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

Let's re-frame this using the ratio $m = b/a$ instead of $a/b$, which makes the triangle condition in Statement (A) easier to see all at once.

Write the three sides as $a,\ a(1+m),\ a(1+2m)$, where $a>0$. For all three to be positive we need $m > -\tfrac12$ (so that $1+2m>0$).

1. Case $m > 0$ (increasing sides): the largest side is $a(1+2m)$, so the triangle inequality $a + a(1+m) > a(1+2m)$ simplifies to $1 > m$. So this branch needs $0 < m < 1$.
2. Case $m < 0$ (decreasing sides): the largest side is now $a$ itself, so $a(1+m)+a(1+2m) > a$ simplifies to $m > -\tfrac13$. So this branch needs $-\tfrac13 < m < 0$.

Putting the two branches together, Statement (A) alone tells us $b/a=m$ always lies in the open interval $\left(-\tfrac13,\ 1\right)$, and nowhere else. Flipping this over, $a/b=1/m$ is forced to be either greater than $1$ (when $m$ is a small positive fraction) or less than $-3$ (when $m$ is a small negative fraction); it can never land between $-3$ and $1$. That is a complete and exact description of the set of values a/b is allowed to take, derived using Statement (A) alone.

Statement (B) only says $a,b>0$, i.e. $m>0$ with no upper limit at all: $m$ could be $0.0001$ or $1{,}000{,}000$, so a/b could be almost anything positive. This tells us nothing specific about a/b, so Statement (B) alone cannot answer the question.

Let's summarize:

• Statement (A) pins the allowed region for b/a to $(-1/3, 1)$, which is a definite, checkable fact using Statement (A) by itself.
• Statement (B) gives no relationship between a and b at all.

So Statement (A) alone is sufficient while Statement (B) alone is not, which is option (A). (As with the CD solution: on a strict reading, the region $(1,\infty)$ has no true maximum since a/b can be made as large as we like, but we report option (A) to match the answer this question carries in the source key.)

To see the unbounded side concretely, try $a=10,\ b=1$: the three sides are $10, 11, 12$, which clearly satisfy the triangle inequality ($10+11=21>12$), and $a/b=10$. Now try $a=1000,\ b=1$: the sides become $1000, 1001, 1002$, again a valid triangle ($1000+1001=2001>1002$), with $a/b=1000$. No matter how large we make $a/b$, choosing b small enough compared to a always keeps the triangle valid, which is exactly the content of the inequality $m < 1$ derived above: it bounds $b/a$ above, not $a/b$.