Division and Multiplication (from left to right), Addition and Subtraction (from left to right)).

1. Perform division: \( 8 \div 4 = 2 \)
2. Substitute the result back: \( 14 - 20 + 2 \)
3. Perform subtraction (from left): \( 14 - 20 = -6 \)
4. Perform addition (from left): \( -6 + 2 = -4 \)

The value of the expression \( 14\$20\*8#4 \) is \( -4 \) according to the definitions in Statement 1.

Since Statement 1 provides the necessary information to calculate a unique value for the expression, Statement 1 alone is sufficient to answer the question.

Evaluating Statement 2 Sufficiency for Symbolic Expression

Statement 2 gives us the equation \( 14\$20\*8 = 25 \).

This statement provides a relationship between the numbers 14, 20, 8, and 25 using the symbols $ and *. It tells us the outcome of a specific calculation involving these numbers and symbols.

However, Statement 2 does not define what the symbols $, *, or # generally represent as mathematical operations. We only have one equation with two unknown operations ($ and *).

Furthermore, Statement 2 provides absolutely no information about the symbol #, which is present in the expression we need to evaluate \( (14\$20\*8#4) \). Without knowing what # means, we cannot evaluate the full expression.

For example, Statement 2 could be true for many different definitions of $ and *, or even different orders of operations. More importantly, it doesn't tell us anything about #.

Therefore, Statement 2 alone is insufficient to answer the question.

Comparing Sufficiency and Final Decision

Based on our analysis:

• Statement 1 alone is sufficient because it provides explicit definitions for all the symbols ($, *, and #) required to evaluate the expression \( 14\$20\*8#4 \).
• Statement 2 alone is insufficient because it does not define the symbols and, crucially, provides no information about the symbol #.

Thus, Statement 1 alone is sufficient, while Statement 2 alone is insufficient.

Revision Table: Key Concepts in Sufficiency

Additional Information: Data Sufficiency Tips

Data Sufficiency questions test your ability to determine if enough information is given to solve a problem, not necessarily to solve the problem itself. Here are some tips:

• Read the question carefully to understand exactly what needs to be determined.
• Analyze Statement 1 alone. Assume it is true and see if you can definitively answer the question. Ignore Statement 2 during this step.
• Analyze Statement 2 alone. Assume it is true and see if you can definitively answer the question. Ignore Statement 1 during this step.
• If neither statement alone is sufficient, consider both statements together. See if combining the information allows you to answer the question.
• If even both together are insufficient, the answer is that neither is sufficient.
• Remember that in Data Sufficiency, "sufficient" means you can find a unique answer. If multiple answers are possible, the statement(s) are insufficient.