RATIO AND PROPORTION

 

📌 What is Ratio?

• A ratio compares two quantities of the same type (like two lengths, two weights).
• It is written as a:b or a/b.
• It shows how many times one value contains another.

Example:
If there are 2 apples and 3 oranges, the ratio of apples to oranges is:
2:3

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✅ Properties of Ratios

• Ratios have no units.
• Ratios must compare quantities in the same unit.
• If a:b = c:d, then the four numbers are in proportion.

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📌 What is Proportion?

• Proportion means two ratios are equal.
  Written as a:b = c:d or a/b = c/d.
• In a:b = c:d:
• a and d are called extremes
• b and c are called means

Example:
2:3 = 4:6 is a valid proportion because:
2 × 6 = 3 × 4 → 12 = 12

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🔁 Types of Proportion

1. Direct Proportion:
• When one increases, the other increases.
• Example: More hours worked = More salary
• Formula: a1/b1 = a2/b2
2. Inverse Proportion:
• When one increases, the other decreases.
• Example: More speed = Less time taken
• Formula: a1 × b1 = a2 × b2

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🧮 Example Problems

1. Find the ratio of 20 and 30:
→ 20:30 = 2:3

2. If 5:9 = x:27, find x:
→ 5/9 = x/27
→ x = (5×27)/9 = 15

3. If A:B = 2:3 and B:C = 4:5, find A:B:C:
→ A:B = 2:3, B:C = 4:5
→ LCM of B = 12 → A:B:C = 8:12:15

4. Divide 1200 in ratio 2:3:
→ Total parts = 5
→ First part = 1200×2/5 = 480, Second = 720

5. If x:y = 4:5 and y:z = 10:7, find x:y:z:
→ x:y = 4:5, y:z = 10:7
→ y = 10 → x = 8, z = 7
→ x:y:z = 8:10:7

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📝 30 Practice Questions

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🟢 Easy Level (1–10)

1. Find the ratio of 6 and 18.
2. Simplify the ratio 20:60.
3. Divide 100 in the ratio 1:4.
4. If A:B = 2:5, find B when A = 6.
5. What is 3:9 in simplest form?
6. 16:32 is equal to which of the following? (a) 1:2 (b) 2:1 (c) 4:5
7. If x:y = 1:2 and y = 10, find x.
8. Is 7:0 a valid ratio?
9. If a:b = 4:7 and b = 14, find a.
10. Which is greater: 2:3 or 3:4?

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🟡 Intermediate Level (11–20)

11. If x:y = 5:7 and y:z = 14:9, find x:y:z.
12. Find the fourth term in 3:4 = 6:x.
13. If x:40 = 7:8, find x.
14. Divide 1000 in the ratio 3:5:2.
15. A and B earn in the ratio 4:5. If A earns ₹8000, how much does B earn?
16. A bag contains red and blue balls in the ratio 3:2. If total is 50, how many are red?
17. x:y = 3:4 and y:z = 8:5, find x:z.
18. Find x: If x/12 = 2/3.
19. In what ratio must ₹1200 be divided between A and B so that A gets ₹300 more than B?
20. If x:y = 7:9 and y = 36, find x and y.

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🔴 Difficult Level (21–30)

21. If A:B = 2:3, B:C = 4:5, and C:D = 6:7, find A:B:C:D.
22. A mixture has milk and water in 5:3. If 16 liters of water is added, ratio becomes 5:7. Find original quantity.
23. If (x+3):(x−3) = 7:5, find x.
24. ₹2400 is divided among A, B, C in the ratio 3:4:5. Find C’s share.
25. A map has scale 1:25000. If map distance is 3 cm, what is actual distance?
26. Two numbers are in ratio 5:7. Their LCM is 420. Find the numbers.
27. In a college, boys:girls = 3:4. Total students = 560. Find boys.
28. A:B = 4:5, B:C = 6:7. Find A:B:C.
29. ₹7200 is divided among A, B, C such that A:B = 2:3 and B:C = 4:5. Find each share.
30. x:y = 3:4, y:z = 2:5, z:w = 6:7. Find x:y:z:w.