If p:q = 2:3 and q:r = 4:5, find p:r.

Ratio is the comparison of two quantities by division: a:b = a/b. Proportion is an equation stating two ratios are equal: a:b = c:d. Key concepts: Compounded Ratio, Duplicate Ratio a squared:b squared. Ratio and Proportion questions appear in SSC CGL, IBPS, CAT, and Railways as mixture problems, partnership problems, age problems, and direct/inverse proportion questions.

Properties of Proportion (a:b = c:d): Product of Means = Product of Extremes (ad = bc). Componendo: (a+b):b = (c+d):d. Dividendo: (a-b):b = (c-d):d. Componendo-Dividendo: (a+b):(a-b) = (c+d):(c-d). If a/b = c/d = e/f, then each ratio equals (a+c+e)/(b+d+f). Mean Proportional of a and b equals root(ab). These properties are essential for solving complex proportion problems.

Alligation Rule: When two mixtures of different concentrations are mixed, the ratio of their quantities is inversely proportional to the differences from the mean concentration. Formula: Ratio = (C2 - Mean):(Mean - C1). Example: Mix milk at Rs 15 and Rs 20 per liter to get mixture at Rs 18. Ratio = (20-18):(18-15) = 2:3. This topic appears frequently in IBPS and SSC exams.

Partnership: When partners invest capital for the same time period, profit is divided in ratio of their investments. When invested for different time periods, profit ratio = (Capital1 x Time1):(Capital2 x Time2). Example: A invests Rs 2000 for 6 months and B invests Rs 3000 for 4 months, ratio = 12000:12000 = 1:1. Working partner may get extra salary before profit division.

Direct Proportion: When one quantity increases, the other increases in the same ratio (y = kx). Example: More workers, more earnings. Inverse Proportion: When one quantity increases, the other decreases (xy = constant). Example: More workers, fewer days to complete work. These concepts are foundational for time-work, speed-distance, and pipeline problems in all competitive exams.