Ratio & Proportion – Study Notes
(Madhya Pradesh TET – Quantitative Aptitude)
1️⃣ Simple Explanation
- Ratio – A comparison of two quantities by division.
a : b = a ÷ b - Proportion – An equation that states two ratios are equal.
a : b = c : d(read: a is to b as c is to d)
Key Idea:
If two ratios are equal, the cross‑multiplication of their terms gives the same product.
2️⃣ Key Points & Formulae
| Concept | Notation | Formula | Example |
|---|---|---|---|
| Direct Proportion | y ∝ x |
y = kx (k = constant) |
Speed ∝ Time |
| Inverse Proportion | y ∝ 1/x |
y = k/x |
Power ∝ 1/Size |
| Cross‑Multiplication | a:b = c:d |
a·d = b·c |
2:3 = 4:6 → 2·6 = 3·4 |
| Finding Missing Term | a:b = c: ? |
? = (b·c)/a |
3:5 = 15:? → ? = (5×15)/3 = 25 |
| Compound Ratio | a:b = c:d = e:f |
a·d·f = b·c·e |
2:3 = 4:6 = 8:12 → 2·6·12 = 3·4·8 |
| Proportion with Percentage Change | a : b = (100‑p)% : 100% |
b = a·100/(100‑p) |
20% decrease → b = a×100/80 |
| Mixture Problem | A₁·C₁ + A₂·C₂ + … = Total × Final Cost |
Solve for unknown mixture amount | See worked example below |
3️⃣ Worked Example
Problem
A 5 kg bag of rice costs ₹200.
If the price per kg is increased by 20%,
what will be the new price of the same 5‑kg bag?
Step‑by‑Step Solution
-
Identify the ratio of old price to new price.
Old price per kg = ₹200 ÷ 5 = ₹40
New price per kg = ₹40 × (1 + 20/100) = ₹48Ratio of old to new = ₹40 : ₹48
Simplify → 40:48 = 5:6 -
Use the ratio to find the new total cost.
Since 5 kg were bought before, the new total cost is: [ \text{New total} = \frac{6}{5} \times \text{Old total} ]
Old total = ₹200
New total = (\frac{6}{5} × 200 = 240) -
Answer
The new price of the 5‑kg bag is ₹240.
Cross‑Multiplication Check
[ \frac{5}{6} = \frac{\text{Old total}}{\text{New total}} ]
[ 5 × \text{New total} = 6 × 200 ;\Rightarrow; \text{New total} = 240 ]
✔️ ✅
4️⃣ Quick Revision Summary
| Topic | Formula / Rule | Quick Tip |
|---|---|---|
| Direct Proportion | (y = kx) | Multiply by the same constant. |
| Inverse Proportion | (y = k/x) | Multiply by the reciprocal. |
| Cross‑Multiply | (a·d = b·c) | Check proportion equality. |
| Find Missing Term | (? = (b·c)/a) | Use cross‑multiplication in parts. |
| Compound Ratio | (a·d·f = b·c·e) | Works for 3 or more ratios. |
| Percentage Change | (b = a·100/(100‑p)) | For decrease; use (1 + p/100) for increase. |
| Mixture | (A₁C₁ + A₂C₂ = T·F) | Set up cost equation. |
Mnemonic for Ratio & Proportion
“Cross‑Multiply, Divide, and Remember: if the products are equal, the ratios are in balance.”
Final Tips for TET Exam
- Always simplify ratios before setting up equations.
- Check units: In direct proportion problems, the constant (k) has the same units as the product of the two variables.
- Quick Cross‑Check: After solving, cross‑multiply the original ratios to verify the answer.
- Time Management: Ratio problems are usually quick; focus on the key quantity and use the simplest method (cross‑multiplication or substitution).
Good luck with your preparation!