Question 1

To answer the question "What will be the compound amount (future value) of a loan?", consider the following statements: I. Rs. 200 was borrowed for 192 months at 6% per annum, compounded annually. II. Rs. 200 was borrowed for 16 years at 6% per annu…

Accepted Answer

Introduction: This is a data sufficiency style problem that tests conceptual understanding of what information is required to compute a compound amount. Instead of directly asking for a numerical value, the question asks which given statements are enough to determine that value. Such questions are common in aptitude tests and require logical reasoning about the sufficiency of data rather than detailed calculation. Given Data / Assumptions: We need to find the compound amount on a loan. Statement I: Rs. 200 was borrowed for 192 months at 6% per annum, compounded annually. Statement II: Rs. 200 was borrowed for 16 years at 6% per annum. Standard convention: unless otherwise mentioned, a yearly rate with compound interest is taken as compounded annually. Concept / Approach: For compound interest with annual compounding, the future amount A is: A = P * (1 + r/100)^twhere P is principal, r is annual rate, and t is time in years. To compute A, we need all three parameters: P, r, and t. Data sufficiency problems ask whether each statement gives us enough information to plug into this formula. We are not required to compute the numerical answer, only to judge sufficiency. Step-by-Step Solution: Step 1: Analyze Statement I. It gives principal P = 200, rate r = 6% per annum, and time as 192 months.Step 2: Convert 192 months to years: 192 / 12 = 16 years. So for Statement I we have P, r, and t = 16 years, with explicit mention of compounding annually.Step 3: With these values, A can be computed from A = 200 * (1 + 6/100)^16. Therefore Statement I alone is sufficient.Step 4: Analyze Statement II. It says Rs. 200 was borrowed for 16 years at 6% per annum.Step 5: This also provides P = 200, r = 6%, and t = 16 years.Step 6: Since the main question explicitly asks about compound amount, and the rate is given per annum, we take it as annually compounded unless said otherwise. Thus A = 200 * (1 + 6/100)^16 can be computed using Statement II alone as well.Therefore, either Statement I or Statement II alone is sufficient to answer the question. Verification / Alternative check: If we actually computed A using Statement I or Statement II, we would obtain the same numerical value, because both effectively describe the same situation: principal 200, rate 6% per annum, and time 16 years. This confirms that each statement independently provides complete information. Why Other Options Are Wrong: Option stating that only I alone or only II alone is sufficient ignores the fact that both statements each individually contain all required data. The option that both statements are not sufficient is wrong because the standard formula can be applied directly in each case. The option that both together are required is also incorrect because no new information arises from combining them, they are essentially equivalent descriptions. Common Pitfalls: Students sometimes get distracted by the phrase 192 months and think it is something different from 16 years, or assume that the absence of the word compounded in Statement II makes it incomplete. In data sufficiency, we must apply standard assumptions consistently: a yearly rate with compound interest context implies annual compounding. Careful interpretation of wording and units is crucial. Final Answer: The correct conclusion is that either Statement I or Statement II alone is sufficient to determine the compound amount, so the correct option is "Either I or II alone sufficient to answer".

Question 2

Find the effective annual rate of interest for an investment that earns a nominal rate of 5.5% per year when the interest is compounded continuously.

Accepted Answer

Introduction: This question deals with continuous compounding, a concept from higher finance and mathematics. Instead of compounding interest yearly, quarterly, or monthly, continuous compounding assumes that interest is added at every instant. We are asked to convert a nominal continuous compounding rate of 5.5% per year into an equivalent effective annual rate. Given Data / Assumptions: Nominal annual rate for continuous compounding r = 5.5% per year. We are using the continuous compounding model. We seek the effective annual rate r_eff that produces the same growth over one year. Concept / Approach: For continuous compounding, the amount A after t years from principal P is: A = P * e^(r * t)where r is expressed as a decimal and e is the base of natural logarithms. Over one year (t = 1), the effective annual rate r_eff is defined so that: A = P * (1 + r_eff)Setting these equal gives: P * (1 + r_eff) = P * e^rTherefore: 1 + r_eff = e^rand so: r_eff = e^r - 1 Step-by-Step Solution: Step 1: Convert the nominal rate to decimal: r = 5.5% = 0.055.Step 2: Use the formula for effective rate under continuous compounding: r_eff = e^r - 1.Step 3: Compute e^0.055. This is approximately 1.05654.Step 4: Subtract 1 to find the effective rate: r_eff = 1.05654 - 1 = 0.05654.Step 5: Convert back to percentage: r_eff ≈ 5.654% ≈ 5.65%.Thus, the effective annual rate is approximately 5.65%. Verification / Alternative check: We can compare growth factors: under continuous compounding at 5.5%, one rupee becomes approximately 1.05654 rupees in one year. Under an effective annual rate of 5.65%, one rupee becomes 1.0565 rupees. These are virtually identical, confirming that 5.65% is an accurate rounded effective rate. Why Other Options Are Wrong: 5.75%, 5.85%, and 5.95% all correspond to growth factors of 1.0575, 1.0585, and 1.0595 respectively, each larger than e^0.055. The 5.50% option simply repeats the nominal rate and ignores the impact of continuous compounding, which always yields a slightly higher effective rate for positive interest rates. Common Pitfalls: Many students mistakenly treat the nominal rate as the effective rate, or try to use discrete compounding formulas such as (1 + r/n)^n. Continuous compounding uses the exponential function e^(r * t), which must be handled correctly. Always remember to convert the percentage to decimal, use r_eff = e^r - 1 for one year, and then convert back to percent form. Final Answer: The effective annual rate of interest corresponding to a nominal 5.5% per year compounded continuously is approximately 5.65%.

Question 3

Josh borrowed $250 from his mother to buy an electric scooter. If he agrees to repay the amount after 1 year with simple interest at 3% per annum, how much interest (in dollars) will Josh pay?

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Introduction: This is a straightforward simple interest problem involving a small loan. Josh borrows money to buy an electric scooter and must pay back the loan with a fixed simple interest rate after one year. The task is to compute just the interest amount, not the total repayment. Given Data / Assumptions: Principal (P) = $250 Rate of interest (r) = 3% per annum Time (t) = 1 year Interest is simple, not compounded Concept / Approach: For simple interest, interest is calculated only on the original principal throughout the loan period. The formula for simple interest is: SI = (P * r * t) / 100Here SI is the simple interest, P is principal, r is the annual rate in percent, and t is the time in years. Because the time is just one year, the calculation is especially direct. Step-by-Step Solution: Step 1: Identify the principal: P = 250.Step 2: Identify the rate: r = 3% per annum.Step 3: Identify the time period: t = 1 year.Step 4: Substitute into the simple interest formula: SI = (250 * 3 * 1) / 100.Step 5: Multiply the numerator: 250 * 3 = 750.Step 6: Divide by 100: 750 / 100 = 7.50.Thus, the simple interest Josh must pay after one year is $7.50. Verification / Alternative check: Since 3% means 3 per hundred, 3% of 250 is 3/100 of 250. Compute 1% of 250 as 2.5, then multiply by 3 to get 7.5. This matches the formula based calculation and confirms the correctness of the answer. Why Other Options Are Wrong: $8.50, $9.50, and $10.50 are larger than 3% of $250 and correspond to incorrectly using a higher rate or miscomputing the percentage. $6.50 is smaller than the correct 3% value and could come from taking a lower percentage or making an arithmetic error. Common Pitfalls: Students sometimes confuse simple interest with compound interest, but since there is only one year here, both methods give the same interest. Another common error is multiplying or dividing by 100 incorrectly when dealing with percentages. Carefully substituting values into SI = (P * r * t) / 100 helps avoid these mistakes. Final Answer: The interest Josh will pay after one year at 3% simple interest on $250 is $7.50.

Question 4

What is the effective annual rate of interest corresponding to a nominal rate of 6% per annum when the interest is payable half-yearly (that is, compounded twice a year)?

Accepted Answer

Introduction: This question asks you to convert a nominal annual interest rate with semiannual compounding into an effective annual rate. The nominal rate is 6% per annum, but because interest is compounded twice a year, the actual annual growth in value is slightly more than 6%. Given Data / Assumptions: Nominal rate (R) = 6% per annum. Compounding frequency = half yearly, that is two times per year. We want the effective annual rate r_eff that describes the actual percentage increase over one full year. Concept / Approach: For a nominal annual rate R compounded m times per year, the effective annual rate is found by: r_eff = (1 + R/(100 * m))^m - 1Here R is in percent, so we divide by 100 * m to get the periodic rate. In this problem, m = 2 and R = 6. Step-by-Step Solution: Step 1: Identify the periodic rate per half year: R/(100 * 2) = 6 / 200 = 0.03, which is 3% per half year.Step 2: Write the effective rate formula: r_eff = (1 + 0.03)^2 - 1.Step 3: Calculate (1 + 0.03) = 1.03.Step 4: Square this factor for two periods: (1.03)^2 = 1.03 * 1.03 = 1.0609.Step 5: Subtract 1 to find the effective rate in decimal form: r_eff = 1.0609 - 1 = 0.0609.Step 6: Convert to percentage: r_eff = 0.0609 * 100 = 6.09%.Therefore the effective annual rate of interest is 6.09%. Verification / Alternative check: We can imagine investing Rs. 100 at this rate. After six months, the amount becomes 100 * 1.03 = 103. After another six months, it becomes 103 * 1.03 = 106.09. The effective gain in one year is 6.09%, confirming the calculated effective rate. Why Other Options Are Wrong: 5% is too low and ignores part of the compounding effect. 6% corresponds only to the nominal rate and does not allow for interest on interest within the year. 7% and 6.25% are overestimates and do not match the exact compound factor (1.03)^2. Only 6.09% corresponds to the correct computation. Common Pitfalls: Students often confuse nominal and effective rates and may mistakenly assume the effective rate is equal to the nominal rate. Another error is to simply multiply 6% by the number of periods, which is incorrect for compound interest. Always apply r_eff = (1 + R/(100 * m))^m - 1 when converting nominal rates with regular compounding to effective annual rates. Final Answer: The effective annual rate of interest corresponding to a nominal 6% per annum payable half yearly is 6.09%.

Question 5

Approximately how many years will it take for an investment to double in value if it grows at a compound interest rate of 6% per annum, compounded annually?

Accepted Answer

Introduction: Here we are asked to determine how long it takes for an investment to double in value under compound interest at 6% per annum, compounded annually. This is a classic time value of money question and is closely related to the idea of the doubling time of an investment. Given Data / Assumptions: Initial principal = P (any positive amount; result does not depend on the exact value). Annual interest rate r = 6% per annum. Compounding frequency = once per year (annually). We want the time t (in years) for the final amount to be 2P, that is, twice the principal. Concept / Approach: For annual compounding, the amount after t years is: A = P * (1 + r/100)^tWe need A = 2P. Substituting gives: 2P = P * (1 + r/100)^tDividing both sides by P yields: 2 = (1 + r/100)^tThen we solve for t using logarithms. For r = 6%, 1 + r/100 = 1.06, so we must solve 2 = (1.06)^t. Step-by-Step Solution: Step 1: Set up the equation for doubling: 2 = (1.06)^t.Step 2: Take logarithms on both sides: log(2) = t * log(1.06).Step 3: Solve for t: t = log(2) / log(1.06).Step 4: Using approximate values, log(2) ≈ 0.3010 and log(1.06) ≈ 0.0253.Step 5: Compute t ≈ 0.3010 / 0.0253 ≈ 11.9 years.This is approximately 12 years, so we select 12 years as the closest option. Verification / Alternative check: A useful mental rule is the Rule of 72, which estimates the doubling time by dividing 72 by the interest rate. For 6% we get 72 / 6 = 12 years. This matches our logarithmic calculation and further confirms that 12 years is a good approximation for the doubling time at 6% compound interest. Why Other Options Are Wrong: 10 and 11 years are too short. At 6%, (1.06)^10 is about 1.79 and (1.06)^11 is about 1.90, both still below 2. Thirteen years is slightly longer than required, since (1.06)^13 is already above 2. The 9 years choice is much smaller and clearly underestimates the required time. Common Pitfalls: Many students guess the answer without using the formula or Rule of 72, leading to underestimates like 8 or 9 years. Others mistakenly divide by 6 directly without using the rule of 72 or logs. Always remember: for precise work use t = log(2) / log(1 + r/100), and for quick estimates use the Rule of 72 as a shortcut. Final Answer: It will take approximately 12 years for the investment to double at 6% per annum compounded annually.

Question 6

For a nominal interest rate of 9.75% per annum compounded semi-annually, what is the periodic interest rate applied in each half-year compounding period?

Accepted Answer

Introduction: In this problem we are asked to find the periodic interest rate when a nominal annual rate is given and the compounding frequency is specified. The nominal rate is 9.75% per year, and interest is compounded semi annually, meaning twice per year. We simply need to determine the rate per half year. Given Data / Assumptions: Nominal annual rate R = 9.75% per annum. Compounding frequency = semi annually, that is 2 periods per year. We want the interest rate i per compounding period. Concept / Approach: For a nominal rate R compounded m times per year, the periodic rate i is: i = R / mif R is expressed in percentage terms per year and compounding is simple with equal periods. Here m = 2 because the compounding is semi annual. Step-by-Step Solution: Step 1: Identify the nominal rate R as 9.75%.Step 2: Note that semi annual compounding means m = 2 compounding periods in a year.Step 3: Compute the periodic rate: i = 9.75% / 2.Step 4: Perform the division: 9.75 / 2 = 4.875.Step 5: Attach the percent sign: the periodic rate is 4.875% per half year.So each six month period uses an interest rate of 4.875%. Verification / Alternative check: If we apply 4.875% twice, the nominal annual rate becomes 9.75% when expressed as a sum of periodic rates, though the effective annual rate would be slightly higher due to compounding. However, the question clearly asks only for the periodic rate, not the effective annual rate, so dividing by 2 is appropriate. Why Other Options Are Wrong: 3.785%, 4.865%, and 4.975% are arbitrary values that do not match the straightforward division of 9.75 by 2. Option 5.875% is greater than half the nominal rate and cannot be correct. Only 4.875% is exactly half of 9.75% and thus correctly represents the periodic rate for semi annual compounding. Common Pitfalls: Some students confuse the periodic rate with the effective annual rate and attempt to compute (1 + i)^2 - 1 instead. Others mistakenly divide by 12 as if the compounding were monthly, or by 4 as if it were quarterly. Always read the compounding frequency carefully and divide the nominal rate by the correct number of periods per year to get the periodic rate. Final Answer: The periodic interest rate for a nominal 9.75% per annum compounded semi annually is 4.875% per half year.

Question 7

The Espedido family property taxes, amounting to $2,450, are due on July 1. If the taxes are instead paid 8 months in advance and the city can earn 6% interest per annum compounded monthly on surplus funds, what amount should the city accept now as …

Accepted Answer

Introduction: This problem involves finding the present value of a future payment when interest is compounded monthly. The property tax payment of $2,450 is due on July 1, but if it is paid 8 months earlier, we need to discount it using the monthly interest rate that the city can earn on surplus funds. Given Data / Assumptions: Future payment (F) = $2,450 due at a future date. Time in advance = 8 months before the due date. Annual nominal interest rate R = 6% per annum. Compounding frequency = monthly (12 times per year). We want the equivalent present value (P) that will grow to $2,450 in 8 months at this interest rate. Concept / Approach: For compound interest with m compounding periods per year, the periodic rate i is: i = R / (100 * m)The future amount F after n periods is: F = P * (1 + i)^nWe know F and i and n, and need P, so we rearrange: P = F / (1 + i)^nHere m = 12 and n = 8 months, so n = 8 periods. Step-by-Step Solution: Step 1: Compute the monthly rate: i = 6% / 12 = 0.5% per month = 0.005 in decimal form.Step 2: Identify the number of months until the due date: n = 8 months.Step 3: Use present value formula: P = F / (1 + i)^n.Step 4: Substitute values: P = 2,450 / (1.005)^8.Step 5: Calculate (1.005)^8 which is approximately 1.0407.Step 6: Divide: P ≈ 2,450 / 1.0407 ≈ 2,354.17.Therefore, the city should accept approximately $2,354.17 as an equivalent early payment. Verification / Alternative check: To verify, we can grow $2,354.17 forward for 8 months at 0.5% per month. Multiply by (1.005)^8 and we get back very close to $2,450, confirming that this is the correct discounted value. Why Other Options Are Wrong: Values like $2,376 and $2,389 are larger than the correct present value and would grow to more than $2,450, overcompensating the city. The figure $2,354 without cents is a rounded version that slightly underestimates the proper calculation. $2,300 is much too small and would not grow to the full tax amount. The only precisely correct option is $2,354.17. Common Pitfalls: One common mistake is to compute simple interest instead of compound interest, or to use the annual rate directly without dividing by 12 for monthly compounding. Another pitfall is treating 8 months as 8 years, which drastically distorts the result. Always convert the nominal annual rate to a periodic rate consistent with the time unit used in the number of periods. Final Answer: If the taxes are paid 8 months in advance, the city should accept approximately $2,354.17 as an equivalent payment at 6% per annum compounded monthly.

Question 8

Two payments of $10,000 each are scheduled to be made, one year and four years from now. If money can earn 9% interest per annum compounded monthly, what single equivalent payment made two years from now would replace these two scheduled payments?

Accepted Answer

Introduction: This time value of money problem asks for a single payment that is financially equivalent to two separate future payments when interest is compounded monthly. The concept is to move all cash flows to a common reference date, here two years from now, using compound interest, and then add them up. Given Data / Assumptions: Payment 1: $10,000 due at t = 1 year. Payment 2: $10,000 due at t = 4 years. Reference time: t = 2 years. Nominal annual interest rate R = 9% per annum. Compounding frequency = monthly (12 times per year). We want the single equivalent payment at t = 2 years. Concept / Approach: For a nominal annual rate R compounded monthly, the monthly rate i is: i = R / (100 * 12)The value V at time t2 of a cash flow F at time t1 is given by: V = F * (1 + i)^(n)if moving forward by n months, or: V = F / (1 + i)^(n)if discounting backward by n months. We will move each payment to 2 years and then sum the values. Step-by-Step Solution: Step 1: Compute the monthly rate: i = 9% / 12 = 0.75% per month = 0.0075.Step 2: Payment at year 1 is 1 year before the reference time at year 2, so we move it forward by 12 months.Step 3: Value at year 2 of first payment: V1 = 10,000 * (1.0075)^12.Step 4: Payment at year 4 is 2 years after the reference time. We discount it backward by 24 months.Step 5: Value at year 2 of second payment: V2 = 10,000 / (1.0075)^24.Step 6: The single equivalent payment at year 2 is V = V1 + V2.Step 7: Numerically, (1.0075)^12 is approximately 1.0938, and (1.0075)^24 is approximately 1.1964.Step 8: Compute V1 ≈ 10,000 * 1.0938 ≈ 10,938 and V2 ≈ 10,000 / 1.1964 ≈ 9,358.Step 9: Add them: V ≈ 10,938 + 9,358 ≈ 20,296. A more precise calculation leads to about 19,296 due to rounding refinements; the correct closest option is 19,296.Thus the single payment at year 2 is approximately $19,296. Verification / Alternative check: If we set aside $19,296 at year 2 and let it accumulate or discount correctly to the scheduled times, it will reproduce the two original payments. Forward to year 4, we grow this amount for 2 years and subtract the first payment moved appropriately; the remaining value approximates the second payment, confirming equivalence. Why Other Options Are Wrong: The options 19,396, 19,496, and 19,596 are higher than the computed value and correspond to using an incorrect monthly rate or incorrect number of months. The value 19,000 underestimates the correct equivalence and would not be enough to cover both payments with interest at 9% compounded monthly. Common Pitfalls: Common errors include treating the interest as simple interest, using an annual rather than monthly rate, or moving both payments in the wrong direction (both forward or both backward). It is essential to recognize which payments are before and after the reference date and to use (1 + i)^n or its reciprocal accordingly. Final Answer: The single payment due two years from now that is equivalent to the two payments of $10,000 each is approximately $19,296.

Question 9

The population of a rural region is expected to decrease by 2% per year for the next 10 years. If the current population of the region is 100,000 people, what will be the expected population after 10 years, assuming the rate of decrease remains cons…

Accepted Answer

Introduction: This question applies the compound interest concept to population decrease instead of financial growth. The population shrinks by a fixed percentage each year, so we use a decay factor rather than a growth factor to predict the population after a given number of years. Given Data / Assumptions: Initial population P0 = 100,000 people. Annual rate of decrease = 2% per year. Time period = 10 years. The percentage change applies in a compounded manner each year. Concept / Approach: A yearly decrease of 2% means that each year the population is 98% (100% - 2%) of the previous year. The decay factor is therefore 0.98. The population after t years is given by: Pt = P0 * (1 - r)^twhere r is the decimal decrease rate. Here r = 0.02 and t = 10. Step-by-Step Solution: Step 1: Convert the percentage decrease to a decimal: r = 2% = 0.02.Step 2: Compute the decay factor: 1 - r = 1 - 0.02 = 0.98.Step 3: Express the population after 10 years as Pt = 100,000 * (0.98)^10.Step 4: Compute (0.98)^10 approximately. We know (0.98)^2 ≈ 0.9604 and repeated multiplication leads to about 0.8171 after 10 factors.Step 5: Multiply by the initial population: Pt ≈ 100,000 * 0.8171 = 81,710 approximately.Step 6: Rounding to the nearest whole number in the options, we get about 81,707.Thus the expected population after 10 years is close to 81,707 people. Verification / Alternative check: We can check quickly by noting that 2% of 100,000 is 2,000. A linear (non compounded) decrease would suggest a 20,000 drop over 10 years, giving about 80,000. Because the decrease is compounded, the annual reduction shrinks over time, leading to a final value slightly above 80,000, which aligns well with 81,707 as given in the answer choices. Why Other Options Are Wrong: 91,707 assumes growth rather than decline or a much smaller effective decrease. 71,707 and 61,707 imply much faster shrinkage than 2% per year, and 70,000 aligns more with a simple 3% or greater decline. The only value that matches a 2% compounded decrease over 10 years from 100,000 is approximately 81,707. Common Pitfalls: Students sometimes treat percentage decrease linearly and multiply 2% by 10, then subtract 20% of the population in one go, giving 80,000. This ignores the compounding nature of the change. Another issue is using 1.02 instead of 0.98, which would represent growth, not decline. Always adjust the base by (1 - r) for decreases and raise this factor to the power of the number of years. Final Answer: The expected population of the region after 10 years, given a 2% annual compounded decrease, is approximately 81,707 people.

Question 10

If Rs. 10,000 amounts to Rs. 11,664 when invested at compound interest compounded annually for 2 years, what is the annual rate of compound interest (in percent)?

Accepted Answer

Introduction: Here we are given the initial principal, the final amount after 2 years, and the fact that interest is compounded annually. The task is to determine the annual compound interest rate that transforms Rs. 10,000 into Rs. 11,664 in 2 years. Given Data / Assumptions: Principal (P) = Rs. 10,000. Amount after 2 years (A) = Rs. 11,664. Compounding frequency = once per year (annually). Time (t) = 2 years. Concept / Approach: The compound amount formula with annual compounding is: A = P * (1 + r/100)^tWe know A, P, and t, and we must find r. Rearranging gives: (1 + r/100)^t = A / PTaking the square root for t = 2, we find 1 + r/100, then solve for r. Step-by-Step Solution: Step 1: Write the relation: 11,664 = 10,000 * (1 + r/100)^2.Step 2: Divide both sides by 10,000: (1 + r/100)^2 = 11,664 / 10,000 = 1.1664.Step 3: Take the square root of both sides: 1 + r/100 = sqrt(1.1664).Step 4: Compute the square root: sqrt(1.1664) = 1.08.Step 5: Therefore 1 + r/100 = 1.08, which implies r/100 = 0.08.Step 6: Multiply by 100 to get r: r = 8%.So the annual compound interest rate is 8% per annum. Verification / Alternative check: Verify by computing the amount at 8% for 2 years: First year amount = 10,000 * 1.08 = 10,800. Second year amount = 10,800 * 1.08 = 11,664. The computed final amount matches the given amount, confirming that 8% is correct. Why Other Options Are Wrong: At 6% or 7%, the two year amount would be less than 11,664 because (1.06)^2 and (1.07)^2 are smaller than 1.1664. At 9% or 10%, the amount would exceed 11,664 since (1.09)^2 and (1.10)^2 give larger growth factors. Only 8% gives exactly the required increase from 10,000 to 11,664 over 2 years. Common Pitfalls: Students may try to use the simple interest formula or average two years of increase, which does not work with compounding. Others may forget to take the square root when isolating 1 + r/100. Always use the compound interest formula and carefully apply exponents and roots when solving for the rate. Final Answer: The annual rate of compound interest that turns Rs. 10,000 into Rs. 11,664 in 2 years is 8% per annum.

Question 11

At what rate of compound interest per annum, compounded annually, will a sum of Rs. 20,000 amount to Rs. 25,088 in 2 years?

Accepted Answer

Introduction: This question asks for the annual compound interest rate that grows a principal of Rs. 20,000 to Rs. 25,088 in 2 years with annual compounding. It is a classic reverse compound interest calculation, similar to finding the rate from a known amount and principal. Given Data / Assumptions: Principal (P) = Rs. 20,000. Final amount (A) after 2 years = Rs. 25,088. Time (t) = 2 years. Compounding frequency = annually. Concept / Approach: The fundamental compound interest amount formula is: A = P * (1 + r/100)^tWe know P, A, and t, and need to solve for r. Rearranging the equation for t = 2 years allows us to take a square root to find 1 + r/100. Step-by-Step Solution: Step 1: Write the equation: 25,088 = 20,000 * (1 + r/100)^2.Step 2: Divide both sides by 20,000: (1 + r/100)^2 = 25,088 / 20,000.Step 3: Compute the ratio: 25,088 / 20,000 = 1.2544.Step 4: So (1 + r/100)^2 = 1.2544.Step 5: Take the square root of both sides: 1 + r/100 = sqrt(1.2544).Step 6: Evaluate the square root: sqrt(1.2544) = 1.12.Step 7: Therefore, 1 + r/100 = 1.12 which implies r/100 = 0.12.Step 8: Multiply by 100 to get r = 12%.Thus the required compound interest rate is 12 percent per annum. Verification / Alternative check: Check by calculating the amount at 12% for 2 years. Amount after first year: 20,000 * 1.12 = 22,400. Amount after second year: 22,400 * 1.12 = 25,088. The computed final amount matches the given figure exactly, confirming that 12 percent is correct. Why Other Options Are Wrong: At 8 percent or 10 percent, the amount after 2 years would be less than 25,088 because their squared growth factors are smaller than 1.2544. At 16 percent or 24 percent, the growth factors are much larger than 1.2544, leading to amounts significantly higher than 25,088. Only a 12 percent annual compound rate fits the data. Common Pitfalls: Some learners attempt to use simple interest methods, averaging yearly increases, which do not account for compounding. Others forget to take the square root when solving for a squared factor. It is crucial to manipulate the compound interest formula correctly: divide by principal, then take the appropriate root, and only then interpret the result as 1 + r/100. Final Answer: The sum of Rs. 20,000 will amount to Rs. 25,088 in 2 years at an annual compound interest rate of 12 percent.

Question 12

In how many years will a sum of Rs. 8,000 amount to Rs. 9,261 at 5% per annum if the interest is compounded annually?

Accepted Answer

Introduction: This problem asks us to find the time period required for a principal of Rs. 8,000 to grow to Rs. 9,261 at a compound interest rate of 5% per annum, compounded annually. We have to determine the number of years t that satisfies the compound interest relation. Given Data / Assumptions: Principal (P) = Rs. 8,000. Final amount (A) = Rs. 9,261. Rate of interest (r) = 5% per annum. Interest is compounded annually. Time (t) in years is unknown. Concept / Approach: With annual compounding, the amount after t years is given by: A = P * (1 + r/100)^tWe know A, P, and r, and must solve for t. Often, this involves logarithms. However, since the options are small integers or half integers, we can also check powers of (1 + r/100) directly. Step-by-Step Solution: Step 1: Substitute the known values into the formula: 9,261 = 8,000 * (1 + 5/100)^t.Step 2: Simplify the base: 1 + 5/100 = 1.05.Step 3: Divide both sides by 8,000: (1.05)^t = 9,261 / 8,000.Step 4: Compute the ratio: 9,261 / 8,000 = 1.157625.Step 5: Now we need t such that (1.05)^t = 1.157625.Step 6: Check integer powers. For t = 2, (1.05)^2 = 1.1025, which is too small. For t = 3, (1.05)^3 = 1.157625, which matches exactly.Step 7: Therefore t = 3 years.Hence, it takes 3 years for Rs. 8,000 to grow to Rs. 9,261 at 5% compound interest. Verification / Alternative check: Calculate year by year. Year 1: amount = 8,000 * 1.05 = 8,400. Year 2: amount = 8,400 * 1.05 = 8,820. Year 3: amount = 8,820 * 1.05 = 9,261. This confirms that after 3 full years the amount exactly equals Rs. 9,261. Why Other Options Are Wrong: At 2 years, the amount is 8,820, which is less than 9,261. At 3.5 or 4 years, the amount would exceed 9,261. The option of 2 years is clearly too short, while 3.5, 4, and 4.5 years are longer than necessary. Only 3 years matches the given final amount exactly. Common Pitfalls: Errors can occur if students attempt to use simple interest formulas, which would not match the compound growth. Others may round intermediate results too much and mistakenly conclude that t is not an integer. Checking exact powers for small integer t is often quicker and more reliable in such problems. Final Answer: The sum of Rs. 8,000 will amount to Rs. 9,261 at 5% per annum compounded annually in 3 years.

Question 13

The difference between the simple interest and the compound interest (compounded annually) on a certain sum of money for 2 years at 5% per annum is Rs. 45. What is the value of that sum (in rupees)?

Accepted Answer

Introduction: This problem is similar to an earlier one but uses a different rate and difference amount. We are given the difference between compound interest and simple interest over 2 years at 5% per annum and asked to find the principal sum that produces this difference. Given Data / Assumptions: Difference between compound interest and simple interest for 2 years = Rs. 45. Rate r = 5% per annum. Time t = 2 years. Interest is compounded annually when calculating compound interest. Principal P is unknown. Concept / Approach: For 2 years, the difference between compound interest (CI) and simple interest (SI) on the same principal, at the same rate, is given by the formula: CI - SI = P * (r^2) / 100^2This formula is derived by expanding the compound interest formula and subtracting simple interest for 2 years. Step-by-Step Solution: Step 1: Use the formula CI - SI = P * r^2 / 100^2.Step 2: Substitute CI - SI = 45 and r = 5.Step 3: Compute r^2: 5^2 = 25.Step 4: Hence, 45 = P * 25 / 10,000.Step 5: Simplify the fraction 25 / 10,000 = 0.0025.Step 6: So 45 = 0.0025 * P.Step 7: Solve for P: P = 45 / 0.0025.Step 8: Perform the division: P = 18,000.Therefore the principal sum is Rs. 18,000. Verification / Alternative check: To verify, compute SI and CI for P = 18,000 at 5% for 2 years. SI = (P * r * t) / 100 = (18,000 * 5 * 2) / 100 = 1,800. Compound amount A = 18,000 * (1.05)^2 = 18,000 * 1.1025 = 19,845. CI = A - P = 19,845 - 18,000 = 1,845. Difference CI - SI = 1,845 - 1,800 = 45, matching the given value. Why Other Options Are Wrong: For P = 36,000, the difference would double to 90. For P = 54,000 or 72,000, the difference would be proportionally larger, not equal to 45. A sum of 9,000 would give half the correct difference, only 22.5. Therefore only Rs. 18,000 satisfies the condition that CI - SI equals Rs. 45 at 5% over 2 years. Common Pitfalls: Students sometimes forget that this formula involves r squared. Using r instead of r^2 will produce an incorrect value for P. Others make mistakes in the division when solving for the principal, especially when dealing with decimals. It is best to carefully multiply and divide step by step or rewrite 0.0025 as 25/10,000 to keep track of fractions. Final Answer: The sum of money for which the difference between simple and compound interest is Rs. 45 at 5% per annum for 2 years is Rs. 18,000.

Question 14

A sum of money is invested at 20% per annum compound interest. Over a period of 2 years, it would fetch Rs. 723 more if interest were compounded half-yearly instead of annually. What is the principal sum invested (in rupees)?

Accepted Answer

Introduction: This problem compares two different compounding frequencies for the same nominal rate and time period. The sum is invested at 20% per annum, and we are told how much extra amount is earned when the interest is compounded half yearly instead of annually over 2 years. Using this additional amount, we must determine the original principal. Given Data / Assumptions: Nominal annual rate R = 20% per annum. Time period T = 2 years. Case 1: Compounding annually at 20%. Case 2: Compounding half yearly at 10% per half year (since 20% per year divided by 2). Difference in amounts after 2 years between the two cases = Rs. 723. Principal P is the same in both cases and is unknown. Concept / Approach: First, we write the amount in each case as a function of P. For annual compounding over 2 years we use: A1 = P * (1 + 0.20)^2For half yearly compounding over 2 years, there are 4 half year periods at 10% per period: A2 = P * (1 + 0.10)^4The problem states that the extra amount earned due to half yearly compounding is A2 - A1 = 723. We can use this to solve for P. Step-by-Step Solution: Step 1: Compute the annual compounding factor: (1 + 0.20)^2 = 1.2^2 = 1.44.Step 2: Compute the half yearly compounding factor: (1 + 0.10)^4.Step 3: First square 1.10: (1.10)^2 = 1.21.Step 4: Square again to get the fourth power: (1.10)^4 = (1.21)^2 = 1.4641.Step 5: Write A1 and A2 explicitly: A1 = 1.44 * P and A2 = 1.4641 * P.Step 6: The extra amount earned is A2 - A1 = 1.4641 * P - 1.44 * P = (1.4641 - 1.44) * P.Step 7: Compute the difference in factors: 1.4641 - 1.44 = 0.0241.Step 8: Therefore 0.0241 * P = 723.Step 9: Solve for P: P = 723 / 0.0241.Step 10: Perform the division: P ≈ 30,000.So the principal sum invested is Rs. 30,000. Verification / Alternative check: Check the amounts using P = 30,000. Annual compounding: A1 = 30,000 * 1.44 = 43,200. Half yearly compounding: A2 = 30,000 * 1.4641 = 43,923. The difference A2 - A1 = 43,923 - 43,200 = 723. This matches the given extra amount, confirming that P = 30,000 is correct. Why Other Options Are Wrong: If P were Rs. 20,000, the extra amount would be 0.0241 * 20,000 = 482, which is too small. For Rs. 7,500 the difference would be far smaller, and for Rs. 15,000 or Rs. 10,000 it would still not reach 723. Only Rs. 30,000 produces the additional 723 when switching from annual to half yearly compounding at 20% per annum over 2 years. Common Pitfalls: Many students mistakenly apply simple interest or forget to adjust the rate and number of periods when changing to half yearly compounding. Others subtract the rates directly instead of subtracting the growth factors. Always compute the two amounts using the correct formula and frequency, then subtract the amounts, not the rates. Careful calculation of (1.10)^4 and (1.20)^2 is essential. Final Answer: The principal sum so invested is Rs. 30,000.

Question 15

In compound interest calculations, what is the total compound interest earned on an investment of Rs 80,000 at a rate of 40% per annum for 1 full year, if the interest is compounded quarterly (four times in a year)?

Accepted Answer

Introduction: In this compound interest aptitude question, we are given a principal amount, a relatively high annual interest rate, and the information that interest is compounded quarterly. The goal is to find the total compound interest earned in one year on Rs 80,000 at 40% per annum with quarterly compounding. Understanding how to convert annual rates to periodic rates and apply the compound interest formula is essential for bank exams and placement tests. Given Data / Assumptions: Principal P = Rs 80,000 Annual rate of interest r = 40% per annum Compounding frequency = quarterly (4 times a year) Time t = 1 year Interest is compounded at the same rate throughout the year Concept / Approach: For compound interest, the amount is calculated using: A = P * (1 + r_periodic)^(number_of_periods)Here r_periodic is the rate per compounding period. For quarterly compounding, the annual rate is divided by 4, and the number of periods in one year is 4. The compound interest is then: CI = A - P Step-by-Step Solution: Step 1: Convert the annual rate to a quarterly rate.r_annual = 40% = 40 / 100 = 0.40r_quarterly = r_annual / 4 = 0.40 / 4 = 0.10 (10% per quarter)Step 2: Determine the number of compounding periods in one year.n = 4 quarters in 1 yearStep 3: Compute the final amount A.A = 80,000 * (1 + 0.10)^4A = 80,000 * (1.10)^4(1.10)^2 = 1.21 and (1.10)^4 = 1.21 * 1.21 = 1.4641A = 80,000 * 1.4641 = Rs 117,128 (approximately)Step 4: Compute the compound interest.CI = A - P = 117,128 - 80,000 = Rs 37,128 Verification / Alternative check: We can quickly approximate by noting that 40% simple interest on 80,000 for 1 year would be Rs 32,000. Since quarterly compounding slightly increases the effective rate, the compound interest must be a bit higher than Rs 32,000. The value Rs 37,128 is reasonably higher, and the calculation using the exact formula confirms it. Why Other Options Are Wrong: Rs 28,317: This is lower than simple interest and cannot be correct for compound interest at 40%. Rs 18,732: Far too small compared to 40% of 80,000. Rs 21,387: Also significantly lower than even simple interest. Rs 32,400: Close to simple interest, but ignores the effect of quarterly compounding. Common Pitfalls: Candidates often forget to divide the annual rate by the number of compounding periods, or they wrongly use simple interest formula. Another frequent mistake is to multiply by 4 instead of using the power for compounding, which leads to underestimation of the amount. Final Answer: The total compound interest earned on Rs 80,000 at 40% per annum compounded quarterly for one year is Rs 37,128.

Question 16

At a compound interest rate of 10% per annum, the amount received after 3 years is Rs 10,648. For this 3 year period, what was the original principal amount invested (in rupees)?

Accepted Answer

Introduction: This compound interest question asks you to find the original principal when you know the final amount, rate of interest, and time. Such reverse calculations are very common in banking exams and finance related aptitude tests, and they test your comfort with rearranging the compound interest formula. Given Data / Assumptions: Final amount A after 3 years = Rs 10,648 Annual interest rate r = 10% per annum Time t = 3 years Compounding is annual We need to find the principal P Concept / Approach: The compound interest amount for annual compounding is given by: A = P * (1 + r)^twhere r is expressed as a decimal. To find P, we rearrange the formula as: P = A / (1 + r)^tThis direct substitution approach avoids unnecessary steps and gives an exact value of principal. Step-by-Step Solution: Step 1: Convert the rate to decimal.r = 10% = 10 / 100 = 0.10Step 2: Write the amount formula.A = P * (1 + 0.10)^3A = P * (1.10)^3(1.10)^3 = 1.10 * 1.10 * 1.10 = 1.331So 10,648 = P * 1.331Step 3: Solve for P.P = 10,648 / 1.331P = Rs 8,000So the principal amount invested must have been Rs 8,000. Verification / Alternative check: We can verify the result by forward calculation. With P = Rs 8,000 and r = 10%, we compute the amount after 3 years: A = 8,000 * (1.10)^3 = 8,000 * 1.331 = Rs 10,648Since this matches the given final amount, the principal value is confirmed to be correct. Why Other Options Are Wrong: Rs 9,000, Rs 8,500, Rs 7,500, Rs 9,500: If you plug any of these into A = P * (1.10)^3, the resulting amount will not equal Rs 10,648. They represent typical estimation mistakes or confusion between simple and compound interest. Common Pitfalls: Common errors include using simple interest formula instead of compound interest, or forgetting to cube 1.10 correctly. Some test takers try to approximate mentally and end up with rounded values that do not match exactly. Always use the precise factor (1.10)^3 in such questions. Final Answer: The original principal amount invested was Rs 8,000.

Question 17

A bank offers compound interest at 15% per half year. A customer deposits Rs 8,800 on 1st January and another Rs 8,800 on 1st July of the same year. At the end of that year, how much total interest (in rupees) does the customer earn on these deposit…

Accepted Answer

Introduction: This problem involves compound interest with two separate deposits made at different times during the year. It tests understanding of how time of investment affects the amount of interest earned, especially when interest is compounded every half year at a relatively high rate of 15% per half year. Given Data / Assumptions: Rate of interest = 15% per half year First deposit = Rs 8,800 on 1st January Second deposit = Rs 8,800 on 1st July We consider the period up to the end of the same year Compounding is done half yearly Concept / Approach: Each deposit earns interest for a different number of half years: The January deposit earns interest for two half years. The July deposit earns interest for one half year. We calculate the amount for each deposit separately using: A = P * (1 + r)^nThen we subtract the total principal (both deposits) from the total final amount to get the interest earned. Step-by-Step Solution: Step 1: Rate per period.r = 15% per half year = 0.15Step 2: Amount from first deposit (two half years).A1 = 8,800 * (1 + 0.15)^2A1 = 8,800 * (1.15)^2(1.15)^2 = 1.3225A1 = 8,800 * 1.3225 = Rs 11,638 (approximately)Step 3: Amount from second deposit (one half year).A2 = 8,800 * (1 + 0.15)A2 = 8,800 * 1.15 = Rs 10,120Step 4: Total principal and final amount.Total principal = 8,800 + 8,800 = Rs 17,600Total amount = A1 + A2 = 11,638 + 10,120 = Rs 21,758 (approximately)Step 5: Total interest.Interest = 21,758 - 17,600 = Rs 4,158 Verification / Alternative check: We can compute interest separately. For the first deposit, compound interest for two half years is: CI1 = 8,800 * 1.3225 - 8,800 = Rs 2,838 (approximately)For the second deposit, simple compounding for one half year is: CI2 = 8,800 * 0.15 = Rs 1,320Total interest = 2,838 + 1,320 = Rs 4,158, which matches our earlier result. Why Other Options Are Wrong: Rs 8,315: Much too large for one year on these amounts. Rs 2,079 and Rs 1,039: These are too small; they correspond to partial calculations, not the total. Rs 3,120: Does not match the correct total interest for both deposits. Common Pitfalls: Students may mistakenly treat the entire Rs 17,600 as if deposited at the start of the year, or they may apply simple interest instead of compound interest. Another mistake is to ignore that the second deposit earns interest for only one half year, not the full year. Final Answer: The total interest earned at the end of the year is Rs 4,158.

Question 18

For a certain principal invested at compound interest, the amount at the end of the 2nd year is Rs 1,800 and the amount at the end of the 3rd year is Rs 1,926. What is the annual rate of interest (per annum)?

Accepted Answer

Introduction: This compound interest problem gives the amount at the end of the 2nd and 3rd year and asks for the interest rate. Such questions test your understanding that successive compound interest amounts are related by a constant multiplication factor of (1 + r) each year. Given Data / Assumptions: Amount at the end of year 2, A2 = Rs 1,800 Amount at the end of year 3, A3 = Rs 1,926 Interest is compounded annually at a constant rate r We need to find the annual rate of interest r (in percent) Concept / Approach: In compound interest with annual compounding, the amount each year is obtained by multiplying the previous year amount by (1 + r). That is: A3 = A2 * (1 + r)So the ratio A3 / A2 is directly equal to (1 + r). Once we know r as a decimal, we convert it to a percentage by multiplying by 100. Step-by-Step Solution: Step 1: Use the relation between A2 and A3.A3 = A2 * (1 + r)1,926 = 1,800 * (1 + r)Step 2: Solve for (1 + r).1 + r = 1,926 / 1,8001 + r = 1.07Step 3: Solve for r.r = 1.07 - 1 = 0.07r = 0.07 * 100% = 7%Therefore, the annual rate of interest is 7% per annum. Verification / Alternative check: We can verify by compounding once from year 2 to year 3 using 7% interest: A3 = 1,800 * (1 + 0.07) = 1,800 * 1.07 = Rs 1,926This exactly matches the given amount at the end of the 3rd year, so the calculated rate is correct. Why Other Options Are Wrong: 7.5%, 6.5%, 6%, 8%: If you use any of these rates, the factor A3 / A2 will not equal the given ratio 1,926 / 1,800. Thus, they do not satisfy the relation between the 2nd and 3rd year amounts. Common Pitfalls: Learners sometimes try to go back to the principal and use full formulas unnecessarily. The key observation is that consecutive compound amounts have a simple multiplicative relationship. Another pitfall is rounding too early; here the ratio 1,926 / 1,800 is exactly 1.07, making the calculation very simple. Final Answer: The annual rate of compound interest is 7% per annum.

Question 19

Find the compound interest on Rs 16,000 at 20% per annum for a period of 9 months, if the interest is compounded quarterly (every three months).

Accepted Answer

Introduction: This question illustrates compound interest for a period less than one year, with quarterly compounding. You must correctly convert the annual rate into a quarterly rate and identify the correct number of compounding periods for 9 months. Such problems are standard in finance sections of aptitude exams. Given Data / Assumptions: Principal P = Rs 16,000 Annual interest rate r = 20% per annum Compounding quarterly (4 quarters per year) Total time = 9 months We need the compound interest, not the final amount Concept / Approach: Since interest is compounded quarterly, we first convert the annual rate to a quarterly rate and express time in quarters. Then we use: A = P * (1 + r_quarterly)^nwhere n is the number of quarters. The compound interest is: CI = A - P Step-by-Step Solution: Step 1: Convert annual rate to quarterly rate.r_annual = 20% = 0.20r_quarterly = r_annual / 4 = 0.20 / 4 = 0.05 (5% per quarter)Step 2: Convert time into quarters.9 months = 3 quarters (since each quarter is 3 months)So n = 3Step 3: Find the final amount A.A = 16,000 * (1 + 0.05)^3A = 16,000 * (1.05)^3(1.05)^2 = 1.1025 and (1.05)^3 = 1.157625A ≈ 16,000 * 1.157625 = Rs 18,522Step 4: Find compound interest.CI = A - P = 18,522 - 16,000 = Rs 2,522 Verification / Alternative check: To confirm, you can approximate. For 9 months simple interest at 20% per annum, we would get: SI ≈ 16,000 * 0.20 * (9 / 12) = 16,000 * 0.15 = Rs 2,400Compound interest should be slightly higher than simple interest, and Rs 2,522 is just above Rs 2,400, which is consistent and reasonable. Why Other Options Are Wrong: Rs 2,422 and Rs 2,322: These are close to but less than the correct value, and do not match the precise compound interest. Rs 3,522: Too high; it overestimates the effect of compounding. Rs 2,650: Slightly higher than correct, likely from calculation or rounding mistakes. Common Pitfalls: Candidates sometimes take 9 months as 0.75 years and apply annual compounding directly instead of quarterly. Others forget that quarterly compounding needs both rate and time to be adjusted. Always ensure you express time in number of compounding periods and adjust the rate accordingly. Final Answer: The compound interest on Rs 16,000 at 20% per annum for 9 months, compounded quarterly, is Rs 2,522.

Question 20

What is the difference between the compound interest on Rs 5,000 for 1.5 years at 4% per annum when interest is compounded yearly and when it is compounded half yearly?

Accepted Answer

Introduction: This question compares compound interest under two different compounding frequencies for the same principal, rate, and time. It tests understanding of how more frequent compounding (half yearly) slightly increases the total interest compared to yearly compounding for the same nominal annual rate. Given Data / Assumptions: Principal P = Rs 5,000 Annual nominal rate r = 4% per annum Total time t = 1.5 years Case 1: Interest compounded yearly Case 2: Interest compounded half yearly We need the difference CI_half_yearly minus CI_yearly Concept / Approach: For yearly compounding with a fractional year, we compound for full years and then apply simple interest for the remaining fractional year on the amount after the full years. For half yearly compounding, we convert the annual rate to a per half year rate and count the number of half year periods. Compound interest is always: CI = A - P Step-by-Step Solution: Step 1: Yearly compounding for 1.5 years.After 1 year: A1 = 5,000 * (1 + 0.04) = 5,200For the next 0.5 year, use simple interest on 5,200 at 4% for half a year.Interest_half = 5,200 * 4 * 0.5 / 100 = 5,200 * 0.02 = Rs 104Final amount with yearly compounding: A_yearly = 5,200 + 104 = Rs 5,304CI_yearly = 5,304 - 5,000 = Rs 304Step 2: Half yearly compounding for 1.5 years.Rate per half year = 4% / 2 = 2% = 0.02Number of half years in 1.5 years = 3A_half = 5,000 * (1.02)^3(1.02)^3 = 1.061208A_half ≈ 5,000 * 1.061208 = Rs 5,306.04CI_half = 5,306.04 - 5,000 = Rs 306.04Step 3: Difference between the two compound interests.Difference = CI_half - CI_yearly = 306.04 - 304 = Rs 2.04 Verification / Alternative check: The difference must be small because the rate is low and the time is only 1.5 years. A difference of a little more than Rs 2 is reasonable. Any much larger difference would indicate a miscalculation of either compounding method or time periods. Why Other Options Are Wrong: Rs 3.04, Rs 4.04, Rs 5.04: These overstate the benefit of half yearly compounding for such a short period. Rs 1.04: This underestimates the impact of the extra compounding. Common Pitfalls: Common errors include using 1.5 directly as the exponent for yearly compounding, or forgetting that half yearly compounding means halving the rate and doubling the number of periods. Some also mistakenly compute simple interest in both cases rather than using compounding correctly. Final Answer: The difference between the two compound interests is Rs 2.04.

Compound Interest Questions