The simplest reduction to the lowest terms of $\frac{116,690,151}{427,863,887}$ is

### Concept & Strategy For extremely large, nine-digit fractions, the problem is testing your ability to use options and estimation, rather than your ability to calculate rigorous prime factorizations. We treat the multiple-choice options as ratios and check which denominator perfectly scales up to the given large denominator, and verify if the numerator scales up by that exact same multiple. ### Step-by-Step Solution * **Given:** The fraction $\frac{116690151}{427863887}$. * **Calculation / Deduction:** 1. Look at the leading digits to estimate the ratio: $\frac{116}{427}$. 2. Notice that $116 \times 3 = 348$, and $116 \times 4 = 464$. The value is slightly more than $\frac{1}{4}$ ($0.25$). 3. Evaluate the given options based on this rough estimate: (a) $\frac{3}{11} \approx 0.27$ (A strong candidate) (b) $\frac{7}{11} \approx 0.63$ (Incorrect) (c) $\frac{11}{3} = 3.66$ (Incorrect) 4. Let's test option (a) rigorously. If the denominator ratio is $11$, then $1$ "part" is: $$427863887 \div 11 = 38896717$$ 5. Now, multiply this $1$-part value by the numerator ratio $3$: $$38896717 \times 3 = 116690151$$ 6. This matches the original numerator perfectly. ### Exam Strategy & Shortcut **Digit Sum / Divisibility Rule Check:** Let's test divisibility by $11$ on the denominator $427863887$. Rule for $11$: (Sum of odd-placed digits) - (Sum of even-placed digits). Odd places: $4+7+6+8+7 = 32$. Even places: $2+8+3+8 = 21$. Difference: $32 - 21 = 11$. Since $11$ is a multiple of $11$, the denominator is divisible by $11$, making options (a) and (b) highly likely. Combining this with our basic $\frac{1}{4}$ estimation immediately isolates $\frac{3}{11}$. ### Common Pitfall Choosing "None of these" because the numbers look too intimidating to check. Examiners use terrifyingly large numbers specifically to reward candidates who stay calm and apply basic approximation and divisibility rules. ### Final Answer **Therefore, the correct answer is $\frac{3}{11}$.**