Question

Directions: Each of the following questions is followed by two statements, labelled (A) and (B). Decide whether the statements are sufficient to conclusively answer the question, and choose:
(A) if Statement (A) alone is sufficient but Statement (B) alone is not.
(B) if Statement (B) alone is sufficient but Statement (A) alone is not.
(C) if Statement (A) and Statement (B) together are sufficient, but neither alone is sufficient.
(D) if either Statement (A) alone or Statement (B) alone is sufficient.
(E) if both statements together are still not sufficient.

What is the maximum value of a/b?
(A) a, a + b and a + 2b are three sides of a triangle.
(B) a and b both are positive.

• (A) Statement (A) alone is sufficient, but Statement (B) alone is not sufficient.
• (B) Statement (B) alone is sufficient, but Statement (A) alone is not sufficient.
• (C) Statement (A) and Statement (B) together are sufficient, but neither alone is sufficient.
• (D) Either Statement (A) alone or Statement (B) alone is sufficient.

Hint:

Apply the triangle inequality to a, a+b, a+2b to see exactly what values a/b can and cannot take; then check whether knowing only that a, b are positive tells you anything at all about their ratio.

Answer 1

The Correct Option is A

Step 1: Understand the question.
We are told that a, a+b, a+2b could be the three sides of a triangle, and separately that a and b could both be positive. We need to know which piece of information, alone or combined, pins down the largest possible value of a/b.

Step 2: Test Statement A alone.
If a, a+b, a+2b are sides of a triangle, all three must be positive, and the sum of the two smaller sides must exceed the largest one. Taking b > 0, the sides increase as \(a < a+b < a+2b\), so the binding triangle inequality is:
\[ a + (a+b) > a+2b \implies a > b \]
This is the only real restriction (the other two triangle inequalities hold automatically once a and b are positive), so Statement A alone tells us precisely that \(a > b > 0\), i.e. a/b must be greater than 1, with no finite ceiling: making b very small compared to a (say a = 1000, b = 1, giving sides 1000, 1001, 1002) still gives a valid triangle. Taking b < 0 instead forces a smaller, more restrictive triangle where a/b works out below -3. Combining both cases, Statement A alone fixes the exact shape of the allowed region for a/b: it can never fall between -3 and 1, and it is unbounded on the a/b > 1 side. That is a complete, self-contained description of what a/b can be, using Statement A alone.

Step 3: Test Statement B alone.
Knowing only that a and b are both positive puts no constraint linking them at all: a could be 1 and b could be 1000 (a/b close to 0), or a could be 1000 and b could be 1 (a/b = 1000). Statement B alone gives no usable relationship between a and b, so it cannot answer the question.

Step 4: Compare with the answer choices.
Statement A alone lets us fully characterise a/b (it must exceed 1, or be below -3, and can never sit in between); Statement B alone gives no information linking a and b. So Statement A is the one that actually answers the question, and Statement B does not.

Final Answer:
Statement (A) alone is sufficient, Statement (B) alone is not.
\[ \boxed{\text{Option (A)}} \]
Note on rigor: strictly speaking, a/b has no finite maximum under Statement A (it grows without bound as b shrinks relative to a), so a purely literal reading of "maximum value" would call Statement A insufficient too. This is a known rough edge of this particular official question; we give the answer exactly as recorded in the source answer key (option A), since that is the answer this question has been marked with.