1. In a joint business venture, Samir, Idrish, and Anthony invested ₹4000, ₹5000, and ₹6000 respectively. After 4 months, Anthony withdrew half of his capital. At the end of the year, the total profit was ₹3900. Determine how much profit each person will receive.
2. A rectangular tank measuring 2.1 meters in length and 1.5 meters in width is half-filled with water. If 630 liters of water is poured into the tank, determine how much the water level will rise.
3. A rectangular container has a base in the shape of a rectangle with a length of 60 cm and a width of 45 cm. The container has a height of 20 cm and is half-filled with water. Determine the side length of a metallic cube that, when placed in the container, will cause the water level to reach the brim.
4. At the beginning of the year, Srikanta and Sadananda started a business with investments of 24,000 taka and 30,000 taka, respectively. After 5 months, Srikanta invested an additional 4,000 taka in the business. If the total profit at the end of the year is 27,716 taka, determine how much profit each person will receive.
5. In a partnership business, B's capital is \(1\frac{1}{2}\) times A's capital. After 4 months, B withdraws half of his capital, and after another 2 months, A withdraws \(\frac{1}{4}\) of his capital. If the total profit for the year is ₹6360, determine how much A will receive.
6. Niyamat Chacha and Korobi Didi jointly started a business with initial investments of 30,000 INR and 50,000 INR, respectively. After 6 months, Niyamat Chacha invested an additional 40,000 INR, while Korobi Didi withdrew 10,000 INR for personal needs. At the end of the year, if the total profit is 19,000 INR, determine how much each of them will receive.
7. A trough measuring 21 decimeters in length, 11 decimeters in width, and 6 decimeters in depth is half full of water. If 100 iron spheres, each with a diameter of 21 centimeters, are completely submerged in the trough, determine how much the water level will rise in decimeters.
8. "y is in direct variation with the square of x, and y = 9 when x = 9; express y in terms of x, and when y = 4, find the value of x."
9. If the base area of a right circular cone is \(A\), its height is \(H\), and its volume is \(V\), then express \(H\) in terms of \(V\) and \(A\).
(a) \(H=\cfrac{3V}{A}\) (b) \(H=\cfrac{V}{A}\) (c) \(H=\cfrac{V}{3A}\) (d) \(H=\cfrac{3V}{2A}\)
10. A roof measuring 13 meters in length and 11 meters in width had its drainage pipe closed during rainfall. After the rain, it was found that water had accumulated to a depth of 7 centimeters on the roof. The pipe through which the water drains has a diameter of 7 centimeters and discharges water in the form of a cylindrical stream at a rate of 200 meters in length per minute. Determine how long it will take for all the water to drain out once the pipe is opened.
11. y is equal to the sum of two variables—one that varies directly with x and another that varies inversely with x. When x = y, then y = –1, and when x = 3, then y = 5. Determine the relationship between x and y.
12. 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.
13. A man left ₹28,000 for his 13-year-old son and 15-year-old daughter with the instruction that, at the age of 18, the amount each receives — including simple interest at an annual rate of 10% — should be equal. Determine the amount allocated to each child.
14. The volume of a sphere is directly proportional to the cube of its radius. Three solid spheres with diameters of \(1\frac{1}{2}\) m, 2 m, and \(2\frac{1}{2}\) m are melted and recast into a single solid sphere. Using the concept of proportionality, determine the diameter of the new sphere.
15. Due to a storm, a telegraph post bent at a point above the ground, causing its top end to touch the ground at a distance of \(4\sqrt{3}\) meters from its base and form an angle of 30° with the horizontal. Determine how high above the ground the post was bent, and what the total height of the post was.
16. A vertical hollow cylindrical pipe has an outer radius of 16 cm and an inner radius of 12 cm, with a height of 36 cm. If the pipe is melted and solid cylindrical rods are made from it, each with a diameter of 2 cm and a length of 6 cm, determine how many such solid rods can be produced.
17. Determine the volume ratio of a solid cone, a solid hemisphere, and a solid cylinder with equal base diameters and equal heights.
18. In a joint business, B's capital was \(1\frac{1}{2}\) times A's capital. After 4 months, B withdrew half of his capital, and after another 2 months, A withdrew one-fourth of his capital. At the end of the year, the total profit was ₹6,360. Determine the share of each person's profit.
19. Anil Babu wants to allocate ₹56,000 between his two sons, aged 13 and 15, such that when they turn 18, the simple interest earned at an annual rate of 10% is equal to their respective principal amounts. Determine the amount allocated to each son in the will.
20. In a circle with a radius of 5 cm, AB and AC are two equal chords. The center of the circle is located outside the triangle ABC. If AB = AC = 6 cm, determine the length of the chord BC.
21. A tent in the shape of a right elliptical cone has a base area of 13.86 square meters. The tent requires a tarpaulin worth 5775 rupees for construction, and the cost of one square meter of tarpaulin is 150 rupees. Determine the height of the tent and the volume of air it can hold in liters.
22. Dipu, Rabeya, and Megha started a business with capitals of ₹6500, ₹5200, and ₹9100 respectively. After exactly one year, they earned a profit of ₹14,400. They divide \(\cfrac{2}{3}\) of the profit equally among themselves, and the remaining portion is divided in proportion to their capitals. Determine the profit each person receives.
23. A and B started a business by investing ₹3000 and ₹5000, respectively. After 6 months, A added ₹4000 more, but after another 6 months, A withdrew ₹1000. At the end of the year, the total profit was ₹6175. Determine how the profit should be distributed between A and B.
24. Three oil drums contained 800 liters, 725 liters, and 575 liters of oil, respectively. The oil from these drums was poured into a rectangular container, resulting in a depth of 7 decimeters. Given that the length-to-width ratio of the container is 4:3, determine the length and width of the container.
25. A right circular drum has a radius of 21 cm and a height of 2i. A solid iron sphere has a diameter of 21 cm. Determine the ratio of the volumes of the drum and the solid iron sphere (neglecting the thickness of the drum). Now, if the drum is completely filled with water and the sphere is fully submerged and then removed, determine the new depth of the water in the drum.
26. In a joint business, A, B, and C started with investments of ₹3000, ₹4000, and ₹5000, respectively. If B's profit is ₹275, determine the total profit of the business.
(a) ₹ 550 (b) ₹ 500 (c) ₹ 750 (d) ₹ 825
27. Gautam and Samaresh started a joint business with ₹12,500 and ₹8,500 respectively. They agreed that 40% of the profit would be equally distributed between them, and the remaining profit would be shared in proportion to their investments. If Gautam received ₹1,950 as his share of the profit, determine Samaresh’s profit amount.
28. Two equal circles, each with a radius of 10 cm, intersect, and the length of their common chord is 12 cm. Determine the distance between the centers of the circles.
29. If the mean and median of a statistical distribution are 35 and 33, respectively, determine the mode of the distribution.
30. A tall gas jar with a diameter of 10 cm contains some water. A solid cylindrical iron piece with a diameter of 8 cm and a height of 5 cm is completely submerged in the water. Determine how much the water level rises.
