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An investment fund holds a portfolio consisting entirely of short-term bonds yielding 4%4\% annually and long-term bonds yielding 8%8\% annually. In 2025, the total dollar amount invested in short-term bonds changed by p%p\%, and the total dollar amount invested in long-term bonds changed by q%q\%. Was the portfolio's overall percentage yield higher in 2025 than in 2024?

(1) In 2024, the amount invested in short-term bonds was twice the amount invested in long-term bonds.
(2) In 2025, the annual interest income earned from long-term bonds was greater than the annual interest income earned from short-term bonds.

  1. A
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
  2. B
    Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
  3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.Cevap
  4. D
    EACH statement ALONE is sufficient.
  5. E
    Statements (1) and (2) TOGETHER are NOT sufficient.

Cevap

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Rephrasing the question stem algebraically demonstrates that the overall portfolio yield increases if and only if the relative proportion of the higher-yielding asset (8%8\%) increases compared to the lower-yielding asset (4%4\%), which simplifies to the condition q>pq > p. Statement (1) alone provides only the initial ratio S=2LS = 2L, leaving pp and qq unconstrained. Statement (2) alone yields 0.08L(1+q/100)>0.04S(1+p/100)0.08L(1 + q/100) > 0.04S(1 + p/100), which cannot be simplified without knowing the ratio S/LS/L. Combining both statements allows substituting S=2LS = 2L into Statement (2)'s inequality, yielding 2L(1+q/100)>2L(1+p/100)  ⟹  q>p2L(1 + q/100) > 2L(1 + p/100) \implies q > p, which definitively answers 'Yes'. Thus, both statements together are sufficient.

Adım Adım Çözüm

1
Rephrase the question target algebraically.
Overall yield in 2024 is Y2024=0.04S+0.08LS+LY_{2024} = \frac{0.04S + 0.08L}{S + L}. Overall yield in 2025 is Y2025=0.04S(1+p/100)+0.08L(1+q/100)S(1+p/100)+L(1+q/100)Y_{2025} = \frac{0.04S(1 + p/100) + 0.08L(1 + q/100)}{S(1 + p/100) + L(1 + q/100)}. Since YY is a weighted average of two fixed values (4%4\% and 8%8\%), Y2025>Y2024Y_{2025} > Y_{2024} if and only if the weight of the higher-yielding component (8%8\%) increases relative to the lower-yielding component (4%4\%). That requires L(1+q/100)S(1+p/100)>LS\frac{L(1 + q/100)}{S(1 + p/100)} > \frac{L}{S}, which simplifies to 1+q/100>1+p/100  ⟺  q>p1 + q/100 > 1 + p/100 \iff q > p. The target question is rephrased as: 'Is q>pq > p?'
Simplifying the question stem reveals that the initial investment amounts SS and LL drop out when asking about relative yield changes.
2
Evaluate Statement (1) independently.
Statement (1) gives S=2LS = 2L. This provides information about the baseline 2024 portfolio composition, but provides no information regarding pp or qq. Therefore, we cannot determine whether q>pq > p. Statement (1) alone is NOT sufficient.
Knowing initial weights without growth rates does not allow determination of relative growth rate direction.
3
Evaluate Statement (2) independently.
Statement (2) states that interest from long-term bonds in 2025 exceeded interest from short-term bonds: 0.08L(1+q/100)>0.04S(1+p/100)0.08L(1 + q/100) > 0.04S(1 + p/100), which simplifies to 2L(1+q/100)>S(1+p/100)2L(1 + q/100) > S(1 + p/100). Because the ratio S/LS/L is unknown, this inequality could hold even if q≤pq \le p (if SS is very small relative to LL) or only when q>pq > p (if SS is large). Statement (2) alone is NOT sufficient.
Without knowing S/LS/L, the inequality 2L(1+q/100)>S(1+p/100)2L(1+q/100) > S(1+p/100) cannot be reduced to compare qq and pp.
4
Evaluate Statements (1) and (2) together.
From Statement (1), substitute S=2LS = 2L into Statement (2)'s inequality: 2L(1+q/100)>(2L)(1+p/100)2L(1 + q/100) > (2L)(1 + p/100). Dividing both sides by 2L2L (since L>0L > 0) gives 1+q/100>1+p/100  ⟺  q>p1 + q/100 > 1 + p/100 \iff q > p. This provides a definitive 'Yes' to the rephrased question target. BOTH statements together are SUFFICIENT.
Combining the baseline ratio with the interest inequality allows SS and LL to cancel, establishing q>pq > p definitively.

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