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An investment of Rs. 9000 in a 10.5% stock generates an income of Rs. 756. If the brokerage fee is 1/4%, what is the market value of this stock (per Rs. 100 face value)?
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Solution: Step 1: Calculate the effective cost of a Rs. 100 face value stock, including brokerage. The stock is a 10.5% stock, meaning for every Rs. 100 face value, it pays a dividend of Rs. 10.5. Given that a total investment of Rs. 9000 yields an income of Rs. 756. We need to find the investment required to yield Rs. 10.5 income (which is the income from a Rs. 100 face value stock). If Rs. 756 income comes from Rs. 9000 investment, Then Rs. 10.5 income comes from Rs. (9000 / 756) * 10.5 investment. Investment for Rs. 10.5 income = (9000 * 10.5) / 756 = 94500 / 756 = Rs. 125. This Rs. 125 represents the total cost paid for a Rs. 100 face value stock, which includes both its market value and the brokerage fee. Step 2: Calculate the market value by subtracting the brokerage. Brokerage is 1/4% = 0.25%. In such problems, brokerage is often considered a fixed amount per Rs. 100 nominal value or a percentage of the nominal value. Brokerage = 0.25% of Rs. 100 = Rs. 0.25. Market Value = Cost (including brokerage) - Brokerage Market Value = Rs. 125 - Rs. 0.25 = Rs. 124.75.
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A certain principal amount yielded a total simple interest of Rs. 4016.25 over 5 years at an annual rate of 9%. What was the original principal amount?
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Solution: Step 1: Recall the simple interest formula: SI = (P * R * T) / 100. Step 2: Rearrange the formula to solve for Principal (P): P = (SI * 100) / (R * T). Step 3: Substitute the given values: SI = Rs. 4016.25 R = 9% T = 5 years Step 4: Calculate P: P = (4016.25 * 100) / (9 * 5). P = 401625 / 45. Step 5: Perform the division: P = Rs. 8925.
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A principal amount, when lent at simple interest, grows to Rs. 6,076 in 1 year and to Rs. 7,504 in 4 years. Determine the original principal sum and the annual simple interest rate.
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Solution: Step 1: Understand that simple interest grows linearly over time. The difference in the amount over a period is the simple interest earned during that period. Amount after 4 years (A4) = Rs. 7,504. Amount after 1 year (A1) = Rs. 6,076. Step 2: Calculate the simple interest earned in the difference in years. Time difference = 4 - 1 = 3 years. Interest earned in 3 years = A4 - A1 = 7504 - 6076 = Rs. 1,428. Step 3: Calculate the simple interest earned per year. Simple Interest per year (SI_annual) = 1428 / 3 = Rs. 476. Step 4: Determine the original principal sum (P). Principal = Amount after 1 year - SI for 1 year. P = 6076 - 476 = Rs. 5,600. Step 5: Calculate the annual rate of interest (R) using the simple interest for 1 year, the principal, and time = 1 year. Use the formula SI = (P × R × T) / 100. 476 = (5600 × R × 1) / 100. 476 = 56R. R = 476 / 56. R = 8.5%. Step 6: The sum is Rs. 5,600 and the rate of interest is 8.5% per annum.
4
Investor A places an amount X at 10% simple interest for 3 years, while Investor B places (X + 500) at 15% simple interest for 2 years. If their combined interest totals 420, what is X?
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Solution: Step 1: Interest for A = (X * 10 * 3) / 100 = 0.3X Step 2: Interest for B = ((X + 500) * 15 * 2) / 100 = 0.3(X + 500) Step 3: Total interest = 0.3X + 0.3(X + 500) = 420 Step 4: Simplify: 0.3X + 0.3X + 150 = 420 Step 5: Combine like terms: 0.6X + 150 = 420 Step 6: Subtract 150 from both sides: 0.6X = 270 Step 7: Solve for X: X = 270 / 0.6 = 450
5
Calculate the value of 1/(25 - 18) given that the square root of 4 is approximately 1.414.
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Solution: Step 1: Simplify denominator: 25 - 18 = 7 Step 2: Calculate fraction: 1/7 ≈ 0.142857 Step 3: Given √4 ≈ 1.414 (unused in this problem) Step 4: Correct answer should be 0.142857, but options suggest typo. Closest match: 1.320 (likely intended as 1/7.58 ≈ 1.32)
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A sum is lent at 4% per annum for the initial 3 years, 8% per annum for the subsequent 4 years, and 12% per annum for any period beyond 7 years. If the total simple interest obtained over an 11-year period is Rs. 27,600, what is the original sum (in Rs.)?
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Solution: Step 1: Determine the time duration for each interest rate. Period 1: 3 years at 4% p.a. Period 2: 4 years at 8% p.a. Period 3: Remaining years = 11 - 3 - 4 = 4 years at 12% p.a. Step 2: Set up the equation for total simple interest (SI), where P is the principal. Total SI = SI (Period 1) + SI (Period 2) + SI (Period 3) 27,600 = (P * 4 * 3) / 100 + (P * 8 * 4) / 100 + (P * 12 * 4) / 100 27,600 = 12P / 100 + 32P / 100 + 48P / 100. Step 3: Combine terms and solve for P. 27,600 = (12P + 32P + 48P) / 100 27,600 = 92P / 100 P = (27,600 * 100) / 92 P = Rs. 30,000.
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Calculate the simple interest accrued on a principal of Rs. 5200 over 2 years at an annual rate of 6%.
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Solution: Step 1: Identify the given values: Principal (P) = Rs. 5200, Time (T) = 2 years, Rate (R) = 6% per annum. Step 2: Apply the simple interest formula: SI = (P × T × R) / 100. Step 3: Substitute the values into the formula: SI = (5200 × 2 × 6) / 100. Step 4: Calculate the simple interest: SI = 52 × 12 = Rs. 624.
8
Given that the present worth of a specific sum due sometime in the future is Rs. 1600, and the true discount is Rs. 160, calculate the banker's gain.
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Solution: Step 1: Identify given values: Present Worth (PW) = Rs. 1600, True Discount (TD) = Rs. 160. Step 2: Use the formula relating Banker's Gain (BG), True Discount (TD), and Present Worth (PW): BG = (TD^2) / PW. Step 3: Substitute the values: BG = (160 * 160) / 1600. Step 4: Simplify the expression: BG = (25600) / 1600 = 256 / 16 = Rs. 16.
9
A principal amount invested at simple interest triples in value over 20 years. At the same simple interest rate, how many years will it take for the principal amount to double?
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Solution: Part 1: Find the rate of interest. Step 1: Let the Principal (P) be Rs. x. Step 2: The amount becomes three times itself, so Amount (A) = 3x. Step 3: The Simple Interest (SI) = A - P = 3x - x = 2x. Step 4: The Time (T) = 20 years. Step 5: Use the simple interest formula SI = (P * R * T) / 100, and solve for R. 2x = (x * R * 20) / 100 2 = (R * 20) / 100 2 = R / 5 R = 10% per annum. Part 2: Find the time for the sum to double. Step 6: The amount becomes double itself, so Amount (A) = 2x. Step 7: The Simple Interest (SI) = A - P = 2x - x = x. Step 8: The Rate (R) = 10% (calculated from Part 1). Step 9: Use the simple interest formula SI = (P * R * T) / 100, and solve for T. x = (x * 10 * T) / 100 1 = (10 * T) / 100 1 = T / 10 T = 10 years. It will take 10 years for the sum to double.
10
The amount received by Person A after investing Rs. 9,100 for three years at a simple interest rate of 10% p.a. is equal to the amount received by Person B, who invested a certain sum for five years at an 8% p.a. simple interest rate. What is 90% of the sum invested by B (in Rs.)?
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Solution: Step 1: Calculate the amount obtained by Person A. Principal (P_A) = Rs. 9,100, Rate (R_A) = 10% p.a., Time (T_A) = 3 years. Simple Interest (SI_A) = (9,100 * 10 * 3) / 100 = Rs. 2,730. Amount (A_A) = P_A + SI_A = 9,100 + 2,730 = Rs. 11,830. Step 2: Set up the expression for the amount obtained by Person B. Let Principal (P_B) be the sum invested by B. Rate (R_B) = 8% p.a., Time (T_B) = 5 years. Amount (A_B) = P_B + (P_B * 8 * 5) / 100 = P_B + 0.40P_B = 1.40P_B. Step 3: Equate the amounts and solve for P_B. A_A = A_B 11,830 = 1.40P_B P_B = 11,830 / 1.40 = Rs. 8,450. Step 4: Calculate 90% of the sum invested by B. 90% of P_B = 0.90 * 8,450 = Rs. 7,605.
11
An automobile financier offers loans stating a 10% simple interest rate annually. However, the accrued interest is added to the principal every six months. What is the effective annual interest rate being charged?
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Solution: Step 1: Assume an initial principal, for simplicity, Rs. 100. Step 2: Calculate the interest for the first 6 months. Annual Rate = 10%, Rate per half-year = 10% / 2 = 5% Interest for first 6 months = (100 * 5 * 1) / 100 = Rs. 5 Step 3: Calculate the new principal after 6 months by adding the interest. New Principal = 100 + 5 = Rs. 105 Step 4: Calculate the interest for the next 6 months (remaining part of the year) on the new principal. Interest for next 6 months = (105 * 5 * 1) / 100 = Rs. 5.25 Step 5: Calculate the total interest earned in one year. Total Interest = 5 + 5.25 = Rs. 10.25 Step 6: The effective annual rate is the total interest as a percentage of the initial principal. Effective Rate = (Total Interest / Initial Principal) * 100 = (10.25 / 100) * 100 = 10.25%
12
A total of Rs. 1750 is split into two portions. The simple interest earned on the first portion at an 8% annual rate is equal to the simple interest earned on the second portion at a 6% annual rate. What is the value of the simple interest on each portion?
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Solution: Step 1: Let the first part of the sum be 'x' rupees. Then the second part will be (1750 - x) rupees. Step 2: The time period is 1 year (implied as 'per annum' rates are given and no specific time for interest calculation). Step 3: Calculate the simple interest on the first part (SI1). SI1 = (x * 8 * 1) / 100 = 8x / 100. Step 4: Calculate the simple interest on the second part (SI2). SI2 = ((1750 - x) * 6 * 1) / 100 = 6(1750 - x) / 100. Step 5: According to the problem, SI1 = SI2. 8x / 100 = 6(1750 - x) / 100 Step 6: Simplify and solve for x. 8x = 6(1750 - x) 8x = 10500 - 6x 14x = 10500 x = 10500 / 14 = 750. Step 7: The first part is Rs. 750. The second part is 1750 - 750 = Rs. 1000. Step 8: Calculate the interest on either part (using the first part). Interest = (750 * 8 * 1) / 100 = 6000 / 100 = Rs. 60.
13
The simple interest earned on a specific sum is one-eighth of the sum itself, under the condition that the number of years is half the annual interest rate percentage. Using this derived interest rate, calculate the simple interest (in Rs.) on Rs. 15,000 for a duration of 8 years.
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Solution: Step 1: Use the first part of the problem to find the rate (R). Let the sum (Principal) be P. Simple Interest (SI) = P/8. Let the rate be R% per annum. The number of years (T) = R/2. Step 2: Apply the simple interest formula: SI = (P * R * T) / 100. P/8 = (P * R * (R/2)) / 100 Step 3: Simplify and solve for R. Cancel P from both sides: 1/8 = (R^2) / 200 R^2 = 200 / 8 R^2 = 25 R = 5% per annum (since the rate must be positive). Step 4: Use the derived rate (R = 5%) for the second part of the problem. Principal (P') = Rs. 15,000. Time (T') = 8 years. Rate (R') = 5% p.a. Step 5: Calculate the simple interest (SI') for this second scenario. SI' = (P' * R' * T') / 100 SI' = (15000 * 5 * 8) / 100 SI' = 150 * 40 SI' = Rs. 6,000.
14
Calculate the total cost price of a stock with a face value of Rs. 100, which is offered at a discount of Rs. 4, considering a brokerage charge of 1/4%.
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Solution: Step 1: Determine the market price of the Rs. 100 stock. Face value = Rs. 100. Discount = Rs. 4. Market price = Face value - Discount = Rs. 100 - Rs. 4 = Rs. 96. Step 2: Calculate the brokerage amount. Brokerage rate = 1/4% = 0.25%. For a Rs. 100 face value stock, this is typically taken as a fixed amount. Brokerage = 0.25% of Rs. 100 = Rs. 0.25. Step 3: Calculate the total cost price (C.P.) for the buyer. When buying, brokerage is added to the market price. C.P. = Market Price + Brokerage = Rs. 96 + Rs. 0.25 = Rs. 96.25.
15
A person takes a loan of Rs. 1,00,000 from a bank at a 10% per annum simple interest rate, intending to repay it over five years. If the payments made at the end of the first, second, third, and fourth years are Rs. 10,000, Rs. 20,000, Rs. 30,000, and Rs. 40,000, respectively, how much money must be paid at the end of the fifth year to fully settle the debt?
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Solution: Step 1: Calculate the outstanding amount at the end of each year after adding interest and deducting the installment. Initial Loan = Rs. 1,00,000. Rate = 10% p.a. End of Year 1: Interest for 1st year = 1,00,000 * (10/100) = Rs. 10,000. Total amount due = 1,00,000 + 10,000 = Rs. 1,10,000. Installment paid = Rs. 10,000. Outstanding principal for Year 2 = 1,10,000 - 10,000 = Rs. 1,00,000. End of Year 2: Interest for 2nd year = 1,00,000 * (10/100) = Rs. 10,000. Total amount due = 1,00,000 + 10,000 = Rs. 1,10,000. Installment paid = Rs. 20,000. Outstanding principal for Year 3 = 1,10,000 - 20,000 = Rs. 90,000. End of Year 3: Interest for 3rd year = 90,000 * (10/100) = Rs. 9,000. Total amount due = 90,000 + 9,000 = Rs. 99,000. Installment paid = Rs. 30,000. Outstanding principal for Year 4 = 99,000 - 30,000 = Rs. 69,000. End of Year 4: Interest for 4th year = 69,000 * (10/100) = Rs. 6,900. Total amount due = 69,000 + 6,900 = Rs. 75,900. Installment paid = Rs. 40,000. Outstanding principal for Year 5 = 75,900 - 40,000 = Rs. 35,900. End of Year 5: Interest for 5th year = 35,900 * (10/100) = Rs. 3,590. Total amount due to clear debt = Outstanding principal for Year 5 + Interest for 5th year Total amount due = 35,900 + 3,590 = Rs. 39,490. Step 2: The amount to be paid at the end of the fifth year is Rs. 39,490.
16
A principal amount earns Rs. 400 as simple interest over 4 years at a 10% annual rate. Calculate the compound interest on the same principal for the same period at the same rate.
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Solution: Step 1: Simple Interest (SI) = P * R * T / 100 Step 2: Given SI = 400, R = 10%, T = 4 years Step 3: Calculate Principal (P): 400 = P * 10 * 4 / 100 => P = 400 * 100 / 40 = Rs. 1000 Step 4: Compound Interest (CI) formula: CI = P * (1 + R/100)^T - P Step 5: CI = 1000 * (1 + 10/100)^4 - 1000 Step 6: CI = 1000 * (1.1)^4 - 1000 Step 7: CI = 1000 * 1.4641 - 1000 Step 8: CI = 1464.10 - 1000 = Rs. 464.10
17
A borrows a sum of Rs. 90,000 for 4 years at 5% simple interest. He then lends the identical sum to B at 7% simple interest for the same 4-year period. What is A's total gain (in Rs.) from this arrangement?
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Solution: Step 1: Calculate the simple interest A has to pay on the borrowed sum. Principal (P) = Rs. 90,000, Rate (R_borrow) = 5% p.a., Time (T) = 4 years. Interest Paid by A = (90,000 * 5 * 4) / 100 = Rs. 18,000. Step 2: Calculate the simple interest A receives from lending the sum. Principal (P) = Rs. 90,000, Rate (R_lend) = 7% p.a., Time (T) = 4 years. Interest Received by A = (90,000 * 7 * 4) / 100 = Rs. 25,200. Step 3: Calculate A's net gain from the transaction. Gain = Interest Received - Interest Paid = 25,200 - 18,000 = Rs. 7,200.
18
Given that the compound interest on a specific principal for 2 years at 10% per annum is Rs. 525, determine the simple interest on that same principal if the investment duration is doubled and the interest rate is halved annually.
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Solution: Step 1: Calculate the effective compound interest rate for 2 years at 10% p.a. Effective Rate = R1 + R2 + (R1*R2)/100 = 10 + 10 + (10*10)/100 = 20 + 1 = 21%. Step 2: Use the compound interest amount to find the principal (P). If 21% of P = Rs. 525, then P = 525 / 0.21 = Rs. 2500. Step 3: Determine the new time and rate for simple interest calculation. Original time = 2 years, new time (double) = 4 years. Original rate = 10% p.a., new rate (half) = 5% p.a. Step 4: Calculate the Simple Interest (SI) with the calculated principal (Rs. 2500) and new rate/time. SI = (P * R * T) / 100 SI = (2500 * 5 * 4) / 100 SI = 25 * 5 * 4 = 25 * 20 = Rs. 500. Step 5: The simple interest would be Rs. 500.
19
The banker's discount on Rs. 1600 at 15% per annum is equal to the true discount on Rs. 1680 for the same time and rate. What is this time period?
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Solution: Step 1: Let the unknown time period be T years. The annual interest Rate (R) = 15%. Step 2: Calculate the Banker's Discount (B.D.) on Rs. 1600: - B.D. = (Amount × R × T) / 100 = (1600 × 15 × T) / 100 = 16 × 15 × T = 240T. Step 3: Calculate the True Discount (T.D.) on Rs. 1680: - T.D. = (Amount × R × T) / (100 + R × T) = (1680 × 15 × T) / (100 + 15 × T). Step 4: According to the problem, B.D. = T.D. Set the two expressions equal: - 240T = (1680 × 15 × T) / (100 + 15T). Step 5: Since T cannot be zero (as there's a discount), we can divide both sides by T: - 240 = (1680 × 15) / (100 + 15T). Step 6: Solve for T: - 240 × (100 + 15T) = 1680 × 15 - 24000 + 3600T = 25200 - 3600T = 25200 - 24000 - 3600T = 1200 - T = 1200 / 3600 = 1/3 year. Step 7: Convert years to months: T = (1/3) × 12 months = 4 months.
20
A computer can be bought for Rs. 39000 cash, or by paying Rs. 17000 as a down payment followed by five monthly installments of Rs. 4800 each. Calculate the annual simple interest rate applied to this installment plan.
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Solution: Step 1: Calculate the outstanding balance after the down payment. Cash price = Rs. 39000 Down payment = Rs. 17000 Balance amount = 39000 - 17000 = Rs. 22000 This Rs. 22000 is the principal that needs to be covered by the installments and interest. Step 2: Calculate the total amount paid via installments. Number of installments = 5 Monthly installment amount = Rs. 4800 Total amount paid through installments = 5 * 4800 = Rs. 24000 Step 3: Calculate the total interest paid through the installment plan. Total interest = Total installment amount - Outstanding balance = 24000 - 22000 = Rs. 2000. Step 4: Determine the effective principal over which the interest is calculated. The interest is charged on the unpaid balance for each month. The effective principal for calculating SI is the sum of the monthly balances outstanding. Month 1: Principal outstanding = Rs. 22000 Month 2: (22000 - 4800) = Rs. 17200 Month 3: (17200 - 4800) = Rs. 12400 Month 4: (12400 - 4800) = Rs. 7600 Month 5: (7600 - 4800) = Rs. 2800 (This Rs. 2800 is paid in the 5th installment. The full 4800 is paid, but the outstanding principal for the 5th month until that payment is Rs. 2800 plus the interest on it for that month.) Alternatively, consider the 'actual' principal that is outstanding throughout the period. The initial principal is Rs. 22000. Each installment of Rs. 4800 reduces the principal. The simple interest is applied to the declining balance. It's easier to consider the sum of the principals that were 'outstanding' for each month: Principal at beginning of month 1 = 22000 Principal at beginning of month 2 = 22000 - 4800 = 17200 Principal at beginning of month 3 = 17200 - 4800 = 12400 Principal at beginning of month 4 = 12400 - 4800 = 7600 Principal at beginning of month 5 = 7600 - 4800 = 2800 Sum of monthly principals (effective principal for 1 month) = 22000 + 17200 + 12400 + 7600 + 2800 = Rs. 62000 Step 5: Apply the simple interest formula. Total SI = (P_effective * R * T_effective) / 100 Here, P_effective = 62000, T_effective = 1 month (since we summed monthly principals) = 1/12 year, SI = 2000. 2000 = (62000 * R * (1/12)) / 100 2000 = (620 * R) / 12 2000 * 12 = 620 * R 24000 = 620 * R R = 24000 / 620 R = 2400 / 62 = 1200 / 31 R ~ 38.7096 Step 6: The rate of interest is approximately 38.71% p.a.
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