In concrete mix design using fineness modulus (FM), if X, Y, and Z are the fineness moduli of the coarse aggregate, the fine aggregate, and the final combined aggregate respectively, determine the percentage P of fine aggregate in the total combined aggregate.

Introduction / Context:
Fineness modulus (FM) is a single-number index that captures the overall particle size distribution of aggregates. In practical mix design, the FM of the final combined aggregate is adjusted by blending coarse and fine aggregates in suitable proportions. This question tests the core relationship between the fineness moduli of the coarse (X), fine (Y), and combined aggregate (Z) to compute the percentage of fine aggregate required.

Given Data / Assumptions:

• X: fineness modulus of the coarse aggregate fraction.
• Y: fineness modulus of the fine aggregate fraction.
• Z: fineness modulus of the combined aggregate after blending.
• P: percentage (by mass) of fine aggregate in the total combined aggregate.

Concept / Approach:
The fineness modulus of a blend is the weighted average of the component FMs, weighted by their proportions. Let p be the fraction of fine aggregate (0 to 1). Then the combined FM satisfies Z = p * Y + (1 - p) * X. This linearity arises because FM is computed from cumulative sieve retention which is itself linear with respect to proportioning of components.

Step-by-Step Solution:
Start from Z = p * Y + (1 - p) * X.Expand: Z = pY + X - pX = X + p * (Y - X).Solve for p: p = (Z - X) / (Y - X).Convert to percentage: P = 100 * p = 100 * (Z - X) / (Y - X).

Verification / Alternative check:
If Z = X (combined equals coarse), then P = 0% (no fine), which the formula gives. If Z = Y (combined equals fine), then P = 100%, which the formula also gives. These boundary checks confirm correctness.

Why Other Options Are Wrong:

• P = 100 * (X - Z) / (Y - Z): mixes variables incorrectly; fails boundary checks.
• P = 100 * (Y - Z) / (Y - X): gives 0% when Z = Y (should be 100%).
• P = 100 * (Z - Y) / (X - Y): sign and placement errors produce negative or inconsistent values.
• P = 100 * (X + Y - 2Z) / (Y - X): not derived from linear blending law.

Common Pitfalls:
Confusing which FM is larger. Typically Y (fine) < X (coarse) is not always true; many conventions list FM of fine sand around 2.4–3.2 and coarse aggregate FM higher. Always verify magnitudes. Also, ensure proportions are by mass when using standard FM blending.

Final Answer:
P = 100 * (Z - X) / (Y - X)