Question 1

₹ 6200 amounts to ₹ 9176 in 4 years at simple interest. If the interest rate is increased by 3% it would amount to how much?

Accepted Answer

Concept & Incremental Simple Interest: When the interest rate increases by a certain percentage, you do not need to calculate the original rate. You only need to calculate the *extra* interest generated by the *extra* rate and add it to the original total amount. Extra Interest = \frac{P × Increase in Rate × T}{100} Step-by-Step Solution: • Given Principal, P = ₹ 6200 • Given Original Amount, A_1 = ₹ 9176 • Given Time, T = 4 years • Given Rate Increase = 3% p.a. • Calculate the Extra Interest generated by the 3% increase: Extra SI = (6200 × 3 × 4/100) Extra SI = 62 × 12 = ₹ 744 • Add the Extra Interest to the Original Amount to find the New Amount: New Amount = 9176 + 744 = ₹ 9920 Exam Strategy & Shortcut: Skip finding the original rate entirely. Multiply the rate increase by the years to get the total extra percentage (3% × 4 = 12%). Then find 12% of 6200, which is 12 × 62 = 744. Add 744 to 9176 to get 9920. Common Pitfall: A massive time-waster is calculating the initial rate first (SI = 2976 \rightarrow R = 12%), then adding 3% to get 15%, and then recalculating the entire amount from scratch. Final Answer: Therefore, the correct answer is ₹ 9920.

Question 2

Simple interest on ₹ 500 for 4 years at 6.25% per annum is equal to the simple interest on ₹ 400 at 5% per annum for a certain period of time. The period of time is

Accepted Answer

Concept & Equating Simple Interests: When two simple interest scenarios are equal, their respective (P × R × T/100) calculations can be set equal to one another to solve for the missing variable. SI_1 = SI_2 \implies (P_1 × R_1 × T_1/100) = (P_2 × R_2 × T_2/100) Step-by-Step Solution: • Scenario 1: P_1 = ₹ 500, T_1 = 4 years, R_1 = 6.25% SI_1 = (500 × 6.25 × 4/100) = 5 × 25 = ₹ 125 • Scenario 2: P_2 = ₹ 400, R_2 = 5%, T_2 = unknown SI_2 = (400 × 5 × T_2/100) = 20 × T_2 • Equate them: 20 × T_2 = 125 T_2 = (125/20) = (25/4) = 6.25 years • Convert to fraction: 6.25 = 6 (1/4) years Exam Strategy & Shortcut: Equate directly and cancel terms without full multiplication: 500 × 4 × 6.25 = 400 × 5 × T_2 2000 × 6.25 = 2000 × T_2 Divide both sides by 2000: T_2 = 6.25. No complex arithmetic is required! Common Pitfall: A student might perform unnecessary, long multiplication on the left side before realizing the right side also contains factors (400 × 5 = 2000) that easily cancel out the left side (500 × 4 = 2000). Final Answer: Therefore, the correct answer is 6 (1/4) years.

Question 3

A sum of ₹ 18750 is left by a will by a father to be divided between the two sons, 12 and 14 years of age, so that when they attain maturity at 18, the amount (principal + interest) received by each at 5 per cent simple interest will be the same. Fi…

Accepted Answer

Concept & Equating Amounts: When a sum is divided into parts such that their final amounts (Principal + Interest) are equal, the ratio of the principals is inversely proportional to (100 + RT) for each part. A = P × (1 + (R × T/100)) Step-by-Step Solution: • Given: Total Sum = ₹ 18,750. Rate (R) = 5% p.a. • Younger son's age = 12 years. Time (T_1) to reach 18 = 18 - 12 = 6 years. • Older son's age = 14 years. Time (T_2) to reach 18 = 18 - 14 = 4 years. • Calculation: Let the younger son's share be x. The older son's share is (18750 - x). • Amount for younger son = x + (x × 5 × 6/100) = x + 0.30x = 1.30x • Amount for older son = (18750 - x) + ((18750 - x) × 5 × 4/100) = (18750 - x) × 1.20 • Equate the amounts: • 1.30x = 1.20(18750 - x) • 1.3x = 22500 - 1.2x • 2.5x = 22500 • x = (22500/2.5) = 9000 • Younger son's share = ₹ 9,000. • Older son's share = 18750 - 9000 = ₹ 9,750. Exam Strategy & Shortcut: Use the direct ratio formula for equal amounts: P_1 : P_2 = (100 + R_2 T_2) : (100 + R_1 T_1). P_1 : P_2 = (100 + 5 × 4) : (100 + 5 × 6) = 120 : 130 = 12 : 13. Younger son's share (P_1) = (12/25) × 18750 = 9000. Older son's share = (13/25) × 18750 = 9750. Common Pitfall: Students often mistakenly equate the simple interests instead of the total amounts, ignoring the "+ principal" part of the condition. Final Answer: Therefore, the correct answer is ₹ 9000, ₹ 9750.

Question 4

A sum of money becomes \frac{7}{6} of itself in 3 years at a certain rate of simple interest. The rate per annum is

Accepted Answer

Concept & Amount in Fractions: When a sum "becomes" a fraction of itself, that fraction represents the Amount (A). Simple Interest (SI) is the difference between Amount and Principal: SI = A - P Step-by-Step Solution: • Given: Amount (A) = (7/6)P, Time (T) = 3 years. • Calculation: First, calculate the Simple Interest (SI): • SI = (7/6)P - P = ((7/6) - 1)P = (1/6)P • Now, plug SI into the simple interest formula: • SI = (P × R × T/100) • (1/6)P = (P × R × 3/100) • Cancel P from both sides: • (1/6) = (3R/100) • Solve for R: • 100 = 18R • R = (100/18) = (50/9) • Convert to a mixed fraction: • (50/9) = 5(5/9)% Exam Strategy & Shortcut: Let Principal (P) be 6 units. Then Amount (A) is 7 units. Interest earned = 7 - 6 = 1 unit. This 1 unit of interest is earned in 3 years. Interest per year = (1/3). Rate % = (\frac{Interest per year}{P}) × 100 = ((\frac{1/3)}{6}) × 100 = (1/18) × 100 = (50/9) = 5(5/9)%. Common Pitfall: A common mistake is using (7/6)P as the Simple Interest rather than the Amount, which results in heavily inflated and incorrect rate calculations. Final Answer: Therefore, the correct answer is 5(5/9)%.

Question 5

Simple interest on a certain amount is $\frac{9}{16}$ of the principal. If the numbers representing the rate of interest in percent and time in years be equal, then time, for which the principal is lent out, is

Accepted Answer

Concept & Equal Rate and Time: When the numerical value of Time (T) equals the numerical value of Rate (R), we can substitute R with T in the standard Simple Interest formula: SI = (P × T × T/100) = (P × T^2/100) Step-by-Step Solution: • Given: Simple Interest (SI) = (9/16) of Principal (P), and R = T. • Calculation: Substitute the known values into the modified formula: • (9/16)P = (P × T^2/100) • Divide both sides by P: • (9/16) = (T^2/100) • Multiply by 100 to solve for T^2: • T^2 = (900/16) • Take the square root of both sides: • T = \sqrt{(900/16)} = (30/4) = (15/2) • Convert the improper fraction to a mixed fraction: • T = 7(1/2) years. Exam Strategy & Shortcut: Whenever SI = (a/b) × P and R = T, the formula simplifies to: T = \sqrt{(a/b) × 100} Here, T = \sqrt{(9/16) × 100} = (30/4) = 7.5 = 7(1/2) years. Common Pitfall: Forgetting to take the square root of the denominator as well as the numerator, or calculating for Rate (R) but forgetting that it has the same numerical value as Time (T) and stopping too early. Final Answer: Therefore, the correct answer is 7(1/2) years.

Question 6

Two equal sums of money were lent at simple interest at 11% p.a. for $3\frac{1}{2}$ years and $4\frac{1}{2}$ years respectively. If the difference in interests for two periods was ₹ 412.50, then each sum is:

Accepted Answer

Concept & Difference in Time: When identical sums are invested at the same rate but for different time periods, the difference in the simple interest earned is solely due to the interest earned during the difference in time. \Delta SI = (P × R × \Delta T/100) Step-by-Step Solution: • Given: P_1 = P_2 = P, Rate (R) = 11%, T_1 = 3.5 years, T_2 = 4.5 years, Difference in Interest = ₹ 412.50. • Calculation: Find the difference in the time periods (\Delta T): • \Delta T = 4.5 - 3.5 = 1 year. • This means the extra interest of ₹ 412.50 was generated in exactly 1 year on the principal P at 11%. • Apply the simple interest formula for this 1 year difference: • 412.50 = (P × 11 × 1/100) • Multiply by 100: • 41250 = 11P • Solve for P: • P = (41250/11) = 3750. Exam Strategy & Shortcut: Observe that the time difference is exactly 1 year. This implies that 11% of the principal is exactly equal to the difference in interest. 11% of P = 412.50 P = 412.50 × (100/11) = 3750. This skips writing out the full equation twice and subtracting. Common Pitfall: Calculating the total interest for 3.5 years and 4.5 years as two separate long equations creates unnecessary room for arithmetic errors. Use the difference in time instead. Final Answer: Therefore, the correct answer is ₹ 3750.

Question 7

What will be the ratio of simple interest earned by certain amount at the same rate of interest for 6 years and that for 9 years ?

Accepted Answer

Concept & Formula: Simple interest is directly proportional to the time period when the principal and rate of interest remain constant. SI = (Principal × Rate × Time/100) (SI_1/SI_2) = (Time_1/Time_2) Step-by-Step Solution: 1. Let the principal be P and the rate of interest be R. 2. Simple Interest for 6 years (SI_1) = (P × R × 6/100) 3. Simple Interest for 9 years (SI_2) = (P × R × 9/100) 4. Required Ratio = (SI_1/SI_2) = (\frac{P × R × 6/100)}{(P × R × 9/100)} 5. Simplify the fraction: (6/9) = (2/3) 6. The ratio is 2 : 3. Since this is not present in options (a), (b), or (c), the answer is "None of these". Exam Strategy & Shortcut: Since principal and rate are constant, skip writing the full formula. The ratio of the simple interests will simply be the ratio of the time periods: 6:9, which simplifies to 2:3. Common Pitfall: A common mistake is selecting "Data inadequate" because the principal and rate are not given. However, since we are only asked for a ratio, the unknown constants cancel out. Final Answer: Therefore, the correct answer is None of these.

Question 8

A sum of money at a simple rate of interest $i_1$, doubles in 5 years. At another simple rate of interest $i_2$, it becomes three times in 12 years. Then, the two rates of interest $i_1$ and $i_2$ respectively are

Accepted Answer

Concept & Simple Interest: Let principal be P. Case 1: Amount = 2P \Rightarrow S.I. = 2P - P = P Case 2: Amount = 3P \Rightarrow S.I. = 3P - P = 2P R = (100 × S.I./P × T) Step-by-Step Solution: • Case 1: S.I. = P, T = 5 years. i_1 = ((100 × P/P × 5))% = 20% • Case 2: S.I. = 2P, T = 12 years. i_2 = ((100 × 2P/P × 12))% = (200/12)% = (50/3)% = 16(2/3)% Exam Strategy & Shortcut: Use formula R = (100(n - 1)/T). i_1 = (100(2 - 1)/5) = 20%. i_2 = (100(3 - 1)/12) = (200/12) = 16(2/3)%. Common Pitfall: Mixing up the formulas for simple and compound interest. The problem specifies simple interest. Final Answer: Therefore, the correct answer is 20%, 16\(2/3)%.

Question 9

A sum of money at simple interest amounts to ₹ 815 in 3 years and to ₹ 854 in 4 years. The sum is

Accepted Answer

Concept & Simple Interest: Simple interest is constant for each year. By finding the difference between the amounts for different time periods, we can find the interest for the intervening period. Amount = Principal + Simple Interest Step-by-Step Solution: • Amount in 4 years = Principal + S.I. for 4 years = ₹ 854 • Amount in 3 years = Principal + S.I. for 3 years = ₹ 815 • Subtracting the two equations gives S.I. for 1 year = ₹ 854 - ₹ 815 = ₹ 39 • Now, calculate the S.I. for 3 years: ₹ 39 × 3 = ₹ 117 • Principal (Sum) = Amount in 3 years - S.I. for 3 years • Principal = ₹ 815 - ₹ 117 = ₹ 698 Exam Strategy & Shortcut: Find the 1-year interest directly by subtracting the amounts (854 - 815 = 39). Then subtract 3 years of interest from the 3-year amount: 815 - (3 × 39) = 698. Common Pitfall: Mistakenly using the amount as the principal or calculating the interest for the wrong number of base years. Final Answer: Therefore, the correct answer is ₹ 698.

Question 10

A sum of money lent out at simple interest amounts to ₹ 720 after 2 years and to ₹ 1020 after a further period of 5 years. The sum is (S.S.C., 2004)

Accepted Answer

Concept & Simple Interest: The difference in the amounts over a certain period gives the simple interest for that specific duration. Simple Interest = (Principal × Rate × Time/100) Step-by-Step Solution: • The problem states the amount is ₹ 1020 after a "further period of 5 years". This means the total time is 2 + 5 = 7 years. • Amount in 7 years = ₹ 1020 • Amount in 2 years = ₹ 720 • S.I. for 5 years (which is 7 - 2) = ₹ 1020 - ₹ 720 = ₹ 300 • S.I. for 1 year = (300/5) = ₹ 60 • S.I. for 2 years = ₹ 60 × 2 = ₹ 120 • Principal = Amount in 2 years - S.I. for 2 years = ₹ 720 - ₹ 120 = ₹ 600 Exam Strategy & Shortcut: Recognize "further period" means the interest earned in those specific 5 years is 1020 - 720 = 300. So, yearly interest is 60. Subtract 2 years of interest (120) from the 2-year amount (720) to get 600. Common Pitfall: Assuming ₹ 1020 is the amount after 5 years total, rather than after a *further* 5 years (7 years total). Final Answer: Therefore, the correct answer is ₹ 600.

Question 11

A sum of money amounts to ₹ 5200 in 5 years and to ₹ 5680 in 7 years at simple interest. The rate of interest per annum is : (S.S.C., 2007)

Accepted Answer

Concept & Simple Interest Rate Calculation: Once the principal and the interest for a specific period are known, the rate of interest can be calculated using the fundamental simple interest formula. Rate = (100 × S.I./Principal × Time) Step-by-Step Solution: • Amount in 7 years = ₹ 5680 • Amount in 5 years = ₹ 5200 • S.I. for 2 years (7 - 5) = ₹ 5680 - ₹ 5200 = ₹ 480 • S.I. for 1 year = (480/2) = ₹ 240 • S.I. for 5 years = ₹ 240 × 5 = ₹ 1200 • Principal = Amount in 5 years - S.I. for 5 years = ₹ 5200 - ₹ 1200 = ₹ 4000 • Now, calculate the rate: Rate = (100 × 240/4000 × 1) • Rate = (24000/4000) = 6% Exam Strategy & Shortcut: Determine 1-year interest: (5680 - 5200/2) = 240. Find Principal: 5200 - (5 × 240) = 4000. Rate percentage is the 1-year interest over Principal: (240/4000) × 100 = 6%. Common Pitfall: Forgetting to calculate the principal first and trying to use the amount directly in the rate formula. Final Answer: Therefore, the correct answer is 6%.

Question 12

A sum becomes its double in $10$ years. Find the annual rate of simple interest.

Accepted Answer

Concept & Sum Multipliers: When a sum becomes "n" times itself, the simple interest earned is equal to (n-1) times the original principal. R = ((n - 1) × 100/T) Step-by-Step Solution: - Let the principal sum be P. - The sum doubles, so the Amount (A) = 2P. - Simple Interest (SI) = Amount - Principal = 2P - P = P. - Time (T) = 10 years. - Using the standard SI formula: - SI = (P × R × T/100) - P = (P × R × 10/100) - 1 = (R × 10/100) - 1 = (R/10) - R = 10% Exam Strategy & Shortcut: If money doubles, it means the interest earned is exactly 100% of the principal. To find the rate, simply divide this 100% by the number of years: (100/10) = 10%. Common Pitfall: A common error is to set the interest equal to 2P instead of the amount being 2P, leading to double the actual interest rate. Final Answer: Therefore, the correct answer is 10%.

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