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Compound Interest Quiz 11 (10 MCQs)

This set of multiple-choice questions evaluates understanding of compound interest, including annual, monthly, and daily compounding techniques. It tests the ability to calculate present and future values, apply interest rate conversions, and use the compound interest formula for interest calculation and investment growth.

Quiz Instructions

Select an option to see the correct answer instantly.

1. The amount of money which should be invested at 3.5% per annum compound interest, compounding monthly, if you require $ 35 000 in three years time, is closest to

Correct Answer: Correct answer is: (E) $ 31 516.

Exam Relevance: CFA, CPA, GMAT, GRE

Difficulty: Moderate

Concept notes: The formula for compound interest with monthly compounding is given by: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] where: - $ A $ is the future value of the investment. - $ P $ is the principal amount (the initial investment). - $ r $ is the annual interest rate (decimal). - $ n $ is the number of times interest is compounded per year. - $ t $ is the time the money is invested for in years. In this problem, we need to find the principal amount $ P $ given the future value $ A = 35000 $, the annual interest rate $ r = 0.035 $, the number of times interest is compounded per year $ n = 12 $, and the time $ t = 3 $ years.

Common Mistakes: A common mistake is to use the simple interest formula instead of the compound interest formula, or to forget to adjust the interest rate and time for monthly compounding.

Explanations: To find the principal amount $ P $, we rearrange the compound interest formula to solve for $ P $: \[ P = \frac{A}{\left(1 + \frac{r}{n}\right)^{nt}} \] Substitute the given values: \[ P = \frac{35000}{\left(1 + \frac{0.035}{12}\right)^{12 \times 3}} \] Calculate the monthly interest rate: \[ \frac{0.035}{12} = 0.00291667 \] Calculate the exponent: \[ 12 \times 3 = 36 \] Now, calculate the term inside the parentheses: \[ 1 + 0.00291667 = 1.00291667 \] Raise this to the power of 36: \[ (1.00291667)^{36} \approx 1.111616 \] Finally, calculate $ P $: \[ P = \frac{35000}{1.111616} \approx 31516.04 \] Thus, the principal amount $ P $ is approximately $31,516.

Option Analysis:

Option A: $ 29 886. This is incorrect because it does not account for the compounding effect over the three years.
Option B: $ 30 000. This is incorrect because it does not account for the compounding effect over the three years.
Option C: $ 31 325. This is incorrect because it does not fully account for the compounding effect over the three years.
Option D: $ 31 674. This is incorrect because it slightly overestimates the effect of compounding.
Option E: $ 31 516. This is the correct answer as it accurately accounts for the compounding effect over the three years.

2. Chu has a credit card debt of $ 3456 that will accrue interest at the rate of 17.4% per annum compounding daily. After one month (30 days), the amount he will owe will be closest to:

Correct Answer: Correct answer is: (C) $ 3505.80.

Exam Relevance: CFA, CPA, GMAT, GRE

Difficulty: Moderate

Concept notes: The concept of compound interest involves calculating interest on the initial principal and the accumulated interest over time. When interest is compounded daily, the interest is calculated each day and added to the principal, which then becomes the new principal for the next day's interest calculation.

Common Mistakes: A common mistake is to assume that the interest is compounded annually or monthly instead of daily, leading to an incorrect calculation of the final amount.

Explanations: To find the amount owed after 30 days with daily compounding interest, we use the formula for compound interest: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] where: - $ A $ is the amount of money accumulated after n periods, including interest. - $ P $ is the principal amount (initial amount of money). - $ r $ is the annual interest rate (decimal). - $ n $ is the number of times interest is compounded per year. - $ t $ is the time the money is invested or borrowed for, in years. Given: - $ P = 3456 $ - $ r = 0.174 $ - $ n = 365 $ (since interest is compounded daily) - $ t = \frac{30}{365} $ (since we are calculating for 30 days) Substitute the values into the formula: \[ A = 3456 \left(1 + \frac{0.174}{365}\right)^{365 \times \frac{30}{365}} \] \[ A = 3456 \left(1 + \frac{0.174}{365}\right)^{30} \] \[ A = 3456 \left(1 + 0.0004767123\right)^{30} \] \[ A = 3456 \left(1.0004767123\right)^{30} \] \[ A \approx 3456 \times 1.01427 \] \[ A \approx 3505.80 \] Thus, the amount owed after 30 days is approximately $3505.80.

Option Analysis:

Option A: This option assumes no interest is accrued, which is incorrect.
Option B: This option assumes a very small interest accrual, which is incorrect.
Option C: This option correctly calculates the amount owed after 30 days with daily compounding interest.
Option D: This option assumes a much higher interest accrual, which is incorrect.
Option E: This option assumes an even higher interest accrual, which is incorrect.

Mnemonic: Daily Interest, Daily Growth

3. Compounding technique is .....

Correct Answer: Correct answer is: (C) Exactly opposite of discounting technique.

Exam Relevance: CFA, CPA, FRM, MBA Finance Exams

Difficulty: Moderate

Concept notes: Compounding and discounting are two fundamental concepts in finance and investment. Compounding involves calculating the future value of an investment by adding interest to the principal and then earning interest on the new total. Discounting, on the other hand, involves calculating the present value of a future sum of money by subtracting interest from the future value.

Common Mistakes: A common misunderstanding is that compounding and discounting are similar processes. However, they are actually opposite processes: compounding calculates future value, while discounting calculates present value.

Explanations: Compounding and discounting are exactly opposite techniques. Compounding calculates the future value of an investment by adding interest to the principal and then earning interest on the new total. Discounting, conversely, calculates the present value of a future sum of money by subtracting interest from the future value. Therefore, the correct answer is that compounding is exactly opposite of discounting.

Option Analysis:

Option A: Slightly different from discounting technique. This is incorrect because compounding and discounting are not just slightly different; they are fundamentally opposite processes.
Option B: Same as discounting technique. This is incorrect because compounding and discounting are not the same; they are opposite processes.
Option C: Exactly opposite of discounting technique. This is correct because compounding calculates future value, while discounting calculates present value, making them opposite processes.
Option D: More or less equal to discounting technique. This is incorrect because compounding and discounting are not equal; they are opposite processes.

4. The opposite of compounding is known as discounting. The discount factor can be thought of as the reciprocal of the interest rate and is the factor by which a future value must be multiplied to get the present value.

Correct Answer: Correct answer is: (A) True.

Exam Relevance: CFA, CPA, GMAT, GRE

Difficulty: Moderate

Concept notes: In the context of compound interest, the discount factor is the reciprocal of the interest rate and is used to find the present value of a future sum of money.

Common Mistakes: A common misunderstanding is that discounting and compounding are unrelated processes, when in fact they are inverses of each other.

Explanations: The discount factor is the reciprocal of the interest rate and is used to convert a future value to its present value. This is the opposite of compounding, which calculates the future value of a present sum. Therefore, the statement that the opposite of compounding is discounting, and that the discount factor is the reciprocal of the interest rate, is true.

Option Analysis:

Option A: This option is correct because the discount factor is indeed the reciprocal of the interest rate and is used to find the present value of a future sum of money.
Option B: This option is incorrect because it contradicts the relationship between compounding and discounting.

Mnemonic: Discount, Reciprocal, Present Value

5. Find compound interest on Rs. 7500 at 4% per annum for 2 years, compounded annually

Correct Answer: Correct answer is: (D) Rs 612.

Exam Relevance: CAT, GMAT, GRE, SAT, Banking Exams

Difficulty: Moderate

Concept notes: The formula for compound interest is $ A = P \left(1 + \frac{r}{n}\right)^{nt} $, where $ A $ is the amount after time $ t $, $ P $ is the principal amount, $ r $ is the annual interest rate, $ n $ is the number of times interest is compounded per year, and $ t $ is the time the money is invested for in years. The compound interest is then calculated as $ CI = A - P $.

Common Mistakes: A common mistake is to use the simple interest formula instead of the compound interest formula, which would yield an incorrect result.

Explanations: Given: - Principal $ P = 7500 $ Rs - Annual interest rate $ r = 4\% = 0.04 $ - Time $ t = 2 $ years - Compounded annually, so $ n = 1 $ First, calculate the amount $ A $ after 2 years: \[ A = 7500 \left(1 + \frac{0.04}{1}\right)^{1 \times 2} \] \[ A = 7500 \left(1 + 0.04\right)^2 \] \[ A = 7500 \left(1.04\right)^2 \] \[ A = 7500 \times 1.0816 \] \[ A = 8112 \] Next, calculate the compound interest $ CI $: \[ CI = A - P \] \[ CI = 8112 - 7500 \] \[ CI = 612 \] Thus, the compound interest is Rs 612.

Option Analysis:

Option A: Rs 312. This is incorrect as it does not match the calculated compound interest.
Option B: Rs 412. This is incorrect as it does not match the calculated compound interest.
Option C: Rs 512. This is incorrect as it does not match the calculated compound interest.
Option D: Rs 612. This is the correct answer as it matches the calculated compound interest.

6. Suppose that you deposit $ 750.00 in an account paying 4% compound interest, with interest compounded annually (at the end of each year). How much interest will your account earn over the first 5 years?

Correct Answer: Correct answer is: (D) $ 162.49.

Exam Relevance: SAT, ACT, GRE, GMAT

Difficulty: Moderate

Concept notes: The concept of compound interest involves calculating the interest earned on an initial principal and the accumulated interest over time. The formula for compound interest is $ A = P(1 + r)^n $, where $ A $ is the amount after $ n $ years, $ P $ is the principal amount, $ r $ is the annual interest rate, and $ n $ is the number of years.

Common Mistakes: A common mistake is to use simple interest instead of compound interest, which would result in a lower interest amount. Another mistake is to miscalculate the number of compounding periods or the interest rate.

Explanations: To find the interest earned over the first 5 years, we use the compound interest formula: 1. $ P = 750 $ 2. $ r = 0.04 $ 3. $ n = 5 $ First, calculate the amount after 5 years: \[ A = 750(1 + 0.04)^5 \] \[ A = 750(1.04)^5 \] \[ A = 750 \times 1.2166529 \] \[ A \approx 912.49 \] Next, calculate the interest earned: \[ \text{Interest} = A - P \] \[ \text{Interest} = 912.49 - 750 \] \[ \text{Interest} = 162.49 \] Thus, the interest earned over the first 5 years is $162.49.

Option Analysis:

Option A: Incorrect. This is the total amount after 5 years, not the interest earned.
Option B: Incorrect. This is the initial principal amount, not the interest earned.
Option C: Incorrect. This is a simple interest calculation, not compound interest.
Option D: Correct. This is the interest earned over the first 5 years using compound interest.

7. What is the compound interest on Rs. 2500 for 2 years at rate of interest 4% per annum?

Correct Answer: Correct answer is: (B) Rs. 204.

Exam Relevance: GMAT, GRE, CAT, Bank PO

Difficulty: Moderate

Concept notes: The formula for compound interest is given by $ A = P \left(1 + \frac{r}{100}\right)^n $, where $ A $ is the amount after $ n $ years, $ P $ is the principal amount, $ r $ is the annual interest rate, and $ n $ is the number of years. The compound interest is then calculated as $ CI = A - P $.

Common Mistakes: A common mistake is to use the simple interest formula instead of the compound interest formula, which would yield a different result.

Explanations: Given: - Principal amount $ P = 2500 $ Rs. - Annual interest rate $ r = 4\% $ - Number of years $ n = 2 $ First, calculate the amount $ A $ after 2 years using the compound interest formula: \[ A = 2500 \left(1 + \frac{4}{100}\right)^2 \] \[ A = 2500 \left(1 + 0.04\right)^2 \] \[ A = 2500 \left(1.04\right)^2 \] \[ A = 2500 \times 1.0816 \] \[ A = 2704 \] Next, calculate the compound interest $ CI $: \[ CI = A - P \] \[ CI = 2704 - 2500 \] \[ CI = 204 \] Thus, the compound interest on Rs. 2500 for 2 years at a rate of 4% per annum is Rs. 204.

Option Analysis:

Option A: Rs. 180 is incorrect because it does not account for the compounding effect over the two years.
Option B: Rs. 204 is correct as it accurately reflects the compound interest calculation.

8. What is another term for future value computations?

Correct Answer: Correct answer is: (A) Compounding.

Exam Relevance: CFA, CPA, CFP, Financial Mathematics Exams

Difficulty: Moderate

Concept notes: Future value computations involve calculating the value of an investment at a future date, taking into account the interest earned over time. This process is also known as compounding.

Common Mistakes: A common misunderstanding is confusing compounding with discounting. Discounting is the process of determining the present value of a future sum of money, whereas compounding is the process of determining the future value of a present sum of money.

Explanations: Future value computations involve the process of compounding, where the interest earned on an investment is reinvested to generate additional interest over time. This leads to exponential growth of the investment, which is the essence of compounding.

Option Analysis:

Option A: Compounding is the correct term for future value computations, as it involves the reinvestment of interest to generate additional interest over time.
Option B: Discounting is the process of determining the present value of a future sum of money, which is not the same as future value computations.

Mnemonic: FV = Future Value, C = Compounding

9. Your $ 440 gets 5.8% interest compounded annually for 8 years. What will your $ 440 be worth in 8 years?

Correct Answer: Correct answer is: (B) $ 690.78.

Exam Relevance: SAT, ACT, GRE, GMAT

Difficulty: Moderate

Concept notes: The concept of compound interest involves calculating the future value of an investment where interest is added to the principal at regular intervals, and the subsequent interest is calculated on the new principal amount.

Common Mistakes: A common mistake is to use simple interest instead of compound interest, which would result in a lower final amount.

Explanations: To find the future value of $440 with 5.8% interest compounded annually for 8 years, we use the compound interest formula: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] where: - $ A $ is the amount of money accumulated after n years, including interest. - $ P $ is the principal amount (initial investment). - $ r $ is the annual interest rate (decimal). - $ n $ is the number of times interest is compounded per year. - $ t $ is the time the money is invested for in years. Given: - $ P = 440 $ - $ r = 0.058 $ - $ n = 1 $ (compounded annually) - $ t = 8 $ Substitute the values into the formula: \[ A = 440 \left(1 + \frac{0.058}{1}\right)^{1 \times 8} \] \[ A = 440 \left(1 + 0.058\right)^8 \] \[ A = 440 \left(1.058\right)^8 \] Calculate $ (1.058)^8 $: \[ (1.058)^8 \approx 1.570 \] Now, multiply by the principal: \[ A = 440 \times 1.570 \] \[ A \approx 690.80 \] Rounding to two decimal places, the future value is approximately $690.78.

Option Analysis:

Option A: Incorrect, as it is significantly lower than the correct value.
Option B: Correct, as it matches the calculated future value.
Option C: Incorrect, as it is much higher than the correct value.
Option D: Incorrect, as it is higher than the correct value but not as high as option C.

10. A savings account was set up with an original deposit of $ 3800. The account earned 2% compound interest, with interest compounded at the end of each year. Which function gives the total value of the account after x years?

Correct Answer: Correct answer is: (B) $y_2=3800\left(1.02\right)^x$.

Exam Relevance: SAT, ACT, GRE, GMAT

Difficulty: Moderate

Concept notes: The formula for compound interest is given by $ A = P(1 + r)^n $, where $ A $ is the amount of money accumulated after $ n $ years, $ P $ is the principal amount (initial deposit), $ r $ is the annual interest rate, and $ n $ is the number of years.

Common Mistakes: A common mistake is to confuse compound interest with simple interest, where the interest is not compounded. Simple interest would be calculated as $ A = P + P \cdot r \cdot n $, which is not the case here.

Explanations: The correct formula for compound interest is $ A = P(1 + r)^n $. Here, the principal $ P $ is $3800, the annual interest rate $ r $ is 2% or 0.02, and $ n $ is the number of years $ x $. Therefore, the total value of the account after $ x $ years is given by $ y_2 = 3800(1.02)^x $.

Option Analysis:

Option A: This option represents simple interest, not compound interest. The formula $ y_1 = 3800 + 3800(0.02x) $ does not account for the compounding effect.
Option B: This option correctly represents the compound interest formula $ y_2 = 3800(1.02)^x $.
Option C: This option incorrectly uses $ 0.02 $ as the base of the exponent, which does not represent the correct growth factor for compound interest.
Option D: This option is incorrect as the correct function is provided in Option B.

Frequently Asked Questions

What is compound interest?

Compound interest is the interest calculated on the initial principal and also on the accumulated interest of previous periods. It allows for exponential growth of the principal amount over time.

How does the compound interest formula work?

The compound interest formula is A = P(1 + r/n)^(nt), where A is the amount of money accumulated after n years, including interest, P is the principal amount, r is the annual interest rate, n is the number of times that interest is compounded per year, and t is the time the money is invested for in years.

What is the difference between annual and monthly compounding?

Annual compounding calculates interest once per year, while monthly compounding calculates interest 12 times per year. Monthly compounding results in a higher final amount due to more frequent interest calculations.

How can I calculate the future value of an investment using compound interest?

To calculate the future value, use the formula A = P(1 + r/n)^(nt), where A is the future value, P is the principal amount, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the time in years.

What is the discount factor in the context of compound interest?

The discount factor is used to calculate the present value of a future sum of money. It is the reciprocal of the future value factor and is used in discounting techniques to find the current value of a future amount

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