An alloy, Alloy K, is made by mixing certain quantities of iron and lead. The total weight of the alloy...

Understanding the Question

We need to find the ratio of the weight of iron to the weight of lead in Alloy K.

Given Information

• Alloy K contains only iron and lead
• Total weight = 50 kg
• If I = weight of iron and L = weight of lead, then \(\mathrm{I + L = 50}\)

What We Need to Determine

We need to find the ratio \(\mathrm{I:L}\) (or equivalently, \(\mathrm{I/L}\)). Since we know \(\mathrm{I + L = 50}\), if we can determine either I or L uniquely, we can find the other and thus determine the ratio.

For this question to be sufficient, we need to arrive at one specific ratio value.

Analyzing Statement 1

Statement 1: If 4 kg of iron were to be replaced with 4 kg of lead, the percentage of lead, by weight, in Alloy K would become 40%.

What Statement 1 Tells Us

This statement describes a hypothetical scenario. After replacing 4 kg of iron with 4 kg of lead:

• New amount of lead = \(\mathrm{L + 4}\) kg
• New amount of iron = \(\mathrm{I - 4}\) kg
• Total weight remains 50 kg (we're replacing, not adding)
• Lead would constitute exactly 40% of the total weight

Working Through the Logic

Since lead would be 40% of 50 kg after the replacement:

• \(\mathrm{L + 4 = 0.40 \times 50 = 20}\) kg
• Therefore, \(\mathrm{L = 16}\) kg (current amount of lead)
• Since \(\mathrm{I + L = 50}\), we get \(\mathrm{I = 34}\) kg (current amount of iron)
• The ratio \(\mathrm{I:L = 34:16 = 17:8}\) (dividing both by 2)

[STOP - Sufficient!] We can determine a unique ratio.

Conclusion

Statement 1 is sufficient.

This eliminates choices B, C, and E.

Analyzing Statement 2

Now let's forget Statement 1 completely and analyze Statement 2 independently.

Statement 2: The weight of iron per 100 cc is 0.7 times the weight of lead per 100 cc.

What Statement 2 Provides

This tells us about the density relationship between iron and lead:

• Density of iron = \(0.7 \times \mathrm{Density\,of\,lead}\)

In other words, iron is less dense than lead (which makes sense physically).

What We Still Don't Know

While we know the density relationship, we don't know:

• The actual volumes of iron and lead in the alloy
• Any constraint that would help us determine unique weights

Think of it this way: If we have 50 kg total, we could have:

• 30 kg iron + 20 kg lead (ratio \(3:2\))
• 25 kg iron + 25 kg lead (ratio \(1:1\))
• 40 kg iron + 10 kg lead (ratio \(4:1\))

The density relationship alone doesn't tell us which combination we actually have, because we don't know the volumes involved.

Conclusion

Statement 2 is NOT sufficient.

This eliminates choices B and D.

The Answer: A

Since Statement 1 alone is sufficient to determine the ratio, but Statement 2 alone is not sufficient, the answer is A.

Answer Choice A: "Statement 1 alone is sufficient, but Statement 2 alone is not sufficient."