If X, Y and Z are fineness modulus of coarse, fine and combined aggregates respectively, the percentage (P) of fine aggregates to combined aggregates is:
Correct Answer: B. P = ((X – Z) / (X – Y)) × 100
📚 Detailed Explanation: FM Blending Formula Derivation
Why B is correct: Let P = % fine aggregate, (100−P) = % coarse aggregate. The combined FM is the weighted average:
P/100 × Y + (1 − P/100) × X = Z
Solving for P gives the formula.
P/100 × Y + (1 − P/100) × X = Z
Solving for P gives the formula.
P/100 × Y + (1 − P/100) × X = Z
PY/100 + X − PX/100 = Z
P(Y − X)/100 = Z − X
P = (Z − X) / (Y − X) × 100
P = (X − Z) / (X − Y) × 100 ✔ [multiply numerator & denominator by −1]
PY/100 + X − PX/100 = Z
P(Y − X)/100 = Z − X
P = (Z − X) / (Y − X) × 100
P = (X − Z) / (X − Y) × 100 ✔ [multiply numerator & denominator by −1]
Example Application
| Given | Value |
|---|---|
| X = FM of coarse | 6.8 |
| Y = FM of fine | 2.6 |
| Z = FM of combined | 4.7 |
| P = ((6.8−4.7)/(6.8−2.6)) × 100 | = (2.1/4.2) × 100 = 50% fine aggregate |
- Formula: P = [(X − Z) / (X − Y)] × 100
- Where X = FM coarse, Y = FM fine, Z = FM of desired combined aggregate.
- The formula comes from the weighted-average definition of fineness modulus blending.