Let's analyze the problem and each of the statements provided.
The problem is: Will the new fraction [tex]\((a + 7) / (b + 7)\)[/tex] be less than the original fraction [tex]\(a / b\)[/tex]?
To solve this, we should consider the statements given:
### Statement-I: [tex]\(a = 73, b = 103\)[/tex]
Given this statement, we can directly plug in these values into our fractions:
- Original fraction: [tex]\(\frac{73}{103}\)[/tex]
- New fraction: [tex]\(\frac{73 + 7}{103 + 7} = \frac{80}{110}\)[/tex]
We can compare these fractions directly, which shows:
- [tex]\(\frac{73}{103} \approx 0.7087\)[/tex]
- [tex]\(\frac{80}{110} \approx 0.7273\)[/tex]
Since [tex]\(0.7273\)[/tex] is not less than [tex]\(0.7087\)[/tex], the new fraction is not less than the original fraction.
### Statement-II: The average of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] is less than [tex]\(b\)[/tex].
The average of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] is [tex]\(\frac{a + b}{2}\)[/tex].
If the average is less than [tex]\(b\)[/tex], this means:
[tex]\[ \frac{a + b}{2} < b \implies a + b < 2b \implies a < b \][/tex]
This information tells us that [tex]\(a\)[/tex] is less than [tex]\(b\)[/tex], but does not provide specific values, so we cannot directly compare the fractions without knowing the exact values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex].
### Statement-III: [tex]\(a - 5\)[/tex] is greater than [tex]\(b - 5\)[/tex].
This implies that [tex]\(a > b\)[/tex], but this is inconsistent with Statement-II where [tex]\(a < b\)[/tex]. Therefore, one of these statements is incorrect or redundant.
Given Statement-II ([tex]\(a < b\)[/tex]) and noting that [tex]\(a = 73\)[/tex] and [tex]\(b = 103\)[/tex] indeed satisfies [tex]\(a < b\)[/tex], we can discard Statement-III since it conflicts with the known facts (as per Statement-I).
### Decision on Redundancy:
With Statement-I alone, we were able to determine the comparison between the original and the new fraction. Therefore, the other statements are redundant for answering the question.
### Conclusion:
Option b. "Only I or III" is the correct answer since only Statement-I is needed to determine whether the new fraction is less than the original one, and Statement-III is incorrect and therefore redundant.
