A certain nutritional study defined the N-score for one serving of a food to be P-Q, where P is the...

Understanding the Question

We need to find how many grams of fiber one serving of Food X provides.

Given Information:

• N-score formula: \(\mathrm{N = P - Q}\), where:
  • \(\mathrm{P = 10 \times (grams\ of\ fat) + 0.1 \times (kilocalories)}\)
  • \(\mathrm{Q = 8 \times (grams\ of\ fiber)^2}\)
• Food X specifics:
  • Provides m kilocalories of energy
  • Contains n grams of fat
  • Has N-score = -28

What We Need to Determine:
Setting up the equation with f = grams of fiber:

• \(\mathrm{-28 = [10n + 0.1m] - 8f^2}\)
• Rearranging: \(\mathrm{8f^2 = 10n + 0.1m + 28}\)

To find f uniquely, we need to know both m and n, or find another way to determine f. Since we have one equation with three unknowns (f, m, and n), we need additional constraints.

Analyzing Statement 1

Statement 1: One serving of Food X provides exactly 400 kilocalories of energy.

This gives us m = 400. Substituting into our equation:

• \(\mathrm{8f^2 = 10n + 0.1(400) + 28}\)
• \(\mathrm{8f^2 = 10n + 40 + 28}\)
• \(\mathrm{8f^2 = 10n + 68}\)

We still have two unknowns (f and n) with only one equation. Without knowing the fat content n, we cannot determine a unique value for f.

For example:

• If n = 2: then \(\mathrm{8f^2 = 88}\), so \(\mathrm{f \approx 3.32\ grams}\)
• If n = 6: then \(\mathrm{8f^2 = 128}\), so \(\mathrm{f = 4\ grams}\)

Different fat contents lead to different fiber amounts.

[STOP - Not Sufficient!] Statement 1 alone is NOT sufficient.

This eliminates choices A and D.

Analyzing Statement 2

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

Statement 2: A food with 120 kilocalories, 3 grams of fiber, and the same fat content as Food X has an N-score of 0.

For this comparison food:

• Kilocalories = 120
• Fiber = 3 grams
• Fat = n grams (same as Food X)
• N-score = 0

Using the N-score formula for the comparison food:

• \(\mathrm{0 = [10n + 0.1(120)] - 8(3)^2}\)
• \(\mathrm{0 = 10n + 12 - 72}\)
• \(\mathrm{0 = 10n - 60}\)
• \(\mathrm{10n = 60}\)
• \(\mathrm{n = 6\ grams}\)

Great! Now we know Food X has 6 grams of fat. But returning to our original equation for Food X:

• \(\mathrm{8f^2 = 10(6) + 0.1m + 28}\)
• \(\mathrm{8f^2 = 60 + 0.1m + 28}\)
• \(\mathrm{8f^2 = 88 + 0.1m}\)

Without knowing m (the kilocalories in Food X), we still cannot determine f uniquely.

For example:

• If m = 400: then \(\mathrm{8f^2 = 128}\), so \(\mathrm{f = 4\ grams}\)
• If m = 520: then \(\mathrm{8f^2 = 140}\), so \(\mathrm{f \approx 4.18\ grams}\)

Different calorie contents lead to different fiber amounts.

[STOP - Not Sufficient!] Statement 2 alone is NOT sufficient.

This eliminates choice B.

Combining Statements

From both statements together:

• Statement 1: m = 400 kilocalories
• Statement 2: n = 6 grams of fat

Substituting both values into our equation:

• \(\mathrm{8f^2 = 10(6) + 0.1(400) + 28}\)
• \(\mathrm{8f^2 = 60 + 40 + 28}\)
• \(\mathrm{8f^2 = 128}\)
• \(\mathrm{f^2 = 16}\)
• \(\mathrm{f = 4\ grams}\) (taking the positive value since fiber content cannot be negative)

With both pieces of information, we can determine that Food X provides exactly 4 grams of fiber per serving.

[STOP - Sufficient!] The statements together are sufficient.

This eliminates choice E.

The Answer: C

Both statements together provide the values of m and n, allowing us to solve uniquely for the fiber content.

Answer Choice C: "Both statements together are sufficient, but neither statement alone is sufficient."