Introduction / Context:
This question tests your number sense and ability to compare fractions without necessarily converting each one into a long decimal. Such comparisons are common in aptitude exams, where quick approximate reasoning and understanding of numerator denominator relationships are very useful. We must decide which inequality chains are correct.
Given Data / Assumptions:
- Statement I: 3/71 < 5/91 < 7/99.
- Statement II: 11/135 > 12/157 > 13/181.
- We must judge each chain as true or false.
- All numbers are positive fractions less than 1.
Concept / Approach:
Two main methods can be used. First, we can convert each fraction to a decimal with enough precision to compare them confidently. Second, we can cross multiply to compare pairs of fractions directly, which avoids decimal conversions. For clarity and reliability, we will effectively approximate or compare rationally to confirm the ordering in each statement.
Step-by-Step Solution:
1. Consider Statement I: compare 3/71, 5/91, and 7/99.
2. Compute approximate values: 3/71 is about 0.0423, 5/91 is about 0.0549, and 7/99 is about 0.0707.
3. These values clearly satisfy 0.0423 < 0.0549 < 0.0707, so Statement I is true.
4. For a fraction comparison check without decimals, you could also cross multiply. For example, to check whether 3/71 < 5/91, compare 3 * 91 and 5 * 71. We find 273 < 355 so 3/71 < 5/91. Similar checks can be done for 5/91 < 7/99.
5. Now consider Statement II: compare 11/135, 12/157, and 13/181.
6. Compute approximate values: 11/135 ≈ 0.0815, 12/157 ≈ 0.0764, and 13/181 ≈ 0.0718.
7. These satisfy 0.0815 > 0.0764 > 0.0718, so Statement II is also true.
8. Therefore both Statement I and Statement II are correct.
Verification / Alternative check:
Another quick method is to observe that when we increase both numerator and denominator in similar proportion, the value of the fraction usually decreases if the numerator grows slower relative to the denominator. In Statement II, from 11/135 to 12/157 to 13/181, the denominators grow faster compared to the numerators, so the fraction decreases, which matches the given ordering. Combining this reasoning with the approximate decimal values gives a strong confirmation that both chains are valid.
Why Other Options Are Wrong:
Option 1 (Only I) disregards the fact that Statement II is also correctly ordered.
Option 2 (Only II) ignores the clearly correct ordering in Statement I.
Option 4 (Neither I nor II) is not possible because we have explicitly verified that both statements are consistent with the actual numerical values of the fractions.
Common Pitfalls:
A common mistake is to compare only denominators or only numerators without considering both together. Another error is incorrect decimal rounding or truncating too early, which can flip an inequality if the fractions are very close. Cross multiplication is a reliable alternative that avoids decimal rounding errors. Carefully computing products like 3 * 91 and 5 * 71 is usually faster and safer in exam settings.
Final Answer:
Both given inequality statements are correct, so the answer is
Both I and II.
Discussion & Comments