Ratio & Proportion – Study Notes
(For Himachal Pradesh TET – Quantitative Aptitude)
1. What is a Ratio?
A ratio tells you how two numbers compare to each other.
- Example (Pizza):
If you have 3 slices and your friend has 5 slices, the ratio of your slices to your friend’s slices is 3 : 5.
It simply means “for every 3 of yours, there are 5 of theirs.”
Key points
| Point | Short explanation |
|---|---|
| Notation | Use a colon (:) or “÷” – e.g., 3 : 5 or 3 ÷ 5 |
| Same units | Both parts must be of the same kind (slices, coins, etc.) |
| Simplify | Reduce to smallest whole numbers – 6 : 10 = 3 : 5 |
2. What is a Proportion?
A proportion is when two ratios are equal.
- Example (Sharing Cookies):
You have 4 cookies, your brother 6 cookies.
Your sister has 8 cookies, your cousin 12 cookies.
The ratio of your to brother’s cookies (4 : 6) is the same as the ratio of sister’s to cousin’s cookies (8 : 12).
→ 4 : 6 = 8 : 12 – a proportion.
3. Types of Ratios
| Type | Everyday analogy | Quick example |
|---|---|---|
| Simple ratio | Two numbers compared | 2 : 3 |
| Compound ratio | Two or more numbers, like a story with three parts | 4 : 5 : 6 |
| Equivalent ratio | Same relationship in different numbers | 6 : 9 is the same as 2 : 3 |
4. Proportionality (Direct & Inverse)
| Kind | How it works | Quick visual |
|---|---|---|
| Direct proportion | If one number goes up, the other goes up in the same way | If distance doubles, time doubles |
| Inverse proportion | If one number goes up, the other goes down | If speed doubles, time halves |
5. Solving a Proportion – The “Cross‑Multiplication” Rule
If a : b = c : d, then a × d = b × c.
- Write the two ratios.
- Multiply the outer terms (a × d).
- Multiply the inner terms (b × c).
- If they are equal, the proportions are correct.
6. Quick Steps to Solve Ratio Problems
- Read the problem.
- Identify the two ratios (or the one ratio and a missing part).
- Convert to a proportion if needed.
- Use cross‑multiplication to find the unknown.
- Check by plugging back into the ratio.
7. Tiny Examples
7.1 Direct Proportion
Problem: A recipe uses 2 cups of flour for 4 cups of sugar. How many cups of flour are needed for 10 cups of sugar?
- Ratio of flour to sugar = 2 : 4 = 1 : 2.
- For 10 cups sugar, flour = 10 × 1 ÷ 2 = 5 cups.
7.2 Inverse Proportion
Problem: 4 workers finish a task in 5 days. How many days will 6 workers need?
- Work ∝ 1 ÷ (workers).
- 5 days × 4 workers = 20 worker‑days (constant).
- 20 worker‑days ÷ 6 workers = 3.33 days (≈3 days 8 hrs).
7.3 Compounding Ratio
Problem: A mixture contains 3 parts water, 5 parts oil, and 2 parts vinegar. If the total mixture is 200 ml, how much oil is there?
- Total parts = 3 + 5 + 2 = 10 parts.
- One part = 200 ml ÷ 10 = 20 ml.
- Oil = 5 parts × 20 ml = 100 ml.
8. Common Question Types in TET
| Question | What to look for | Quick tip |
|---|---|---|
| Find an unknown ratio | One part missing | Set up a proportion, solve by cross‑multiply |
| Convert a ratio to a proportion | Express as “=” | e.g., 4 : 5 → 4 ÷ 5 = ? |
| Direct or inverse relation | “If X increases, Y does what?” | Use direct if both go same direction, inverse if opposite |
| Mixture problems | Total parts vs. total quantity | Find “parts per unit” and multiply |
9. Quick Tips & Tricks
- Keep units the same – if one part is in grams, the other must also be in grams.
- Simplify early – reducing a ratio makes cross‑multiplication easier.
- Check with a quick test – if the answer feels huge or tiny, double‑check the proportion.
- Remember the “outer = inner” rule – handy for quick mental checks.
- Practice both direct and inverse – they often appear in the same set of questions.