Material A costs $3 per kilogram, and Material B costs $5 per kilogram. If 10 kilograms of Material K consists...

Understanding the Question

We need to determine whether Material A (x kilograms) makes up more than Material B (y kilograms) in a 10-kilogram mixture.

Given Information

• Material A costs $3 per kilogram
• Material B costs $5 per kilogram
• Total weight: \(\mathrm{x + y = 10}\) kilograms
• Both \(\mathrm{x ≥ 0}\) and \(\mathrm{y ≥ 0}\)

What We Need to Determine

Is \(\mathrm{x > y}\)?

Since the total is 10 kilograms, asking "Is \(\mathrm{x > y}\)?" is equivalent to asking "Is \(\mathrm{x > 5}\)?" Because if x is more than 5 kilograms, then y must be less than 5 kilograms, making \(\mathrm{x > y}\).

For this yes/no question to be sufficient, we need to arrive at a definitive YES or definitive NO — the same answer for all possible values that satisfy the given conditions.

Analyzing Statement 1

Statement 1: \(\mathrm{y > 4}\)

This means Material B makes up more than 4 kilograms of the mixture. Since \(\mathrm{x + y = 10}\), we know that \(\mathrm{x < 6}\).

Testing Different Scenarios

Let's check if this gives us a definite answer:

• If \(\mathrm{y = 4.5}\) kg, then \(\mathrm{x = 5.5}\) kg → Since \(\mathrm{5.5 > 4.5}\), we get \(\mathrm{x > y}\) (YES)
• If \(\mathrm{y = 5.5}\) kg, then \(\mathrm{x = 4.5}\) kg → Since \(\mathrm{4.5 < 5.5}\), we get \(\mathrm{x < y}\) (NO)

Different values of y give us different answers to our question.

[STOP - Not Sufficient!]

Conclusion

Statement 1 alone is NOT sufficient.

This eliminates choices A and D.

Analyzing Statement 2

Now we forget Statement 1 completely and analyze Statement 2 independently.

Statement 2: The cost of the 10 kilograms of Material K is less than $40.

Strategic Analysis

Let's think about what this cost constraint means. First, let's establish the cost boundaries:

• If all 10 kg were Material A (cheapest): Cost = \(\mathrm{10 × \$3 = \$30}\)
• If all 10 kg were Material B (most expensive): Cost = \(\mathrm{10 × \$5 = \$50}\)

Since the actual cost is less than $40, we're somewhere between the minimum of $30 and $40.

Here's the key insight: What happens if we have exactly equal amounts (\(\mathrm{x = y = 5}\))?

• Cost = \(\mathrm{5 × \$3 + 5 × \$5 = \$15 + \$25 = \$40}\)

This equals exactly $40! Since our actual cost must be less than $40, we can't have equal amounts or more of the expensive Material B. We must have more of the cheaper Material A.

Therefore, \(\mathrm{x > 5}\) and \(\mathrm{y < 5}\), which means \(\mathrm{x > y}\).

[STOP - Sufficient!]

Conclusion

Statement 2 alone is sufficient — it always gives us a YES answer.

This eliminates choices C and E.

The Answer: B

Statement 2 alone is sufficient because the cost constraint forces us to have more of the cheaper material, while Statement 1 alone allows for different outcomes.

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