1. If 5 farmers can harvest jute from 10 bighas of land in 12 days, then using the theory of proportion, determine how many farmers are needed to harvest jute from 18 bighas of land in 9 days.
2. 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.
3. A right circular cylinder has a height-to-base-radius ratio of 3:1. If the volume of the cylinder is \(1029\pi\) cubic cm, find the total surface area of the cylinder.
4. Which of the following is greater: (a) The ratio of 8 meters to 10 meters (b) 20% of \(4\frac{2}{5}\)
5. 15 farmers can cultivate 18 bighas of land in 5 days. Using the concept of inverse variation, find how many days 10 farmers will take to cultivate 12 bighas of land.
6. If after one year the ratio of the principal to the amount (principal + interest) is 10 : 12, what is the annual rate of simple interest (in percentage)?
7. If the ratio of cost price to selling price is 12:13, Profit percentage = \( \frac{13 - 12}{12} \times 100 = \frac{1}{12} \times 100 \approx 8.33\% \)
(a) \(7\cfrac{1}{3}\)% (b) \(7\cfrac{2}{3}\)% (c) 8% (d) \(8\cfrac{1}{3}\)%
8. 15 farmers can cultivate 18 bighas of land in 5 days. Using the concept of proportion (rule of three), determine how many days 10 farmers will take to cultivate 12 bighas of land.
9. At an annual interest rate of 10%, the ratio of simple interest to compound interest on ₹100 for 2 years will be –.
(a) 10:11 (b) 11:10 (c) 20:21 (d) 21:20
10. If John cultivates 10 bighas of land in 9 days, determine how many days it will take for 25 people to cultivate 30 bighas of land using the method of proportion.
11. Rahim took a loan from a bank at an annual simple interest rate of 10%, and on the same day, Ram took a separate loan from the same bank at the same interest rate. After 2 years, the total amount the bank received from Rahim was exactly the same as the total amount it received from Ram after 3 years. Determine the ratio of Rahim's loan to Ram's loan.
12. 15 farmers can cultivate 18 bighas of land in 5 days. Using the theory of proportion, determine how many days 10 farmers will take to cultivate 12 bighas of land.
13. If 5 farmers can harvest jute from 10 bighas of land in 12 days, then using the theory of variation, determine how many farmers are needed to harvest jute from 18 bighas of land in 9 days.
14. The radius of the first sphere \( = \cfrac{21}{2} \) cm And the radius of the second sphere \( = \cfrac{17.5}{2} \) cm \( = \cfrac{175}{2 \times 10} \) cm \( = \cfrac{35}{4} \) cm The ratio of the amount of metal used to make the two spheres = Ratio of their surface areas \( = 4π\left(\cfrac{21}{2}\right)^2 : 4π\left(\cfrac{35}{4}\right)^2 = \cfrac{21 \times 21}{4} : \cfrac{35 \times 35}{16} \) \( = 9 : \cfrac{25}{4} = 36 : 25 \) (Answer)
15. Deepu, Rabea, and Megha started a small business by investing ₹6500, ₹5200, and ₹9100 respectively. Exactly one year later, they made a profit of ₹14,400. They decided to divide part of the profit equally and the remaining part according to their capital ratio. Let us calculate how much profit each person will receive.
16. Three friends purchased a bus by investing ₹1,20,000, ₹1,50,000, and ₹1,10,000 respectively. The first person worked as the driver, and the other two worked as conductors. They agreed to divide \(\frac{2}{5}\) of the total monthly income based on work in the ratio 3:2:2, and the remaining amount based on their capital investment. If the monthly income is ₹29,260, calculate how much each person will receive.
17. Jayanta, Ajit, and Kunal jointly started a partnership business with a total capital of ₹15,000. At the end of the year, they earned profits of ₹800, ₹1,000, and ₹1,200 respectively. Profit ratio = 800 : 1000 : 1200 = 4 : 5 : 6 Total capital = ₹15,000 Sum of ratio parts = 4 + 5 + 6 = 15 Jayanta’s capital = ₹\(\frac{4}{15} × 15,000 = 4,000\) Ajit’s capital = ₹\(\frac{5}{15} × 15,000 = 5,000\) Kunal’s capital = ₹\(\frac{6}{15} × 15,000 = 6,000\)
18. 15 farmers can cultivate 18 bighas of land in 5 days. Using the theory of proportion, determine how many days it will take for 10 farmers to cultivate 12 bighas of land.
19. \25 farmers of a cooperative society cultivate 2400 bighas of land in 36 days. After the society buys a tractor, it is found that half of the land can be cultivated in 30 days. Using the concept of ratio (ভেদতত্ত্ব), determine the equivalent farming capacity of one tractor in terms of number of farmers.
20. Three friends invested capital amounts of ₹6500, ₹5200, and ₹9100 respectively to start a business. After exactly one year, the business made a profit of ₹14,400. If \(\frac{2}{3}\) of this profit is divided equally among them and the remaining portion is shared according to their capital ratio, calculate how much profit each person receives.
21. If \( 2\sqrt{6} \) is a rationalizing factor of \( \sqrt{2x} \), what is the value of \( x \) ?
(a) 2 (b) 3 (c) 6 (d) √6
22. If the ratio of the capital investments of A, B, and C is \(\cfrac{1}{2} : \cfrac{1}{3} : \cfrac{1}{4}\) and the total profit is ₹520, then C's share of the profit will be
(a) ₹120 (b) ₹160 (c) ₹180 (d) ₹140
23. If the ratio of the principal and the total amount is 10:11, what is the annual interest rate?
(a) 10% (b) 11% (c) \(10\cfrac{1}{11}\)% (d) 12%
24. At an annual simple interest rate of 12%, if the ratio of principal to interest after \(x\) years is 25:24, what is the value of \(x\)?
(a) 8 (b) 10 (c) 12 (d) 5
25. If the ratio of the principal to its total amount after one year is 25:28, then the annual interest rate is—?
(a) 3% (b) 12% (c) 10\(\frac{5}{7}\)% (d) 8%
26. 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
27. In triangle \( \triangle ABC \), AD is a median. Point E divides AD in the ratio 1:2. The extended line BE intersects AC at point F. If \( AC = 10 \) cm, find the length of \( AF \).
(a) 5 cm (b) 4 cm (c) 2 cm (d) None of the above
28. "Line segments AB and PQ intersect at point O. AP and BQ are perpendiculars to AB. OA = 20 cm, OB = 8 cm, AP = 10 cm. Find the length of BQ."
(a) 4 cm (b) 6 cm (c) 8 cm (d) None of the above
29. What annual rate of simple interest will make the interest on a sum over 10 years equal to \(\cfrac{2}{5}\) of the total amount?
(a) \(6\frac{1}{3}\%\) (b) \(6\%\) (c) \(6\frac{2}{3}\%\) (d) \(2\frac{1}{6}\%\)
30. The curved surface area of a right circular cone is \(\sqrt{10}\) times the area of its base. What is the ratio of the height of the cone to the diameter of its base?
