1. Problem 56: A and B start a business. A invests 80000 for 9 months, B invests 120000 for 6 months. Find B's share of a 45000 profit. 2. Use the formula for profit sharing based on investment and time: \n$$\text{Share} = \frac{\text{Investment} \times \text{Time}}{\text{Total Investment-Time}} \times \text{Total Profit}$$ 3. Calculate A's investment-time product: $$80000 \times 9 = 720000$$ 4. Calculate B's investment-time product: $$120000 \times 6 = 720000$$ 5. Total investment-time: $$720000 + 720000 = 1440000$$ 6. B's share ratio: $$\frac{720000}{1440000} = \frac{1}{2}$$ 7. B's profit: $$\frac{1}{2} \times 45000 = 22500$$ --- 1. Problem 57: Find the average of all integers between 100 and 250 divisible by 11. 2. Find the smallest multiple of 11 greater than or equal to 100: $$11 \times 10 = 110$$ 3. Find the largest multiple of 11 less than or equal to 250: $$11 \times 22 = 242$$ 4. The numbers divisible by 11 are: 110, 121, ..., 242. 5. Count of terms: $$22 - 10 + 1 = 13$$ 6. Average of an arithmetic sequence: $$\frac{\text{first term} + \text{last term}}{2} = \frac{110 + 242}{2} = 176$$ --- 1. Problem 58: Average of 15 numbers is 80. First 6 average 72. Next 6 average 25% more than first 6. 13th number is 8 more than 15th, 14th is 10 less than 15th. Find average of 13th and 14th. 2. Total sum of 15 numbers: $$15 \times 80 = 1200$$ 3. Sum of first 6: $$6 \times 72 = 432$$ 4. Average of next 6: $$72 + 0.25 \times 72 = 90$$ 5. Sum of next 6: $$6 \times 90 = 540$$ 6. Sum of last 3 numbers: $$1200 - (432 + 540) = 228$$ 7. Let 15th number be $$x$$. 8. Then 13th number: $$x + 8$$, 14th number: $$x - 10$$ 9. Sum of last 3: $$(x + 8) + (x - 10) + x = 3x - 2 = 228$$ 10. Solve for $$x$$: $$3x = 230 \Rightarrow x = \frac{230}{3} = 76.67$$ 11. Average of 13th and 14th: $$\frac{(x + 8) + (x - 10)}{2} = \frac{2x - 2}{2} = x - 1 = 76.67 - 1 = 75.67$$ --- 1. Problem 59: Landlord buys flat for 800000. Wants 9% annual return after paying 2000 per month maintenance. Find monthly rent. 2. Annual maintenance cost: $$2000 \times 12 = 24000$$ 3. Desired annual return: $$9\% \times 800000 = 72000$$ 4. Total annual income needed: $$72000 + 24000 = 96000$$ 5. Monthly rent: $$\frac{96000}{12} = 8000$$ 6. Since maintenance is paid separately, rent charged should cover return plus maintenance, so rent = 8000. Final answers: - Q56: 22500 - Q57: 176 - Q58: 75.67 - Q59: 8000