1. If A : B = 3 : 2 and B : C = 3 : 5, then what is the ratio A : B : C?

2. A straight line parallel to side BC of triangle ∆ABC intersects sides AB and AC at points D and E respectively. If AD : BD = 3 : 5, then what is the ratio of the area of triangle ∆ADE to the area of trapezium DBCE?

3. If A : B = 2 : 3, B : C = 5 : 8, and C : D = 6 : 7, then what is the ratio A : D?

(a) 2:7 (b) 7:2 (c) 5:8 (d) 5:14

4. In \(\triangle\)ABC, if DE \(\parallel\) BC and AD:DB = 3:2, then what is the ratio of DE:BC?

5. A, B, and C made a joint business profit of ₹1,500. If A's capital : B's capital = 2 : 3 and B's capital : C's capital = 2 : 5, then C's share of the profit is...

(a) ₹ 900 (b) ₹ 1000 (c) ₹ 1100 (d) ₹ 1200

6. If \((3x - 2y) : (x + 3y) = 5 : 6\), then what is the value of the ratio \(x : y\)?

7. If \(a : b = 2 : 3\) and \(b : c = 4 : 5\), then what is the value of \(a^2 : b^2 : bc\)?

(a) 8:18:21 (b) 16:36:45 (c) 16:20:36 (d) 8:15:18

8. If \(a : b = 2 : 1\) and \(x : y = 3 : 4\), then what is the value of \((2ax - by) : (2by - ax)\)?

(a) 2:1 (b) 3:1 (c) 4:1 (d) 5:1

9. In a business, if the capital ratio of A, B, and C is 2:3:5 and their time of investment ratio is 3:4:5, then in what ratio will the profit be distributed?

(a) 3:4:5 (b) 5:8:9 (c) 8:9:11 (d) 6:12:15

10. In triangle ABC, if line DE is parallel to side BC and the ratio AD:DB = 3:2, What is the ratio of DE:BC?

11. If A : B = 2 : 3, B : C = 4 : 5, and C : D = 6 : 7, then what will be the value of D : A?

(a) 16 : 35 (b) 35 : 16 (c) 2 : 3 (d) 4 : 5

12. If \(a : b = 2 : 3\) and \(b : c = 2 : 3\), then what is \((a + b) : (b + c)\)?

13. If the similar triangles \(\triangle\)ABC and \(\triangle\)PQR have AB:PQ = 3:5, then the area ratio of \(\triangle\)ABC to \(\triangle\)PQR is—

(a) 9.25 (b) 25.9 (c) 3.5 (d) 5.3

14. If \(a:b = 3:2\) and \(b:c = 3:2\), then what is \(a+b : b+c\)?

15. Here’s the English translation: *If the mean of a statistical distribution is 4.1, \(∑f_i.x_i = 132 + 5k\), and \(∑f_i = 20\), then what is the value of \(k\)?* Would you like help solving it too? I’d be glad to walk through it with you.

16. In a right-angled triangle, if the ratio of the perpendicular (opposite side) to the hypotenuse with respect to a positive acute angle \(\theta\) is \(12 : 13\), then determine the ratio of the perpendicular to the base and the ratio of the hypotenuse to the base, and verify that \( \sec^2\theta = 1 + \tan^2\theta \).

17. “If \(\cfrac{a}{b} = \cfrac{3}{2}\) and \(\cfrac{b}{c} = \cfrac{3}{2}\), then what is \((a + b) : (b + c)\)?”

18. If 2a=3b=4c, then what is 2a:5b:7c ?

(a) 2:3:4 (b) 4:15:28 (c) 12:20:21 (d) None of these

19. A straight line parallel to side BC of \(\triangle\)ABC intersects AB and AC at points P and Q, respectively. If AQ = 2AP, then what is the ratio PB:QC?

(a) 1:2 (b) 2:1 (c) 1:1 (d) None of these

20. In trapezium ABCD, AD is parallel to BC. A straight line parallel to BC intersects AB and DC at points P and Q respectively. If \( AP : PB = 2 : 1 \), then what is the ratio \( DQ : QC \)?

(a) 1:1 (b) 1:2 (c) 1:4 (d) 2:1

21. In triangle \( \triangle ABC \), a line parallel to side BC intersects sides AB and AC at points D and E respectively. If \( AB = 20 \) cm and \( BD = 14 \) cm, then what is the ratio \( DE : BC \)?

(a) 7:10 (b) 5:17 (c) 3:10 (d) 7:17

22. In triangle ABC, \(\angle\)BAC = 90° and AD is perpendicular to BC. Given: AB : AC = 3 : 4 Find: What is the ratio BD : DC?

(a) 3:4 (b) 9:16 (c) 2:3 (d) None of the above

23. If \( (3x - 2y) : (3x + 2y) = 4 : 5 \), then what is the value of \(x : y\)?

(a) 1:6 (b) 1:1 (c) 2:1 (d) 6:1

24. In triangle ABC, \(\angle C = 90^\circ\) and AC : BC = 3 : 4, then what is the value of cosec A?

(a) \(\cfrac{3}{4}\) (b) \(\cfrac{5}{3}\) (c) \(\cfrac{5}{4}\) (d) \(\cfrac{3}{5}\)

25. If \(\cfrac{x-3y}{2y}=\cfrac{6x-5y}{5x}\), then what is the value of \(x:y\)?

(a) 5:1 or 2:5 (b) 1:5 or 2:5 (c) 1:5 or 5:2 (d) None of the above

26. If the length, breadth, and height of a right rectangular prism are in the ratio 3:2:1 and its total surface area is 88 square centimeters, then what is its volume?

(a) 120 cubic cm (b) 64 cubic cm (c) 48 cubic cm (d) 24 cubic cm

27. If the volumes of two cones are in the ratio 2:3 and the ratio of their base radii is 1:2, then what is the ratio of their heights?

(a) 3:8 (b) 8:3 (c) 3:4 (d) 4:3

28. If the ratio of A to B is 2:3 and the ratio of B to C is 4:3, then what is the ratio A:B:C?

(a) 8:12:15 (b) 6:9:8 (c) 8:12:9 (d) 8:16:9

29. If \(x^2 + y^2 - 4x - 6y + 13 = 0\), then what is the value of \((x + y) : (y - x)\)?

30. The base length and width of a rectangular reservoir are 15 meters and 12 meters respectively. Water is filled into the reservoir from a nearby pond using a pump. If the pump can fill 36,000 liters of water per hour, then how long will it take for the pump to fill the reservoir up to a height of 7.2 decimeters? [Note: 1 liter = 1 cubic decimeter]