1. If the ratio of three consecutive angles of a cyclic quadrilateral is 1:2:3, then what are the measures of the first and third angles?

2. If the ratio of three consecutive angles of a cyclic quadrilateral is 1:2:3, find the first and third angles in radians.

3. The capital ratio of three people is 4:8:9, and the difference in profit between the first and third person is ₹200. What is the profit share of the second person?

4. If the three consecutive angles of a cyclic quadrilateral are in the ratio \(1:2:5\), then the measure of the fourth angle will be _____.

5. "If the two acute angles of a right-angled triangle are in the ratio 2:3, what are the radian measures of those two angles?

(a) \(\cfrac{π}{5},\cfrac{3π}{10}\) (b) \(\cfrac{π}{10},\cfrac{3π}{5}\) (c) \(\cfrac{π}{5},\cfrac{3π}{20}\) (d) \(\cfrac{π}{5},\cfrac{π}{15}\)

6. PQRS is a cyclic quadrilateral in which side QR is extended up to point T. If the measures of angles ∠SRQ and ∠SRT are in the ratio 4:5, then find the measures of ∠SPQ and ∠SRQ.

7. The ratio of the capital invested by three people is 4:7:9, and the profit of the first person is ₹100 less than the profit of the third person. What is the amount of profit earned by the second person?

8. \[ \frac{1}{(x - 1)(x - 2)} + \frac{1}{(x - 2)(x - 3)} + \frac{1}{(x - 3)(x - 4)} = \frac{1}{6} \] This is a mathematical equation involving rational expressions. Here's the English translation of the statement: **"The sum of the following three fractions equals one-sixth:"** - The first fraction is: one divided by \((x - 1)(x - 2)\) - The second fraction is: one divided by \((x - 2)(x - 3)\) - The third fraction is: one divided by \((x - 3)(x - 4)\) And their total is equal to \(\frac{1}{6}\).

9. In a joint business, the capital ratio of three individuals is 5:8:3, and the second person's profit is ₹100 more than the third person's. What is the total profit in the business?

10. If three consecutive angles of a cyclic quadrilateral are in the ratio \(1:3:4\), then the measure of the fourth angle will be _____.

11. In a partnership business, the capital ratio of three friends is \(\frac{1}{4} : \frac{1}{3} : \frac{1}{2}\). If the total profit after one year is ₹3,900, what will be each of their shares of the profit?

12. A solid lead sphere with a diameter of 12 cm is melted to form three smaller spheres. If the diameters of the smaller spheres are in the ratio 3:4:5, what is the radius of the smallest sphere?

(a) 1.5 cm (b) 3 cm (c) 4 cm (d) 5 cm

13. 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

14. 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

15. In a partnership business, the capital ratio between A and B is 2:3, and the capital ratio between B and C is 6:5. If A's profit share is ₹400, what will be C's profit share?

(a) ₹ 500 (b) ₹ 600 (c) ₹ 700 (d) ₹ 800

16. Two unequal arcs of a circle subtend two angles at the center in the ratio 5:3, and the sexagesimal (degree) measure of the second angle is 45°. What is the radian measure of the first angle?

17. Three friends purchase a bus by investing ₹1,20,000, ₹1,50,000, and ₹1,10,000 respectively. The first friend works as the driver, while the other two work as conductors. They decide that \(\frac{2}{5}\) of the income will be distributed based on work in the ratio 3 : 2 : 2, and the remaining amount will be divided according to their capital investment. If the total income in a certain month is ₹29,260, determine how much each person will receive.

18. A solid lead sphere with a diameter of 12 cm is melted down to form three smaller solid spheres. If the diameters of the smaller spheres are in the ratio 3 : 4 : 5, then what is the radius of each of the smaller spheres?

19. In a business, A and B invest in the ratio 3:2. After donating 5% of the profit, A receives ₹1197 as his share of the profit. What is the total profit?

20. Ajay, Bijay, and Sujay invest capital in a business in the ratio \(\cfrac{1}{2} : \cfrac{1}{3} : \cfrac{1}{5}\). If the total profit of the business is ₹1550, what is Sujay's share of the profit?

(a) ₹ 600 (b) ₹ 300 (c) ₹ 150 (d) ₹ 250

21. In a business, three friends have their capital investment in the ratio 2:3:4. If in the following year they increase their capital by 10%, 15%, and 20%, respectively, in what ratio will the profit be distributed among them at the end of the second year?

(a) 44 : 69 : 96 (b) 69 : 44 : 96 (c) 96 : 69 : 44 (d) 45 : 96 : 63

22. In a business, the capital ratio of A and C is 2:1, and the capital ratio of A and B is 3:2. If the annual profit is ₹1,57,300, what will be B's share of the profit?

(a) ₹ 48,000 (b) ₹ 58,400 (c) ₹ 48,400 (d) ₹68,000

23. A, B, and C are partners in a joint business. Their capital ratio is \(\frac{1}{2} : \frac{1}{3} : \frac{1}{4}\). After 4 months, A withdraws half of his capital. After another 4 months, the total profit is ₹6072. What is A’s share of the profit?

24. In a partnership business, the capital ratio of three individuals is 3:8:5, and the profit of the first person is 70 rupees less than the profit of the third person. What was the total profit in the business?

25. In a business, the ratio of the capital investments of three friends is 5:8:12. If the profit of the first person is Rs. 700 less than the profit of the third person, find the profit of the second person.

26. In a partnership business, the capital ratio of three individuals is 3:8:5, and the profit of the first person is 60 rupees less than the profit of the third person. What is the total profit of the business?

27. In a partnership business, the ratio of the capital investments of A, B, and C is 6 : 5 : 4. After 4 months, A withdraws half of his capital. If the annual profit is ₹1212, what is each person’s share of the profit? Let me know if you'd like help solving it too. I'm ready.

28. In a partnership business, the capital contributions of three individuals are in the ratio 3:8:5. If the profit of the first person is ₹60 less than that of the third person, what was the total profit of the business?

29. If the ratio of the radii of two tall solid circular cones is 2:3 and the ratio of their heights is 5:3, then what is the ratio of their curved surface areas?

(a) 8:07 (b) 2:05 (c) 10:09 (d) 16:09

30. Three household kerosene oil drums contained 800 liters, 725 liters, and 575 liters of oil respectively. The oil from these three drums was poured into a rectangular container, which then had a liquid depth of 7 decimeters. If the ratio of the length to the breadth of the rectangular container is 4:3, calculate and write the length and breadth of the container. Also, if the depth of the container were 5 decimeters instead, determine whether it could hold 1620 liters of oil.