Ratio & proportion: Quantitative Aptitude Notes for Sikkim PSC

Ratio & Proportion – Sikkim PSC Quantitative Aptitude

Quick note – Ratio & Proportion is a core topic in Sikkim PSC. It is used to solve problems about parts of a whole, rate problems, age problems, mixing problems, and distance/time type questions. Mastering the basic rules and formulas will help you score high.

1. Simple Explanation

Concept	What it means	Formula / Rule
Ratio	Comparison of two (or more) numbers.	`a : b` or `a/b`
Proportion	Two ratios are equal.	`a : b = c : d ⇔ a/b = c/d`
Direct Proportion	If one variable increases, another increases in the same proportion.	`y = kx`
Inverse Proportion	If one variable increases, another decreases in the same proportion.	`y = k/x`
Compound Ratio	More than two numbers in a ratio.	`a : b : c`
Part of a Whole	Find a part when the ratio of parts to the whole is known.	`Part = (ratio / sum of ratios) × Total`
Cross‑Multiplication	Method to solve proportions.	`a × d = b × c`

2. Key Points (Bullet‑List)

Always write ratios in simplest form (e.g., 6 : 8 → 3 : 4).
If given a proportion a : b = c : d, you can:
- Solve for an unknown by cross‑multiplication: a × d = b × c.
- Find a missing part of a whole: c = (b/(a+b)) × Total.
Compound ratio – sum the individual parts to get the total parts.
- Example: 2 : 3 : 5 → total parts = 2+3+5 = 10.
Part of a whole:
- If the parts are x : y : z and the whole is T, then:
  - Part_x = (x/(x+y+z)) × T
  - Part_y = (y/(x+y+z)) × T
  - Part_z = (z/(x+y+z)) × T
Rate problems – Treat rate as a ratio of two quantities.
- For example, “5 apples cost ₹25” → rate = ₹5 per apple → 12 apples = 12×5 = ₹60.
Age problems – If age ratio is p : q and one age is known, solve:
- Age_other = (q/p) × Age_known.
Proportion in a line – If a line is divided in a ratio, find the length of a segment by applying the ratio to the whole length.
Mixing / combining – When mixing two substances, use ratios of amounts to find the resulting mixture.

3. Worked Example

Problem (Sikkim PSC style)

Two numbers are in the ratio 3 : 5.
The sum of the numbers is 64.
Find the two numbers.

Step‑by‑Step Solution

Express the numbers in terms of a common variable
Let the two numbers be 3x and 5x (because the ratio is 3:5).
Use the given sum
3x + 5x = 64
→ 8x = 64
Solve for x
x = 64 ÷ 8 = 8
Find the numbers
- First number = 3x = 3 × 8 = 24
- Second number = 5x = 5 × 8 = 40
Check
- Sum = 24 + 40 = 64 ✔️
- Ratio = 24 : 40 = 3 : 5 ✔️

Answer: The numbers are 24 and 40.

4. Quick Revision Summary

Topic	What to remember
Ratio	`a : b` or `a/b`; simplify first
Proportion	`a/b = c/d`; cross‑multiply `a×d = b×c`
Compound ratio	Sum all parts to get total parts
Part of a whole	`Part = (ratio / sum of ratios) × Total`
Rate problems	Treat rate as `value/quantity`; apply directly
Age & distance	Use ratio `age1 : age2` → `age2 = (ratio2/ratio1) × age1`
Mixing	Combine proportions of each component

Tip: Always write a clear equation first and keep track of units (₹, km, years, etc.) – this helps avoid mistakes in the final answer.

Happy studying! 🚀