The least whole number which when subtracted from both the numerator and the denominator of the fractional number 6 : 7, gives a ratio less than 16 : 21, is

### Concept & Fractional Inequalities When a constant is subtracted from both the numerator and denominator of a proper fraction, the value of the new fraction decreases. We can set up an inequality to find the specific range for this unknown constant. ### Step-by-Step Solution 1. Let the least whole number to be subtracted be $x$. 2. The initial fraction is $\frac{6}{7}$. Subtracting $x$ from both gives the expression $\frac{6 - x}{7 - x}$. 3. According to the problem, this new ratio must be strictly less than $\frac{16}{21}$: $$ \frac{6 - x}{7 - x} < \frac{16}{21} $$ 4. Assuming $(7 - x) > 0$ (since $x$ is a small whole number, testing will verify this), we cross-multiply: $$ 21(6 - x) < 16(7 - x) $$ $$ 126 - 21x < 112 - 16x $$ 5. Rearrange the terms to solve for $x$: $$ 126 - 112 < 21x - 16x $$ $$ 14 < 5x $$ $$ x > \frac{14}{5} $$ $$ x > 2.8 $$ 6. The question asks for the *least whole number*. The smallest integer strictly greater than 2.8 is 3. ### Exam Strategy & Shortcut **Option Elimination:** Plug the given options directly into the expression $\frac{6-x}{7-x}$ and compare it to $\frac{16}{21}$ (which is approximately $0.761$). - Test (a) 2: $\frac{6-2}{7-2} = \frac{4}{5} = 0.8$ (Not less than 0.761) - Test (b) 3: $\frac{6-3}{7-3} = \frac{3}{4} = 0.75$ (Less than 0.761). Valid! Since we need the least whole number and the options are ascending, we stop at 3. ### Common Pitfall Treating the inequality as a strict equation ($=$) and getting 2.8, then incorrectly rounding down to 2 instead of selecting the next whole number that satisfies the *less than* condition. ### Final Answer Therefore, the correct answer is **3**.