Proportion solving — In the proportion 10 / 11 = 110 / x, find the value of x using cross-multiplication for exact equality of ratios.

Introduction / Context:
Proportions equate two ratios. The standard method for finding a missing term is cross-multiplication, which preserves equality when all terms are non-zero. This is routinely tested in aptitude arithmetic for speed and accuracy.

Given Data / Assumptions:

• Proportion: 10 / 11 = 110 / x.
• All quantities are positive and non-zero.
• We solve for x.

Concept / Approach:
Use cross-multiplication: (10) * (x) = (11) * (110). Then isolate x by dividing both sides by 10. This direct manipulation is both quick and reliable. Finally, verify by substituting x back to confirm equal ratios.

Step-by-Step Solution:
Apply cross-multiplication: 10 * x = 11 * 110.Compute right-hand side: 11 * 110 = 1210.Solve for x: x = 1210 / 10 = 121.Therefore, x = 121.

Verification / Alternative check:
Replace x with 121: 10 / 11 = 0.90909..., and 110 / 121 = 0.90909..., confirming the proportion holds exactly.

Why Other Options Are Wrong:

• 111 / 1100 / 100 / 11: None satisfy the equality 10 / 11 = 110 / x when substituted; the ratios differ slightly or substantially.

Common Pitfalls:
Swapping numerator and denominator during cross-multiplication; dividing the wrong side; arithmetic slips when multiplying 11 by 110.

Final Answer:
121