Ratio & proportion: Quantitative Aptitude Notes for Tamil Nadu PSC

Ratio & Proportion – Study Notes

Tamil Nadu PSC – Quantitative Aptitude

1. Simple Explanation

Ratio – A comparison of two or more quantities showing how many times one value contains another.
- Written as a : b or a/b.
Proportion – An equation that states two ratios are equal.
- Written as a : b = c : d or a/b = c/d.
In a proportion, the product of the extremes (a·d) equals the product of the means (b·c).

Why it matters for PSC
Many questions ask you to find an unknown part of a whole, compare portions, or maintain a certain ratio while distributing resources.

2. Key Points

#	Concept	Formula / Rule	Quick Example
1	Basic Ratio	`a : b`	3 : 4
2	Equivalent Ratio	Multiply/divide both terms by same number	3 : 4 = 6 : 8
3	Proportion	`a : b = c : d` → `a/b = c/d`	2 : 5 = 4 : ?
4	Cross‑Multiplication	`a·d = b·c`	2·? = 5·4 → ? = 10
5	Finding a Missing Term	Use cross‑multiplication or proportion formula	`x` in `3 : 5 = x : 10` → `x = (3·10)/5 = 6`
6	Scaling	If `a : b = c : d`, then `ka : kb = kc : kd`	2 : 3 = 4 : 6 (k=2)
7	Sum of Parts	If parts `x` and `y` are in ratio `m:n` and `x + y = P`, then: `x = P·m/(m+n)`, `y = P·n/(m+n)`	10 : 15 =? with total 50 → `x = 50·10/25 = 20`, `y = 30`
8	Common Ratio (Geometric)	When a sequence is in geometric progression: `b/a = c/b = r`	2, 6, 18 → r = 3
9	Rate Problems	`Rate = (Quantity/Time)`; Convert to same units before comparing	5 km in 10 min → 30 km/h
10	Percentage from Ratio	`ratio part / whole × 100`	3 : 7 of 200 → `(3/10)·200 = 60`

2.1 Common Mistakes to Avoid

Unit mismatch – Always convert units before cross‑multiplication.
Wrong orientation – In a : b = c : d, a ↔ c, b ↔ d.
Neglecting ‘sum of parts’ formula – Use when total is given.

3. Worked Example

Problem (PSC‑style):

The total expenditure of a shopkeeper on raw materials and labor is ₹ 24,000.
The ratio of expenditure on raw materials to labor is 5 : 3.
Find the amount spent on each.

Step‑by‑Step

Identify parts and ratio
- Let R = raw materials, L = labor.
- Ratio R : L = 5 : 3.
Use sum‑of‑parts formula
- Total T = R + L = ₹ 24,000.
- R = T · 5/(5+3) = 24,000 · 5/8 = 15,000.
- L = T · 3/(5+3) = 24,000 · 3/8 = 9,000.
Check
- 15,000 + 9,000 = 24,000 ✔️
- Ratio 15,000 : 9,000 = 5 : 3 ✔️

Answer:

Raw materials: ₹ 15,000
Labor: ₹ 9,000

3.1 Alternative Approach (Cross‑Multiplication)

Assume R = 5x, L = 3x.
5x + 3x = 24,000 → 8x = 24,000 → x = 3,000.
R = 5x = 15,000; L = 3x = 9,000.
Same result.

4. Quick Revision Summary

Ratio = “part : part” → a : b.
Proportion = “two ratios equal” → a : b = c : d.
Cross‑multiplication: a·d = b·c.
Missing term: x = (product of known terms)/(other known term).
Sum of parts: x = Total·m/(m+n) when x : y = m : n.
Scale up/down: multiply all terms by same factor.
Always check units before comparing or adding.
Practice: Convert all given quantities to a common unit, set up a proportion, solve by cross‑multiplying or using the sum‑of‑parts formula.

Quick Practice Questions (for self‑test)

A factory produces 240 units in 8 hours. How many units will it produce in 12 hours at the same rate?
A mixture is