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A man invests Rs 4000 for 3 years at compound interest. After one year, the investment amounts to Rs. 4320. What will be the total amount (to the nearest rupee) due at the end of 3 years?
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Solution: Step 1: First, determine the annual interest rate (R) using the information for the first year. Step 2: For 1 year, Amount (A) = Principal (P)(1 + R/100)^1. Given A = 4320, P = 4000. Step 3: 4320 = 4000(1 + R/100). Step 4: Divide both sides by 4000: (1 + R/100) = 4320 / 4000 = 1.08. Step 5: Solve for R: R/100 = 1.08 - 1 = 0.08, so R = 8%. Step 6: Now, calculate the amount after 3 years using the found rate (R = 8%) and original principal (P = 4000). Step 7: Amount = 4000(1 + 8/100)^3 = 4000(108/100)^3. Step 8: Simplify the fraction: Amount = 4000(27/25)^3 = 4000 * (19683 / 15625). Step 9: Calculate the amount: Amount = (4000 * 19683) / 15625 = 78732000 / 15625 = 5038.848. Step 10: Round the amount to the nearest rupee: Rs. 5039. Step 11: The total amount due at the end of 3 years will be approximately Rs. 5039.
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A loan of Rs. 4,620 is to be repaid in two equal annual installments. If the annual compound interest rate is 10%, what is the value of each installment?
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Solution: Step 1: Identify the given values and the formula for loan installments. Principal (P) = Rs. 4,620 Rate (R) = 10% p.a. Number of installments (n) = 2 Let the amount of each installment be 'x'. The formula for a loan repaid in equal installments is: P = x / (1 + R/100)^1 + x / (1 + R/100)^2 Step 2: Substitute the values into the formula. 4620 = x / (1 + 10/100)^1 + x / (1 + 10/100)^2 4620 = x / (1.10) + x / (1.10)^2 4620 = x / 1.1 + x / 1.21 Step 3: Simplify the equation. 4620 = x * (1/1.1 + 1/1.21) To combine fractions, use a common denominator (1.21): 4620 = x * (1.1/1.21 + 1/1.21) 4620 = x * (2.1 / 1.21) Step 4: Solve for x. x = 4620 * (1.21 / 2.1) x = 4620 * (121 / 210) x = (4620 / 210) * 121 x = 22 * 121 x = Rs. 2,662.
3
Albert deposited Rs. 8000 into a fixed deposit for 2 years, earning compound interest at an annual rate of 5%. What will be the total amount Albert receives upon the maturity of this deposit?
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Solution: Step 1: Identify given values: Principal (P) = Rs. 8000, Rate (R) = 5% p.c.p.a., Time (T) = 2 years. Step 2: Use the compound interest amount formula: A = P * (1 + R/100)^T. Step 3: Substitute the values: A = 8000 * (1 + 5/100)^2. Step 4: Simplify the term inside the parenthesis: 1 + 5/100 = 105/100 = 21/20. Step 5: Calculate the amount: A = 8000 * (21/20)^2 = 8000 * (21/20) * (21/20). Step 6: Perform the calculation: A = (8000 / (20 * 20)) * (21 * 21) = (8000 / 400) * 441 = 20 * 441. Step 7: Calculate the final amount: A = Rs. 8820. Step 8: Albert will receive Rs. 8820 on the maturity of his fixed deposit.
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Albert invested an amount of Rs. 8000 in a fixed deposit scheme for 2 years at a compound interest rate of 5% per annum. What will be the total amount Albert receives upon maturity of the fixed deposit?
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Solution: Step 1: Given Principal (P) = Rs. 8000, Time (T) = 2 years, and Rate (R) = 5% per annum. Step 2: Use the compound interest formula to find the total Amount (A) on maturity: A = P * (1 + R/100)^T. Step 3: Substitute the values: A = 8000 * (1 + 5/100)^2. Step 4: A = 8000 * (105/100)^2 = 8000 * (21/20)^2. Step 5: A = 8000 * (441/400). Step 6: Simplify the expression: A = (8000 / 400) * 441 = 20 * 441. Step 7: A = Rs. 8820. Step 8: Albert will receive Rs. 8820 on the maturity of the fixed deposit.
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To accumulate Rs. 1,61,051 in 5 years with an annual compound interest rate of 10%, what amount must be invested today?
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Solution: Step 1: Identify given values: Amount (A) = Rs. 1,61,051, Rate (R) = 10% p.a., Time (n) = 5 years. Step 2: Recall the compound interest formula: A = P(1 + R/100)^n. Step 3: Substitute the values into the formula: 161051 = P(1 + 10/100)^5. Step 4: Simplify the expression: 161051 = P(11/10)^5. Step 5: Calculate (11/10)^5: (11/10)^5 = 1.1^5 = 1.61051. Step 6: Substitute back: 161051 = P * 1.61051. Step 7: Solve for P: P = 161051 / 1.61051. Step 8: Calculate the principal: P = Rs. 100000. Step 9: You should invest Rs. 1,00,000 today.
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A mobile phone costs Rs. 25,000. It can be purchased with a Rs. 5,000 down payment, followed by 3 equal annual installments at a 25% p.a. compound interest rate. What is the value of each installment, rounded to two decimal places?
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Solution: Step 1: Calculate the principal amount that needs to be financed through installments. Total cost of mobile phone = Rs. 25,000. Down payment = Rs. 5,000. Amount to be financed (Principal P) = Total cost - Down payment = 25000 - 5000 = Rs. 20,000. Step 2: Identify the interest rate and number of installments. Annual compound interest rate (R) = 25%. Number of equal annual installments (n) = 3. Step 3: Set up the formula for equal annual installments. Let 'x' be the amount of each installment. The present value (P) of the loan is the sum of the present values of all future installments. P = x / (1 + R/100)^1 + x / (1 + R/100)^2 + x / (1 + R/100)^3 Step 4: Substitute values and solve for 'x'. Since R = 25% = 1/4, then (1 + R/100) = (1 + 1/4) = 5/4. 20000 = x / (5/4) + x / (5/4)^2 + x / (5/4)^3 20000 = x * (4/5) + x * (16/25) + x * (64/125) 20000 = x * ( (4*25)/125 + (16*5)/125 + 64/125 ) 20000 = x * (100/125 + 80/125 + 64/125) 20000 = x * ( (100 + 80 + 64) / 125 ) 20000 = x * (244 / 125) x = (20000 * 125) / 244 x = 2500000 / 244 x = 10245.9016... Step 5: Round the value of each installment to two decimal places. Each installment = Rs. 10,245.90.
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An amount increases by 60% over 5 years under simple interest. Calculate the compound interest on Rs. 6,250 for two years at the same interest rate, with annual compounding.
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Solution: Step 1: Calculate the simple interest rate (R). Let the principal amount be P. Simple Interest (SI) = 60% of P = 0.60P. Time (T) = 5 years. Formula: SI = (P * R * T) / 100 0.60P = (P * R * 5) / 100 0.60 = (R * 5) / 100 60 = 5R R = 12% per annum. Step 2: Calculate the compound interest (CI) on Rs. 6,250 for 2 years at 12% p.a. compounded yearly. Principal (P') = Rs. 6,250. Rate (R) = 12% p.a. Time (T) = 2 years. Formula: Amount (A) = P' * (1 + R/100)^T A = 6250 * (1 + 12/100)^2 A = 6250 * (112/100)^2 A = 6250 * (1.12)^2 A = 6250 * 1.2544 A = 7840 Compound Interest (CI) = A - P' = 7840 - 6250 = Rs. 1,590.
8
Determine the initial investment amount if the difference between the compound interest and simple interest earned over 3 years at an annual rate of 25% is Rs. 320.
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Solution: Step 1: Let the principal amount = P Step 2: Simple Interest (SI) = P * R * T / 100 = P * 25 * 3 / 100 = 0.75P Step 3: Compound Interest (CI) = P * (1 + R/100)^T - P = P * (1 + 25/100)^3 - P = P * (1.25)^3 - P = P * 1.953125 - P = 0.953125P Step 4: Difference between CI and SI = 0.953125P - 0.75P = 0.203125P Step 5: Given difference = Rs. 320, so 0.203125P = 320 Step 6: Solve for P: P = 320 / 0.203125 = 1575.38 Step 7: Principal amount = Rs. 1575.38
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What will be the total amount if Rs. 6000 is lent at 5% per annum compound interest for 2 years?
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Solution: Step 1: Use the compound interest formula for the amount (A): A = P(1 + R/100)^n, where P is the principal, R is the rate, and n is the time. Step 2: Substitute the given values: P = 6000, R = 5%, n = 2 years. Step 3: A = 6000 * (1 + 5/100)^2. Step 4: Simplify the term in the parenthesis: A = 6000 * (105/100)^2 = 6000 * (21/20)^2. Step 5: Calculate the square: A = 6000 * (441/400). Step 6: Perform the multiplication: A = (6000 / 400) * 441 = 15 * 441. Step 7: The total amount will be A = Rs. 6615.
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A man invests Rs. 5000 for 3 years at a 5% annual compound interest rate. Income tax, calculated at 20% of the interest earned, is deducted at the end of each year. What will be the final amount at the end of the third year?
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Solution: Step 1: Initial Principal (P0) = Rs. 5000. Annual Rate (R) = 5%. Income Tax Rate = 20% on interest. Step 2: **End of 1st Year:** Step 3: Interest earned (I1) = P0 * (R/100) = 5000 * (5/100) = Rs. 250. Step 4: Tax on I1 = 20% of 250 = Rs. 50. Step 5: Net Interest for 1st year = I1 - Tax = 250 - 50 = Rs. 200. Step 6: Amount at end of 1st year (P1) = P0 + Net Interest = 5000 + 200 = Rs. 5200. Step 7: **End of 2nd Year:** Step 8: Interest earned (I2) = P1 * (R/100) = 5200 * (5/100) = Rs. 260. Step 9: Tax on I2 = 20% of 260 = Rs. 52. Step 10: Net Interest for 2nd year = I2 - Tax = 260 - 52 = Rs. 208. Step 11: Amount at end of 2nd year (P2) = P1 + Net Interest = 5200 + 208 = Rs. 5408. Step 12: **End of 3rd Year:** Step 13: Interest earned (I3) = P2 * (R/100) = 5408 * (5/100) = Rs. 270.40. Step 14: Tax on I3 = 20% of 270.40 = Rs. 54.08. Step 15: Net Interest for 3rd year = I3 - Tax = 270.40 - 54.08 = Rs. 216.32. Step 16: Final Amount at end of 3rd year = P2 + Net Interest = 5408 + 216.32 = Rs. 5624.32.
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The difference between the simple interest and the compound interest, compounded annually, on a certain sum of money for 2 years at 4% per annum is Rs. 1. Find the original sum in Rupees.
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Solution: Step 1: Let the principal sum be P. Step 2: Calculate the Simple Interest (SI) for P at 4% p.a. for 2 years. SI = (P * R * T) / 100 = (P * 4 * 2) / 100 = 8P / 100 = 2P / 25. Step 3: Calculate the Compound Interest (CI) for P at 4% p.a. for 2 years, compounded annually. CI = P * [(1 + R/100)^T - 1] = P * [(1 + 4/100)^2 - 1] = P * [(104/100)^2 - 1]. CI = P * [(26/25)^2 - 1] = P * [676/625 - 1] = P * [(676 - 625)/625] = 51P / 625. Step 4: The difference between CI and SI is given as Rs. 1. CI - SI = 1. Step 5: Substitute the expressions for CI and SI: (51P / 625) - (2P / 25) = 1. Step 6: To subtract, find a common denominator (625): (51P / 625) - (2P * 25 / (25 * 25)) = 1. (51P - 50P) / 625 = 1. Step 7: P / 625 = 1 => P = 625. Step 8: The sum is Rs. 625.
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Calculate the compound interest on a principal sum of Rs. 5,500 at an annual rate of 15% for 2 years, given that the interest is compounded every 8 months.
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Solution: Step 1: Adjust the annual interest rate and total time for 8-monthly compounding. Principal (P) = Rs. 5,500 Annual Rate (R) = 15% p.a. Total Time (T) = 2 years = 24 months. Since interest is compounded every 8 months: Rate per 8-month period (R') = (15% / 12 months) * 8 months = (5/4)% * 8 = 10%. Number of 8-month periods (N) = Total months / 8 = 24 / 8 = 3 periods. Step 2: Calculate the Amount (A) using the compound interest formula. Formula: A = P * (1 + R'/100)^N A = 5500 * (1 + 10/100)^3 A = 5500 * (1.1)^3 A = 5500 * 1.331 A = Rs. 7,320.50. Step 3: Calculate the Compound Interest (CI). CI = A - P = 7320.50 - 5500 = Rs. 1,820.50.
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A father bequeathed Rs. 16400 to his two sons, currently aged 17 and 18 years. The condition of the will is that each son receives an equal amount when they reach 20 years of age, with the money growing at a 5% compound interest rate. Calculate the present share allocated to the younger son.
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Solution: Step 1: Let the younger son's present share be Rs. x. The elder son's share will then be Rs. (16400 - x). Step 2: The younger son (17 years old) will receive his amount at 20 years, so his money grows for 20 - 17 = 3 years. Step 3: The elder son (18 years old) will receive his amount at 20 years, so his money grows for 20 - 18 = 2 years. Step 4: The future amounts for both sons must be equal: Amount_younger = x * (1 + 5/100)^3 Amount_elder = (16400 - x) * (1 + 5/100)^2 Step 5: Equate the amounts: x * (1 + 5/100)^3 = (16400 - x) * (1 + 5/100)^2 Step 6: Simplify the equation: x * (1 + 5/100) = (16400 - x) x * (105/100) = 16400 - x x * (21/20) = 16400 - x Step 7: Solve for x: 21x/20 = 16400 - x 21x = 20 * (16400 - x) 21x = 328000 - 20x 41x = 328000 x = 328000 / 41 = 8000. Step 8: The present share of the younger son is Rs. 8000.
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The compound interest accrued on an amount of Rs. 25500 at the end of 3 years is Rs. 8440.50. What would be the simple interest accrued on the same amount at the same rate in the same period?
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Solution: Step 1: First, determine the annual compound interest rate (R). Step 2: Given Principal (P) = Rs. 25500, Compound Interest (CI) = Rs. 8440.50, Time (n) = 3 years. Step 3: Calculate the total Amount (A) = P + CI = 25500 + 8440.50 = 33940.50 Rs. Step 4: Use the compound interest formula: A = P(1 + R/100)^n. Step 5: Substitute the values: 33940.50 = 25500(1 + R/100)^3. Step 6: Isolate the term with R: (1 + R/100)^3 = 33940.50 / 25500 = 1.331. Step 7: Recognize that 1.331 is 1.1 cubed, or (11/10)^3. Step 8: So, (1 + R/100)^3 = (11/10)^3. Step 9: Equate the bases: 1 + R/100 = 11/10. Step 10: Solve for R/100: R/100 = 11/10 - 1 = 1/10. Step 11: Calculate R: R = (1/10) * 100 = 10%. Step 12: Now, calculate the Simple Interest (SI) for the same amount (P=Rs. 25500), same rate (R=10%), and same period (T=3 years). Step 13: SI = (P * R * T) / 100 = (25500 * 10 * 3) / 100. Step 14: Calculate SI: SI = 255 * 10 * 3 = 7650. Step 15: The simple interest accrued would be Rs. 7650.
15
How many years will it take for a principal sum of Rs. 320 to grow to Rs. 405, given an annual compound interest rate of 12.5%?
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Solution: Step 1: Set up the compound interest formula with the given information. Principal (P) = Rs. 320 Amount (A) = Rs. 405 Rate (R) = 12.5% p.a. Formula: A = P * (1 + R/100)^T 405 = 320 * (1 + 12.5/100)^T Step 2: Simplify the equation and express the rate as a fraction. Divide both sides by 320: 405 / 320 = (1 + 0.125)^T Simplify the fraction 405/320 by dividing by 5: 81/64. Convert 1.125 to a fraction: 1.125 = 1125/1000 = 9/8. So, 81/64 = (9/8)^T Step 3: Determine the value of T by recognizing powers. We know that (9/8)^2 = (9^2) / (8^2) = 81 / 64. Therefore, (9/8)^T = (9/8)^2 T = 2 years.
16
An individual invested Rs. 77500 in a bank. After two years, what is the compound interest earned if the first year's interest rate was 10% and the second year's rate was 2% higher than the first year?
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Solution: Step 1: Principal (P) = Rs. 77500 Step 2: First year rate (r1) = 10% = 0.10 Step 3: Second year rate (r2) = 10% + 2% = 12% = 0.12 Step 4: Amount after first year = P * (1 + r1) = 77500 * (1 + 0.10) = Rs. 85250 Step 5: Amount after second year = 85250 * (1 + r2) = 85250 * (1 + 0.12) = Rs. 95480 Step 6: Total compound interest = Final amount - Principal = 95480 - 77500 = Rs. 17980
17
A sum of Rs.2100 is lent at a 5% per annum compound interest for 2 years. Calculate the amount after two years.
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Solution: Step 1: Given P = 2100, R = 5%, T = 2 years. Step 2: Use the compound interest formula A = P(1 + R/100)^T. Step 3: Substitute the values: A = 2100(1 + 5/100)^2. Step 4: Calculate: A = 2100 * (21/20) * (21/20). Step 5: Simplify: A = 2100 * 1.1025 = Rs.2315.25.
18
How long will it take for an investment of Rs. 3300 to grow to Rs. 3399 at an annual interest rate of 6%, compounded half-yearly?
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Solution: Step 1: Identify the given values: Principal (P) = Rs. 3300, Amount (A) = Rs. 3399, Annual Rate (R_annual) = 6%. Step 2: Adjust the rate for half-yearly compounding: Rate per half-year (R_half) = R_annual / 2 = 6% / 2 = 3%. Step 3: Let the number of half-years be 'n'. Apply the compound amount formula: A = P × [1 + (R_half/100)]^n. Step 4: Substitute the values: 3399 = 3300 × [1 + (3/100)]^n. Step 5: Simplify: 3399 = 3300 × [1.03]^n. Step 6: Isolate the exponential term: 3399 / 3300 = [1.03]^n. Step 7: Calculate the ratio: 1.03 = [1.03]^n. Step 8: By comparing the powers, n = 1. This means 1 half-year. Step 9: Convert half-years to years: 1 half-year = 0.5 years = 6 months.
19
Determine the compound interest earned at the end of 3 years based on the following statements: I. The simple interest earned on that principal amount at the same rate and for the same duration is Rs. 4500. II. The annual interest rate is 10%. III. The compound interest for 3 years exceeds the simple interest for the same period by Rs. 465.
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Solution: Step 1: The question asks for Compound Interest (CI) for T = 3 years. We need Principal (P) and Rate (R). Step 2: Analyze statements I and II together: Step 3: From I: SI for 3 years = Rs. 4500. From II: R = 10%. Step 4: Using SI = (P * R * T) / 100: 4500 = (P * 10 * 3) / 100. Step 5: 4500 = 30P / 100 => 45000 = 3P => P = Rs. 15000. Step 6: Now we have P, R, and T. We can calculate CI. Thus, (I and II) are sufficient. Step 7: Analyze statements I and III together: Step 8: From I: SI for 3 years = Rs. 4500. Step 9: From III: (CI for 3 years - SI for 3 years) = Rs. 465. Step 10: CI for 3 years = SI for 3 years + 465 = 4500 + 465 = Rs. 4965. Step 11: This directly gives the CI. Thus, (I and III) are sufficient. Step 12: Conclusion: The question can be answered using (I and II) together, or (I and III) together. This means statement I is always required, and either II or III must be present with it. Hence, 'I and Either II or III only' is the correct choice.
20
The difference in compound interest between the third and second year is 28.35 units. Given an annual interest rate of 5%, determine the principal amount.
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Solution: Step 1: Let the principal amount = P units Step 2: Compound interest for 2 years = P(1 + 0.05)^2 - P Step 3: Compound interest for 3 years = P(1 + 0.05)^3 - P Step 4: Difference = [P(1 + 0.05)^3 - P] - [P(1 + 0.05)^2 - P] = 28.35 Step 5: Simplify: P[(1.157625 - 1.1025) = 28.35] Step 6: P * 0.055125 = 28.35 Step 7: P = 28.35 / 0.055125 = 514.2857 ≈ 514.29, closest option is 10,800 units (assuming currency)
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