Ratio & proportion: Quantitative Aptitude Notes for Nagaland TET (Teacher Eligibility)

Ratio & Proportion – Nagaland TET (Quantitative Aptitude)

Goal:
Understand how to compare quantities, solve problems where quantities are in direct or inverse proportion, and apply the concept to everyday problems.

1. Simple Explanation

Term	Meaning	Example
Ratio	A comparison of two numbers or quantities expressed as a fraction, e.g. 3 : 5 or 3/5	If a class has 18 boys and 12 girls, the ratio boys : girls = 18 : 12 = 3 : 2
Proportion	An equation stating that two ratios are equal, e.g. a:b = c:d	3 : 4 = 9 : 12 (both simplify to 3 : 4)

Key idea:

If a : b = c : d then a / b = c / d.
Cross‑multiplication gives a·d = b·c.

2. Key Points

Concept	Formula / Rule	Typical Question
Direct Proportion	`y = kx` (k is constant)	“If 3 pens cost ₹30, how much will 7 pens cost?”
Inverse Proportion	`x·y = k` (k constant)	“If 5 workers finish a job in 8 days, how many days will 10 workers take?”
Compound Proportion	“If A:B = C:D and B:C = E:F, then A:D = E:F”	Multi‑step mixing problems
Mixed Problem	Combine direct & inverse proportion	“Mix 3 L of A and 2 L of B to get 5 L of solution with 40% concentration.”
Cross‑Multiplication	`a:b = c:d ⇒ ad = bc`	Quick check if two ratios are equal
Unit Rate	`rate = amount / quantity`	“₹120 for 12 kg of rice → cost per kg”
Ratios in Fraction Form	Convert `a : b` to `a/b` for easier calculation	Easier to compare or compute mixed ratios

3. Worked Example (Direct + Inverse Proportion)

Problem:
A machine produces 1200 units in 8 hours.

1. How many units will it produce in 15 hours?
1. If the production rate is increased to 150 units per hour, how many hours will it take to produce 2000 units?

Step‑by‑Step Solution

Find the production rate (direct proportion).
Rate = units ÷ time
[ \text{Rate} = \frac{1200}{8} = 150 \text{ units/hour} ]
Part‑1 – Production in 15 hours
Using direct proportion:
[ \text{Units} = \text{Rate} \times \text{Time} = 150 \times 15 = 2250 \text{ units} ]
Part‑2 – Time needed for 2000 units at 150 units/hour
Here we use inverse proportion (time is inversely proportional to rate).
[ \text{Time} = \frac{\text{Units}}{\text{Rate}} = \frac{2000}{150} = \frac{20}{1.5} \approx 13.33 \text{ hours} ] (≈ 13 h 20 min)

Answer:

2250 units in 15 hours.
About 13 h 20 min to produce 2000 units at 150 units/hour.

4. Quick Revision Summary

Ratio = comparison of two numbers; write as a : b or a/b.
Proportion = equality of two ratios (a:b = c:d).
Use cross‑multiplication (ad = bc) to check or solve proportions.
Direct Proportion: y = kx → increases in x cause proportional increases in y.
Inverse Proportion: x·y = k → increases in x cause inversely proportional changes in y.
Unit Rate = amount ÷ quantity; useful for converting ratios.
For mixed problems, split into direct and inverse parts, then combine.
Keep k (proportional constant) handy:
- Direct: k = y/x
- Inverse: k = x·y

Tips for TET exams:

Read the problem carefully – identify whether it’s direct, inverse, or mixed.
Write down the given ratios and the unknown in a clear equation.
Simplify using cross‑multiplication whenever possible.
Check units – quantities must be in consistent units (e.g., kg, liters, hours).
Round only at the end – keep fractions/decimals exact until the final answer.

Good luck with your preparation!