Question 1

One can purchase a flat from a house building society for ₹ 55000 cash or on the terms that he should pay ₹ 4275 as cash down payment and the rest in three equal instalments. The society charges interest at the rate of 16% per annum compounded half-…

Accepted Answer

Concept & Half-Yearly Compounding Instalments: If interest is compounded half-yearly, the annual rate must be halved, and the instalments are paid per half-year. The principal to be compounded is the cash price minus the down payment. P = \sum_{t=1}^{n} (x/(1 + \frac{R_{half)}{100})^t} Step-by-Step Solution: • Given: Cash price = ₹ 55000, Down payment = ₹ 4275. • Principal balance P = 55000 - 4275 = 50725. • Rate R = 16% p.a., so half-yearly rate R_{half} = (16%/2) = 8% = (8/100) = (2/25). • The discounting factor is 1 + (2/25) = (27/25). Let each instalment be x. • 50725 = (x/\frac{27){25}} + (x/(\frac{27){25})^2} + (x/(\frac{27){25})^3} • 50725 = x [ (25/27) + (625/729) + (15625/19683) ] • Take LCM of 27, 729, and 19683, which is 19683. • 50725 = x [ (25 × 729 + 625 × 27 + 15625/19683) ] • 50725 = x [ (18225 + 16875 + 15625/19683) ] • 50725 = x [ (50725/19683) ] • The 50725 on both sides cancels out perfectly! • x = 19683. Exam Strategy & Shortcut: In problems structured for competitive exams, the sum inside the bracket's numerator will frequently perfectly cancel out the initial principal. Notice the denominator 27^3 = 19683. It is a very strong hint that the answer will just be the denominator itself. Common Pitfall: Using the full 16% rate as the compounding factor instead of halving it to 8% for half-yearly periods is the most common error in this specific type of question. Final Answer: Therefore, the correct answer is ₹ 19683.

Question 2

What annual payment will discharge a debt of ₹ 1025 due in 2 years at the rate of 5% compound interest?

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Concept & Instalments for Compound Interest: The present value of a debt paid in equal instalments is calculated using the formula for compound interest present worth. P = (x/(1 + \frac{R){100})} + (x/(1 + \frac{R){100})^2} where P is the principal borrowed, x is the annual instalment, and R is the rate of interest. Step-by-Step Solution: • Given: Debt (Principal) P = 1025, Rate R = 5%, Time T = 2 years. • Let the annual instalment be x. • 1025 = (x/1 + \frac{5){100}} + (x/(1 + \frac{5){100})^2} • 1025 = (x/1.05) + (x/(1.05)^2) = (x/\frac{21){20}} + (x/(\frac{21){20})^2} • 1025 = x((20/21) + (400/441)) • 1025 = x((420 + 400/441)) = x((820/441)) • x = (1025 × 441/820) • x = (205 × 441/164) = (5 × 441/4) = (2205/4) = 551.25 Exam Strategy & Shortcut: Recognize that the sum of the series for 2 years at 5% is (820/441). The instalment is simply P × (441/820). Notice that 820 and 1025 are both multiples of 205 (which is 41 × 5), simplifying the calculation quickly to 5 × (441/4). Common Pitfall: A common mistake is treating the "debt due in 2 years" as the future amount rather than the present principal, leading to incorrect applications of the instalment formula. Final Answer: Therefore, the correct answer is ₹ 551.25.

Question 3

A sum of money is borrowed and paid back in two annual instalments of ₹ 882 each allowing 5% compound interest. The sum borrowed was

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Concept & Present Value of Instalments: The sum borrowed (Principal) is the total present value of all future instalments discounted at the given interest rate. P = (x/(1 + \frac{R){100})} + (x/(1 + \frac{R){100})^2} Step-by-Step Solution: • Given: Instalment x = 882, Rate R = 5%. • The fractional rate is 1 + (5/100) = 1 + (1/20) = (21/20). • P = (882/\frac{21){20}} + (882/(\frac{21){20})^2} • P = 882 × (20/21) + 882 × (400/441) • Calculate the first term: 882 \div 21 = 42, so 42 × 20 = 840. • Calculate the second term: 882 \div 441 = 2, so 2 × 400 = 800. • Sum the present values: P = 840 + 800 = 1640. Exam Strategy & Shortcut: Recognize that 882 is exactly 2 × 441, which is a strong hint since 21^2 = 441. This makes the cancellation immediate: 2 × 21 × 20 + 2 × 400 = 840 + 800 = 1640. Common Pitfall: Students sometimes incorrectly multiply the instalment amount by the number of years and then subtract interest, effectively confusing simple interest arithmetic with compound discounting. Final Answer: Therefore, the correct answer is ₹ 1640.

Question 4

The compound interest on a certain sum of money for 2 years at 10% per annum is ₹ 525. The simple interest on the same sum of money for double the time at half the rate percent per annum is?

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Concept & Relating Compound and Simple Interest: First, find the principal using the effective compound interest rate for 2 years. Then apply the simple interest formula with the newly adjusted time and rate constraints. Effective\ CI% = 2R + (R^2/100) SI = (P × R × T/100) Step-by-Step Solution: • Given: CI for 2 years at 10% = 525. • Effective CI rate for 2 years at 10% = 10 + 10 + (10 × 10/100) = 21%. • So, 21% of Principal = 525. • P = (525/0.21) = (52500/21) = 2500. • New Conditions: Double the time T = 2 × 2 = 4 years. Half the rate R = 10 / 2 = 5%. • SI = (2500 × 5 × 4/100) • SI = 25 × 20 = 500. Exam Strategy & Shortcut: The effective CI rate is 21%. The effective SI rate for the new conditions (4 years × 5%) is 20%. Therefore, the SI is just (20/21) of the CI. 525 × (20/21) = 25 × 20 = 500. This skips finding the principal entirely! Common Pitfall: Calculating the principal correctly but forgetting to adjust both the time (double) and the rate (half) before calculating the simple interest. Final Answer: Therefore, the correct answer is ₹ 500.

Question 5

The difference between CI and SI on a certain sum of money for 3 years at 5% p.c.p.a. is ₹ 122. Find the sum invested.

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Concept & Difference Between CI and SI for 3 Years: For a period of exactly 3 years, the difference between compound interest and simple interest on a principal P at rate R% is given by a specific derived formula. Difference = P((R/100))^2((300 + R/100)) Step-by-Step Solution: • Given: Difference D = 122, Rate R = 5%, Time = 3 years. • Substitute the given values directly into the 3-year difference formula: • 122 = P × ((5/100))^2 × ((300 + 5/100)) • 122 = P × ((1/20))^2 × ((305/100)) • 122 = P × ((1/400)) × ((61/20)) • 122 = P × (61/8000) • Solve for P: • P = (122 × 8000/61) • Since 61 × 2 = 122, the equation simplifies: • P = 2 × 8000 = 16000 Exam Strategy & Shortcut: Memorize the ratio fraction for 3 years at 5%. For R = 5%, the fractional rate is 1/20. The multiplier for the difference is (3(1/20) + (1/20)^2/1) = (61/8000) of the principal. Knowing the multiplier (61/8000) allows you to immediately write 122 = P × (61/8000) and solve it in seconds. Common Pitfall: Using the 2-year difference formula P(R/100)^2 instead of the 3-year formula, which would yield a completely incorrect principal. Final Answer: Therefore, the correct answer is ₹ 16000.

Question 6

What is the difference between the compound interests on ₹ 5000 for 1 year at 4% per annum compounded yearly and half yearly?  [SSC—Junior Associates (Pre.) Exam, 2016]

Accepted Answer

Concept & Effect of Compounding Frequency: When interest is compounded half-yearly for one year, the principal effectively earns "interest on the interest" from the first half-year during the second half-year. This "interest on interest" is exactly the difference between annual and half-yearly compounding for a single year. Difference = P (\frac{R_{half}}{100})^2 Step-by-Step Solution: • Given: Principal P = 5000, Annual Rate R = 4%, Time = 1 year. • Yearly Compounding: Rate = 4%, Time = 1 year. Interest = (5000 × 4 × 1/100) = 200. • Half-Yearly Compounding: Rate per half-year = (4%/2) = 2%. Number of periods = 1 × 2 = 2. Interest for 1st half-year = (5000 × 2/100) = 100. Interest for 2nd half-year = Interest on principal + Interest on previous interest 2nd\ half\ interest = 100 + ((100 × 2/100)) = 100 + 2 = 102. Total Half-Yearly CI = 100 + 102 = 202. • Difference: Difference = 202 - 200 = 2. Exam Strategy & Shortcut: For exactly one year, the difference between yearly and half-yearly compounding is simply the interest calculated on the first half-year's interest. First half-year interest = 2% of 5000 = 100. Interest on that 100 at 2% = 2% of 100 = 2. The answer is just 2. Common Pitfall: A time-consuming mistake is manually applying the full compound interest formula A = P(1+R/100)^N for both scenarios, which introduces unnecessary arithmetic steps and risk of calculation errors. Final Answer: Therefore, the correct answer is 2.

Question 7

Directions: Each of the questions given below consists of a question followed by three statements. You have to study the question and the statements and decide which of the statement(s) is/are necessary to answer the question.  What is the rate of i…

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Concept & Evaluating Multiple Statements: When given three statements in Data Sufficiency, test each independently first, then in pairs. The goal is to find the minimum information required to find the rate of interest (R). Step-by-Step Solution: 1. Evaluate III alone: An amount doubles in 5 years at simple interest. This means SI = P. Formula: P = (P × R × 5/100) \Rightarrow R = 20%. Statement III alone is sufficient. 2. Evaluate I and II together: Statement I: SI per annum = ₹ 5300. Thus, (P × R/100) = 5300. Statement II: CI - SI for 2 yrs = ₹ 1060. Thus, P((R/100))^2 = 1060. Divide II by I: (P(\frac{R/100))^2}{(P × R/100)} = (1060/5300) \Rightarrow (R/100) = 0.2 \Rightarrow R = 20%. Statements I and II together are also sufficient. 3. Conclusion: The rate can be found using either III alone, or I and II together. Exam Strategy & Shortcut: Always check the simplest statement first. Statement III is a classic standard property (doubling in 5 years always means R = (100/5) = 20%). Finding this immediately narrows down the options. Common Pitfall: Stopping after finding one sufficient statement (like III) and choosing "Only III" without checking if combinations of the remaining statements also work. Final Answer: Therefore, the correct answer is Only III or I and II.

Question 8

Directions: Each of the questions given below consists of a question followed by three statements. You have to study the question and the statements and decide which of the statement(s) is/are necessary to answer the question.  What will be the diff…

Accepted Answer

Concept & Solving with Two Unknowns: To calculate the difference between CI and SI for 8 years, we need both the Principal (P) and the Rate of interest (R). We have two unknowns, so we need two independent mathematical relationships (equations) involving P and R to solve for them. Step-by-Step Solution: 1. Analyze the Statements: - Statement I: SI for 3 years = 750. This means SI for 1 year = 250. Equation: (P × R/100) = 250 \Rightarrow PR = 25000. - Statement II: CI for 2 years = 512.50. Equation: P[(1 + (R/100))^2 - 1] = 512.50. - Statement III: R = 5%. 2. Evaluate Combinations: - I and III: We know PR = 25000 and R = 5. We can easily find P (P = 5000). With P and R known, we can find the 8-year difference. (Sufficient) - II and III: We know P[(1.05)^2 - 1] = 512.50. We can solve for P. With P and R known, we can find the 8-year difference. (Sufficient) - I and II: From I, SI for 1 year is 250. For 2 years, SI is 500. We are given CI for 2 years is 512.50. The difference for 2 years is 12.50. Using the 2-year difference formula: (R/100) = \frac{CI - SI for 2 years}{SI for 1 year} = (12.50/250) = 0.05 \Rightarrow R = 5%. Once R is known, we can use I (PR = 25000) to find P. (Sufficient) 3. Conclusion: Any combination of two statements provides enough information to determine both P and R. Exam Strategy & Shortcut: Recognize the structural dependencies. You need P and R. - I gives P × R. - II gives a relation of P and R via compound growth. - III gives R directly. Any pair of these distinct relationships forms a solvable system of equations. Common Pitfall: Wasting time calculating the actual 8-year difference. Data sufficiency questions only ask *if* the data is sufficient to find the answer, not what the answer actually is. Final Answer: Therefore, the correct answer is Any two of I, II and III.

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