Question

What is the value of \( \frac{a}{b} \)? Statement (I): \( 7a - 3b = 0 \)
Statement (II): \( b = 5 \)

• A. Statement (I) alone is sufficient.
• B. Statement (II) alone is sufficient.
• C. Both statements (I) and (II) are sufficient.
• D. Neither statement is sufficient.

Hint:

In algebra-based data sufficiency, always check if the given equation can be algebraically manipulated to match the form requested in the question stem.

Answer 1

The Correct Option is A

Concept: A ratio \( \frac{a}{b} \) represents the proportional relationship between two variables. In data sufficiency, to find the numerical value of a ratio, we do not necessarily need the individual values of \( a \) and \( b \), but rather an equation that expresses one variable in terms of the other or defines their fixed ratio.

Step 1: Evaluate Statement (I). We are given the equation: \[ 7a - 3b = 0 \] Rearranging to isolate the terms: \[ 7a = 3b \] To find the ratio \( \frac{a}{b} \), we divide both sides by \( 7b \): \[ \frac{a}{b} = \frac{3}{7} \] Since Statement (I) allows us to determine the specific numerical value of the ratio \( \frac{a}{b} \) without any additional information, this statement is sufficient.

Step 2: Evaluate Statement (II). Statement (II) provides only the value of \( b = 5 \). Substituting this into the ratio: \[ \frac{a}{b} = \frac{a}{5} \] Without knowing the value of \( a \), the ratio remains dependent on an unknown variable. Therefore, Statement (II) is not sufficient. { Statement (I) alone is sufficient.}