Ratio & Proportion – Gujarat TET (Quantitative Aptitude)
Word of the Day – Ratio: A comparison of two quantities expressed as “a : b” or “a/b”.
Proportion: An equation stating that two ratios are equal.
1️⃣ Simple Explanation
| Term | Meaning | Symbol | Example |
|---|---|---|---|
| Ratio | Comparison of two numbers | a : b or a/b |
3 : 4 or 3/4 |
| Proportion | Statement that two ratios are equal | a : b = c : d |
3 : 4 = 6 : 8 |
Why it matters in TET
- Appears in questions about money, time, speed, density, cost‑benefit, and mixed‑rate problems.
- Requires quick mental math, cross‑multiplication, and understanding of “direct” vs “inverse” proportionality.
2️⃣ Key Points
| Concept | What to remember |
|---|---|
| Direct Proportion | a/b = c/d → a·d = b·c. If a increases, c increases proportionally. |
| Inverse Proportion | a·b = c·d. If a increases, b decreases. |
| Cross‑Multiplication | Handy for solving proportions: a:b = c:d → a·d = b·c. |
| Ratio in Decimal | Convert to fraction for easier cross‑multiplication. |
| Unit Conversion | Keep units consistent before comparing ratios. |
| Proportionality Constant (k) | For y = kx → k = y/x. |
| Scaling | If a ratio is a:b, multiplying both by any number n gives na : nb. |
| Simplification | Reduce ratios to lowest terms by dividing by GCD. |
3️⃣ Worked Example
Problem
10 books cost ₹30.
How much would 15 books cost, if the price per book remains the same?
Step‑by‑Step
-
Identify the ratio
Cost per book = ₹30 ÷ 10 = ₹3 per book.
Ratio of cost to books:₹30 : 10→ simplifies to3 : 1. -
Set up proportion
Letxbe the cost for 15 books.
30 : 10 = x : 15(direct proportion). -
Cross‑multiply
30 × 15 = 10 × x
450 = 10x -
Solve for x
x = 450 ÷ 10 = ₹45.
Answer
₹45 for 15 books.
Quick check: ₹3 per book × 15 = ₹45 ✔️
4️⃣ Quick Revision Summary
| Topic | Quick Tips |
|---|---|
| Direct Proportion | Use a·d = b·c. |
| Inverse Proportion | Use a·b = c·d. |
| Cross‑Multiplication | Fastest way to solve a:b = c:d. |
| Simplify Ratios | Divide by GCD before cross‑multiplying. |
| Units | Keep all units identical (e.g., convert kgs to grams). |
| Scaling | Multiply both sides of a ratio by same number keeps proportion. |
| Common Mistake | Forgetting to keep the ratio’s direction when solving inverse proportions. |
📌 Final Tips for Gujarat TET
- Read the question carefully – determine if it's direct or inverse proportion.
- Write the proportion explicitly before manipulating.
- Check with a quick mental calculation to avoid arithmetic errors.
- Practice mixed‑rate problems (e.g., speed–time, density) to reinforce concepts.
Good luck with your preparation! 🚀