Quantitative Aptitude

Percentages

Ratio, proportion, averages, mixtures

• Combine ratios by making the common term equal ($A:B = 2:3$, $B:C = 4:5$ → $8:12:15$).
• Average $= \text{sum}/\text{count}$; work with sums when items are added or removed.
• Alligation: mixing quantities with values $a$ and $b$ to get mean $m$ needs the ratio $(b - m) : (m - a)$.

Time, work, speed

• Work: rates add. If A takes $a$ days and B takes $b$, together $\tfrac{ab}{a + b}$ days. A pipe that empties subtracts its rate.
• Speed: $\text{km/h} \times \tfrac5{18} = \text{m/s}$. Average speed for equal distances at $u$ and $v$ is the harmonic mean $\tfrac{2uv}{u + v}$, not $\tfrac{u + v}{2}$.
• Trains: crossing a pole covers the train's length; crossing a platform covers length + platform. Relative speed: add for opposite directions, subtract for the same direction.

Interest

$$\text{SI} = \frac{PRT}{100}, \qquad A_{CI} = P\left(1 + \frac{R}{100}\right)^T, \qquad \text{CI} - \text{SI}\ (2\text{ yrs}) = P\left(\frac{R}{100}\right)^2.$$

Powers, logarithms, series

• $a^m a^n = a^{m+n}$, $(a^m)^n = a^{mn}$, $a^{-n} = 1/a^n$, $a^{1/n} = \sqrt[n]{a}$.
• $\log(xy) = \log x + \log y$, $\log x^k = k\log x$, $\log_a b = \tfrac{\log b}{\log a}$, $\log_a a = 1$.
• AP: $n$-th term $a + (n-1)d$, sum $\tfrac n2(2a + (n-1)d)$; GP: $ar^{n-1}$, sum $a\tfrac{r^n - 1}{r - 1}$; $\sum_{k=1}^n k = \tfrac{n(n+1)}{2}$, $\sum k^2 = \tfrac{n(n+1)(2n+1)}{6}$.
• Series puzzles: look at differences, ratios, squares/cubes, $n(n+1)$, primes, alternating patterns.

Counting and probability

The counting and probability pages of the main syllabus cover this in depth. In GA expect $\binom nr$, arrangements with repetition, dice and coin problems.

Mensuration and geometry

Scaling lengths by $k$ scales areas by $k^2$ and volumes by $k^3$. Sum of interior angles of an $n$-gon: $(n-2)180°$; diagonals: $\tfrac{n(n-3)}{2}$.

Clocks and calendars

• Angle between hands at $H{:}M$: $|30H - 5.5M|$ (take $360°$ minus it if over $180°$).
• A normal year shifts the weekday by 1, a leap year by 2. Leap years: divisible by 4, except centuries not divisible by 400.

Data interpretation

Read units and bases carefully: "percentage of what?", "lakh vs thousand", "growth over the previous year vs over the base year". Approximate aggressively when options are far apart.

Quick revision