Quantitative Aptitude – Concepts and Questions

1. Number System

• Even ± Even = Even, Odd ± Odd = Even, Even × Any = Even, Odd × Odd = Odd
• LCM × HCF = Product of the two numbers
• Perfect square check: The unit digit of a perfect square can only be 0, 1, 4, 5, 6, or 9

2. Percentage

• Percentage = (Value / Total) × 100
• X% of Y = $\frac{X \times Y}{100}$
• Increase or decrease: New Value = Original × $(1 \pm \frac{r}{100})$, where r is the percentage change
• Successive percentage change: $a + b + \frac{ab}{100}$

3. Profit and Loss

• Profit = SP − CP; Loss = CP − SP
• Profit % = $\frac{\text{Profit}}{\text{CP}} \times 100$; Loss % = $\frac{\text{Loss}}{\text{CP}} \times 100$
• Selling Price (SP): $\frac{(100 \pm \text{Profit or Loss \%}) \times CP}{100}$

4. Simple and Compound Interest

• Simple Interest (SI): $\frac{P \times R \times T}{100}$, where P = Principal, R = Rate, T = Time in years
• Compound Interest (CI): $P \times \left(1 + \frac{R}{100}\right)^n – P$
• Total Amount: $P \times \left(1 + \frac{R}{100}\right)^n$

5. Ratio and Proportion

• Ratio: a : b represents a divided by b
• Proportion: $a : b = c : d$ implies $ad = bc$
• Direct proportion: When one increases, the other increases; x/y = constant
• Inverse proportion: When one increases, the other decreases; x × y = constant

6. Average

• Average = Total Sum / Number of items
• Average speed (equal distances): $\frac{2xy}{x + y}$, where x and y are speeds in different legs of the journey

7. Time, Speed and Distance

• Speed = Distance / Time
• Time = Distance / Speed
• Distance = Speed × Time
• Relative speed:
  • Same direction: Relative Speed = A − B
  • Opposite direction: Relative Speed = A + B

8. Time and Work

• Work = Efficiency × Time
• 1 day’s work = $1/x$ — If a person completes the task in x days, then in one day, they complete $1/x$ of the work.
• Combined work: If A can do it in x days and B in y days, combined 1-day work = $\frac{1}{x} + \frac{1}{y}$
• Efficiency ratio: If A is n times as efficient as B, then A : B = n : 1, and time taken is inversely proportional

9. Algebra

• $(a + b)^2 = a^2 + 2ab + b^2$
• $(a – b)^2 = a^2 – 2ab + b^2$
• $(a + b)(a – b) = a^2 – b^2$
• $a^3 + b^3 = (a + b)(a^2 – ab + b^2)$
• $a^3 – b^3 = (a – b)(a^2 + ab + b^2)$
• Quadratic roots: $x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}$

10. Mensuration

2D Figures:

• Square: Area = $a^2$
• Rectangle: Area = $l \times b$
• Triangle: Area = $\frac{1}{2} \times b \times h$
• Circle: Area = $\pi r^2$, Circumference = $2\pi r$

3D Figures:

• Cube: Volume = $a^3$, Surface Area = $6a^2$
• Cuboid: Volume = $l \times b \times h$
• Cylinder: Volume = $\pi r^2 h$
• Cone: Volume = $\frac{1}{3}\pi r^2 h$
• Sphere: Volume = $\frac{4}{3}\pi r^3$

11. Permutation and Combination

• Permutation: $nP_r = \frac{n!}{(n – r)!}$ — arrangements matter
• Combination: $nC_r = \frac{n!}{r!(n – r)!}$ — selection only
• Factorial: $n! = n \times (n−1) \times (n−2) \dots \times 1$

12. Probability

• Probability = $\frac{\text{Favorable Outcomes}}{\text{Total Outcomes}}$
• Probability value lies between 0 and 1: $0 \leq P \leq 1$

13. Mixtures and Alligation

• Alligation Rule: $\frac{Quantity}{Mean} = \frac{Higher – Mean}{Mean – Lower}$
• Final concentration: $\frac{\text{Old Quantity} \pm \text{Added}}{\text{Total}}$

14. Boats and Streams

• Downstream speed = Boat speed + Stream speed
• Upstream speed = Boat speed − Stream speed
• Speed in still water = $\frac{\text{Downstream} + \text{Upstream}}{2}$
• Speed of stream = $\frac{\text{Downstream} – \text{Upstream}}{2}$

15. Pipes and Cisterns

• Work done by a pipe in 1 hour = $1/x$, if it can fill/empty in x hours
• Combined flow: $\frac{1}{x} \pm \frac{1}{y}$ depending on filling (+) or emptying (−)