1. Define HCF and LCM. HCF (Highest Common Factor) is the greatest number that divides two or more numbers without leaving a remainder. LCM (Least Common Multiple) is the smallest number that is a multiple of two or more numbers. 2. State any two divisibility rules. - A number is divisible by 2 if its last digit is even. - A number is divisible by 3 if the sum of its digits is divisible by 3. 3. Find the HCF of 24, 36, and 60. Prime factors: 24 = $2^3 \times 3$ 36 = $2^2 \times 3^2$ 60 = $2^2 \times 3 \times 5$ Common factors: $2^2 \times 3 = 12$ HCF = 12 4. Find the LCM of 15, 20, and 30. Prime factors: 15 = $3 \times 5$ 20 = $2^2 \times 5$ 30 = $2 \times 3 \times 5$ LCM = $2^2 \times 3 \times 5 = 60$ 5. Simplify: (0.25 + 0.75 + 1.125). Sum = $0.25 + 0.75 + 1.125 = 2.125$ 6. Find the square root of 784. $\sqrt{784} = 28$ 7. Find the cube root of 1728. $\sqrt[3]{1728} = 12$ 8. Find the HCF and LCM of 108, 288, and 360 by prime factorization. 108 = $2^2 \times 3^3$ 288 = $2^5 \times 3^2$ 360 = $2^3 \times 3^2 \times 5$ HCF = $2^2 \times 3^2 = 36$ LCM = $2^5 \times 3^3 \times 5 = 4320$ 9. Simplify and express 4.375 as a fraction. $4.375 = 4 + 0.375 = 4 + \frac{3}{8} = \frac{32}{8} + \frac{3}{8} = \frac{35}{8}$ 10. Find the least number which when divided by 24, 36, and 54 leaves remainder 3. Find LCM of 24, 36, 54: 24 = $2^3 \times 3$ 36 = $2^2 \times 3^2$ 54 = $2 \times 3^3$ LCM = $2^3 \times 3^3 = 216$ Least number = LCM + remainder = $216 + 3 = 219$ 11. Find the square root of 2.25 and cube root of 0.027. $\sqrt{2.25} = 1.5$ $\sqrt[3]{0.027} = 0.3$ 12. A number is divisible by 8 and 12; find the smallest such number greater than 100. LCM of 8 and 12: 8 = $2^3$ 12 = $2^2 \times 3$ LCM = $2^3 \times 3 = 24$ Multiples of 24 greater than 100: 120, 144... Smallest is 120 13. Explain in detail the steps for finding HCF and LCM with solved examples. - HCF: Find prime factors of each number, take the lowest powers of common primes. - LCM: Find prime factors, take the highest powers of all primes. Example: For 24 and 36, 24 = $2^3 \times 3$ 36 = $2^2 \times 3^2$ HCF = $2^2 \times 3 = 12$ LCM = $2^3 \times 3^2 = 72$ 14. Explain divisibility rules for 2, 3, 5, 9, and 11 with examples. - 2: Last digit even (e.g., 14) - 3: Sum of digits divisible by 3 (e.g., 123) - 5: Last digit 0 or 5 (e.g., 45) - 9: Sum of digits divisible by 9 (e.g., 81) - 11: Difference between sum of digits in odd and even places divisible by 11 (e.g., 121) 15. Solve 5 problems on finding square roots and cube roots of decimals. Examples: $\sqrt{0.16} = 0.4$ $\sqrt{1.44} = 1.2$ $\sqrt[3]{0.008} = 0.2$ $\sqrt[3]{0.125} = 0.5$ $\sqrt{0.81} = 0.9$ 16. Define average. Average is the sum of values divided by the number of values. 17. Write the formula for percentage. Percentage = $\frac{\text{Part}}{\text{Whole}} \times 100$ 18. Define profit and loss. Profit is when selling price > cost price. Loss is when cost price > selling price. 19. Define ratio and proportion. Ratio compares two quantities. Proportion states two ratios are equal. 20. A man gains 20% on selling an article for 120. Find the cost price. Let cost price = $x$ Selling price = $x + 0.20x = 1.2x = 120$ $x = \frac{120}{1.2} = 100$ 21. Find the average of 15, 25, 35, 45, and 55. Sum = 175 Average = $\frac{175}{5} = 35$ 22. A person scored 60, 70, 80, and 90 in 4 subjects. Find average percentage if max marks per subject = 100. Total marks = 400 Obtained = 300 Percentage = $\frac{300}{400} \times 100 = 75\%$ 23. A shopkeeper bought an article for 500 and sold it for 650. Find profit % and gain. Profit = 650 - 500 = 150 Profit % = $\frac{150}{500} \times 100 = 30\%$ 24. Divide 1200 in the ratio 3:5. Total parts = 8 Part value = $\frac{1200}{8} = 150$ First part = $3 \times 150 = 450$ Second part = $5 \times 150 = 750$ 25. A’s income is 20% more than B’s. Find the ratio of their incomes. If B’s income = $x$ A’s income = $x + 0.20x = 1.2x$ Ratio A:B = $1.2x : x = 6:5$ 26. Define time and work. Time is duration to complete a task. Work is the task done. 27. Define speed, distance, and time. Speed = distance/time Distance = speed × time Time = distance/speed 28. Write the formula for speed. Speed = $\frac{\text{Distance}}{\text{Time}}$ 29. Define upstream and downstream. Upstream: moving against the current. Downstream: moving with the current. 30. A can do a work in 6 days, B in 8 days. Find how long they take together. Work done per day: A = $\frac{1}{6}$ B = $\frac{1}{8}$ Together = $\frac{1}{6} + \frac{1}{8} = \frac{7}{24}$ Time = $\frac{1}{7/24} = \frac{24}{7} \approx 3.43$ days 31. A and B can do a piece of work in 12 days and 16 days respectively. In how many days can they do it together? Work per day: A = $\frac{1}{12}$ B = $\frac{1}{16}$ Together = $\frac{1}{12} + \frac{1}{16} = \frac{7}{48}$ Time = $\frac{1}{7/48} = \frac{48}{7} \approx 6.86$ days 32. A train travels 360 km in 4 hours. Find its speed. Speed = $\frac{360}{4} = 90$ km/h 33. A boat goes 10 km downstream in 2 hours and returns in 4 hours. Find the speed of the boat and stream. Downstream speed = $\frac{10}{2} = 5$ km/h Upstream speed = $\frac{10}{4} = 2.5$ km/h Boat speed = $\frac{5 + 2.5}{2} = 3.75$ km/h Stream speed = $\frac{5 - 2.5}{2} = 1.25$ km/h 34. A can finish a job in 10 days and B in 15 days. How many days together? Work per day: A = $\frac{1}{10}$ B = $\frac{1}{15}$ Together = $\frac{1}{10} + \frac{1}{15} = \frac{1}{6}$ Time = 6 days 35. A car travels 120 km at 40 km/h and returns at 60 km/h. Find the average speed. Average speed = $\frac{2 \times 40 \times 60}{40 + 60} = 48$ km/h 36. Define simple interest. Simple interest is interest calculated only on the principal amount. 37. Define compound interest. Compound interest is interest calculated on principal plus accumulated interest. 38. Write the formula for simple interest. $SI = \frac{P \times R \times T}{100}$ 39. Define dividend and share. Dividend is profit paid to shareholders. Share is a unit of ownership in a company. 40. Write the formula for compound amount. $A = P \left(1 + \frac{R}{100}\right)^T$ 41. Find the simple interest on 5000 for 3 years at 10% per annum. $SI = \frac{5000 \times 10 \times 3}{100} = 1500$ 42. Find the compound interest on 4000 for 2 years at 5% per annum. $A = 4000 \times (1 + 0.05)^2 = 4000 \times 1.1025 = 4410$ $CI = 4410 - 4000 = 410$ 43. Differentiate between Simple and Compound Interest. Simple interest is calculated on principal only. Compound interest is calculated on principal plus interest. 44. A man invests 10000 in shares of 100 at 125. Find number of shares. Number of shares = $\frac{10000}{125} = 80$ 45. Find the market value of shares paying 8% dividend when yield is 10%. Market value = $\frac{8}{10} \times 100 = 80$ 46. Find the square root of 784. $\sqrt{784} = 28$ 47. Find the cube root of 1728. $\sqrt[3]{1728} = 12$