Q1

Problem Restatement

We want to determine if (zw > 0).
We are given two statements:

1. $(z + w^3 = 20)$
2. $(z > 0)$

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Step 1: Consider Statement (1) Alone

From (1):
$[ z = 20 - w^3 ]$

This equation links $(z)$ and $(w)$, but it doesn’t tell us the sign of either variable individually.

• If $(w = 2)$, then $(z = 20 - 8 = 12)$, so $(zw = 24 > 0)$.
• If $(w = -2)$, then $(z = 20 - (-8) = 28)$, so $(zw = -56 < 0)$.

Thus, Statement (1) alone is insufficient.

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Step 2: Consider Statement (2) Alone

Statement (2) says $(z > 0)$.
But we know nothing about $(w)$.

• If $(w > 0)$, then $(zw > 0)$.
• If $(w < 0)$, then $(zw < 0)$.

So Statement (2) alone is insufficient.

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Step 3: Combine Statements (1) and (2)

From (1): $(z = 20 - w^3)$.
From (2): $(z > 0)$.

So: $[ 20 - w^3 > 0 \quad \Rightarrow \quad w^3 < 20 ]$

This means $(w < \sqrt[3]{20} \approx 2.71)$.
But $(w)$ could still be positive or negative.

• If $(w = 2)$, then $(z = 12)$, so $(zw = 24 > 0)$.
• If $(w = -2)$, then $(z = 28)$, so $(zw = -56 < 0)$.

Even with both statements, we cannot determine the sign of $(zw)$.

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Final Answer

The information given is insufficient to determine whether $(zw)$ is positive.

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Q2

Problem Restatement

We know the ratio of Brand Y to Brand X computers is $(5:6)$.
We want to find the number of Brand Y computers.

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Step 1: Express the Ratio

Let:

• Brand Y computers = $(5k)$
• Brand X computers = $(6k)$

for some positive integer $(k)$.

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Step 2: Analyze Statement (1)

Statement (1): Brand X computers exceed Brand Y computers by 80.
$[ 6k - 5k = 80 \quad \Rightarrow \quad k = 80 ]$
So Brand Y computers = $(5k = 400)$.

✅ Statement (1) alone is sufficient.

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Step 3: Analyze Statement (2)

Statement (2): Total computers = 880.
$[ 5k + 6k = 11k = 880 \quad \Rightarrow \quad k = 80 ]$
So Brand Y computers = $(5k = 400)$.

✅ Statement (2) alone is sufficient.

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Step 4: Combine Statements

Since each statement alone is sufficient, combining them is unnecessary.

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Final Answer

Each statement alone is sufficient.
Correct choice: D (Each statement alone is sufficient).

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