Ratio, Proportion and Simple Equations: Class 6 Study Guide
Ratios compare quantities. Algebra uses letters to stand for numbers we do not know yet.
A ratio compares two quantities of the same kind by division. If a class has 12 boys and 18 girls, the ratio of boys to girls is 12 : 18, which simplifies to 2 : 3 by dividing both by 6. Two ratios that are equal form a proportion: 2 : 3 = 4 : 6.
In algebra, a letter such as x stands for an unknown number. An expression like 3x + 2 is a phrase, and an equation like 3x + 2 = 11 says two sides are equal. To solve it, do the same thing to both sides until x stands alone.
Working with ratios
Simplify a ratio by dividing both terms by their highest common factor. The order matters: 2 : 3 is not the same as 3 : 2. To share an amount in a ratio, add the parts and find one part. Share ₹600 between Asha and Ravi in the ratio 2 : 3: total parts = 5, one part = ₹120, so Asha gets ₹240 and Ravi gets ₹360.
| Ratio | Simplified | Meaning |
|---|---|---|
| 10 : 25 | 2 : 5 | 2 parts to 5 parts |
| 45 : 15 | 3 : 1 | first is 3 times the second |
| 1.5 : 3 | 1 : 2 | multiply by 2, then simplify |
Proportion
If 5 notebooks cost ₹150, how much do 8 cost? Find the cost of one: 150 ÷ 5 = ₹30, so 8 cost 8 × 30 = ₹240. This is called the unitary method. A map scale of 1 : 50 000 means 1 cm on the map is 50 000 cm (500 m) in real life.
Divide.
Scale up.
Rupees, cm, km.
More items cost more.
Simple equations
To solve x + 7 = 15, subtract 7 from both sides: x = 8. To solve 3x = 21, divide both sides by 3: x = 7. To solve 2x + 5 = 19: subtract 5 (2x = 14), then divide by 2 (x = 7). Check by substituting back: 2 × 7 + 5 = 19. Equations can model situations: 'I think of a number, double it and add 3 to get 17' is 2x + 3 = 17, so x = 7.
Worked examples and common mistakes
Example: The ratio of red to blue marbles is 3 : 5. If there are 24 red, blue = 24 ÷ 3 × 5 = 40. Common mistakes: writing a ratio in the wrong order, not simplifying and doing a different operation on each side of an equation. Keep the equation balanced like a pair of scales.
Check your understanding: model answers
1. 2 : 3. 2. ₹15 each, so 7 cost ₹105. 3. 3x = 15, so x = 5.
✨ Key Takeaways
- A ratio compares two quantities.
- Simplify by dividing by the HCF.
- The unitary method: find one, then scale.
- Do the same to both sides of an equation.
💬 Questions for Discussion & Reflection
- Simplify 24 : 36.
- If 4 pens cost ₹60, what do 7 cost?
- Solve 3x − 4 = 11.