Test: Simple And Compound Interest Including Annuity - 1 - CA Foundation MCQ

The formula for Simple Interest (SI) is: SI = (P x R x T) / 100

Where, P = Principal amount (Rs. 3500) R = Rate of interest (12%) T = Time (3 years)

So, SI = (3500 x 12 x 3) / 100 = 1260

Therefore, the simple interest on Rs. 3500 for 3 years at 12% per annum is Rs. 1260. So the answer is 2. Rs. 1260

Calculation of Rate Percent per Annum

- Given:
- Principal amount (P) = Rs. 12000
- Amount (A) = Rs. 16500
- Time (T) = 2 ½ years

- We know that:
- Simple Interest (SI) = A - P
- SI = P * R * T / 100

- Substituting the given values:
- A - P = P * R * T / 100
- 16500 - 12000 = 12000 * R * 2.5 / 100
- 4500 = 300 * R
- R = 4500 / 300
- R = 15%
Therefore, the rate percent per annum is 15%, which corresponds to option A.

SI = Amount - Principal
= 10200 - 8500 = 1700
hence , now put this into the formula S.I = P x R x T/100
1700 = (8500 x 25/2 x t)/100
1700 x 1200 = 212500 x t
t = 1700 / 1062.5 = 1.6 years

Let the principal be P and the rate of interest be R%.
Using the simple interest formula: Amount = Principal + (Principal × Rate × Time) / 100
For 2 years: 6,200 = P + (P × R × 2) / 100 6,200 = P(1 + 2R/100)  ... (1)
For 3 years: 6,800 = P + (P × R × 3) / 100 6,800 = P(1 + 3R/100)  ... (2)
Subtract equation (1) from equation (2):
6,800 - 6,200 = P[(1 + 3R/100) - (1 + 2R/100)] 600 = P(R/100) P = 600 × 100 / R P = 60,000 / R  ... (3)
Substitute P from equation (3) into equation (1):
6,200 = (60,000 / R)(1 + 2R/100) 6,200
=(60,000 / R) + 1,200 6,200 - 1,200
= 60,000 / R 5,000
= 60,000 / R
R = 60,000 / 5,000
R = 12% Now, substitute R back into equation (3):
P = 60,000 / 12 P = 5,000 The principal is ₹5,000 and the rate of interest is 12%. 

• When the sum doubles in 10 years, the total interest is equal to the principal amount (P).
• Step 1: For doubling, interest = P and time = 10 years. This gives the rate of interest as 10%.
• Step 2: Now, for tripling, the total interest needs to be 2P (because the total amount is 3P, and the principal is P).
• The formula for Simple Interest is:
• Interest = Principal × Rate × Time / 100
• Substitute the values for tripling:
• 2P = P × 10% × t
• Cancel P from both sides:
• 2 = (10 × t) / 100
• Solve for t:
• t = 20 years
• Final Answer: It will take 20 years for the sum to triple.

Given:

• Principal (P): Rs. 1000
• Rate of Interest (R): 5% per annum
• Time Period (n): 4 years

Formula for Compound Interest:

A = P × (1 + R/100)ⁿ

Where:

• A = Amount after n years
• P = Principal
• R = Rate of interest
• n = Time period in years

Step-by-Step Calculation:

Step 1: Calculate the Amount (A)

A = 1000 × (1 + 5/100)⁴
A = 1000 × (1 + 0.05)⁴
A = 1000 × 1.21550625
A≈ 1215.51

Step 2: Calculate Compound Interest (C.I.)

C.I. = A - P C.I. = 1215.51 - 1000 C.I. ≈ 215.51

Final Answer:

• Amount (A): Rs. 1215.51 ≈ 1215
• Compound Interest (C.I.): Rs. 215.51 ≈ 215

⇒ Hence, r = 5/100 = 0.05.

⇒ So, A = 100(1 + 0.05/1)^(1×20).

⇒ This yields A = 100(1 + 0.05)^20 = 100(1.05)^20.

⇒ This comes out to A =~ Rs. 265.33.

Therefore, Rs. 100 will become approximately Rs. 265.33 after 20 years at 5% per annum compound interest.

Hence, the correct answer is approximately 265.50.

The amount after 1 year at 8% per annum when interest is compounded half yearly is

=100(1+3/2/100)^2×1=100(101.5/100)²=103.0225
CI for 1 year = 103.0225 – 100 = 3.0225
The effective annual rate of interest is = 3.0225%

Cost of machine (P) = Rs 1,00,000

Scrap value (A) = Rs 30,000

Rate of Depreciation = 20% per annum on reducing value

The effective life of the machine in years is the number of years in which P (Rs 1,00,000) would reduce to A (scrap value Rs 30,000) reducing at the rate of 20% per annum of the value at the start of that year year.

Value of the machine at time t= 0 years = P

The depreciated cost at end of one year = P[1 — 20%] = P[1 — 0.2] = P × 0.8

At the end of second year = P × 0.8²

At the end of 3rd year = P × 0.8³

And so on.

Let after n years the value depreciate to scrap value. We are required to find n.

P(0.8)^n = A

1,00000 (0.8)^n = 30,000

=> (0.8)^n = (30,000)/(1,00,000) = 0.3

Taking log of both sides

n log (0.8) = log (0.3)

=>n × (-0.09691) = (-0.52288)

=> n = (-0.52288)/(-0.09691)= 5.396 year ~5.4 years

To determine the number of years it takes for the population to increase by 40% given a 2% annual increase, we can use the compound interest formula:

Thus, the number of years it takes for the population to increase by 40% is approximately 17 years.

Let the principal be P. The rate of interest is 6% per annum, and the time period is 3 years.

So, the sum of money is Rs. 10,000.

• The nominal interest rate is 6% per annum, compounded half-yearly.
• This means the interest is applied twice a year at a rate of 3% (6% / 2).
• To find the effective annual rate (EAR), use the formula: EAR = (1 + r/n)^(nt) - 1, where r is the nominal rate, n is the number of compounding periods per year, and t is the number of years.
• Plugging in the values: EAR = (1 + 0.03)^2 - 1 = 0.0609 or 6.09%.

• Principal (P): Rs. 16,000
• Rate of Interest (R): 10% per annum (compounded half-yearly)
• Time Period (t): 1½ years

Formula for Compound Interest:

A = P × (1 + R/100)ⁿ

Where:

• A = Amount after n periods
• P = Principal
• R = Rate of interest per period
• n = Number of periods (half-yearly)

Step-by-Step Calculation:

Step 1: Convert the rate to half-yearly

Rate per half-year = 10% / 2 = 5%

Step 2: Apply the Compound Interest formula

A = 16000 × (1 + 5/100)³
A = 16000 × (1 + 0.05)³
A = 16000 × 1.157625
A ≈ 18522

Step 3: Calculate Compound Interest (C.I.)

C.I. = A - P
C.I. = 18522 - 16000
C.I. ≈ 2522

Final Answer:

• Compound Interest (C.I.): Rs. 2522

Given: b( birth rate)=39.4/1000, d(death rate) =19.4/1000

b-d= 39.4–19.4=20/1000 i.e.0.02

P( population) = 2P( population)

To find: t=?

Solution :Let's assume P=1000

applying formula:

An= p(1+i)ⁿ

Putting values:

2000=1000(1+0.02)ⁿ

2=1.02ⁿ

n=35 years.

Calculation of Compound Interest
 

• Principal amount (P) = Rs. 4000


• Rate of interest (R) = 12% per annum


• Time period (T) = 6 months = 0.5 years


• Interest compounded quarterly, so n = 4


• Using the formula for compound interest: A = P(1 + R/n)^(nt)


• Substitute the values: A = 4000(1 + 0.12/4)^(4*0.5)


• Calculating further, A = 4000(1 + 0.03)^2


• A = 4000(1.03)^2 = 4000(1.0609) = Rs. 4243.60


• Compound Interest (CI) = A - P = 4243.60 - 4000 = Rs. 243.60


 

Therefore, the compound interest on Rs. 4000 for 6 months at 12% per annum payable quarterly is Rs. 243.60. So, option A is the correct answer.

A = PMT × [(1 + r)ⁿ - 1] / r

Where:

• A = Amount of the annuity after n years
• PMT = Annual Payment
• r = Annual Interest Rate (in decimal)
• n = Number of years

Step-by-Step Calculation:

Step 1: Convert the rate to decimal

r = 3.5 / 100 = 0.035

Step 2: Apply the formula

A = 150 × [(1 + 0.035)¹² - 1] / 0.035
A = 150 × [(1.035)¹² - 1] / 0.035

Step 3: Calculate (1.035)¹²

(1.035)¹² ≈ 1.49957

Step 4: Substitute the value into the formula

A = 150 × [1.49957 - 1] / 0.035
A = 150 × 0.49957 / 0.035 A ≈ 150 ×14.271
A ≈ 2140.65

• The formula to calculate the Equated Monthly Installment (EMI) is:
• EMI = (P × r × (1 + r)ⁿ) / ((1 + r)ⁿ - 1)
• Where:
• P = Principal loan amount = Rs. 10,000
• r = Annual interest rate (as a decimal) = 0.04 (4%)
• n = Number of installments = 30
• Step-by-Step Calculation:
• 1. Substitute the values into the formula:
• EMI = (10000 × 0.04 × (1 + 0.04)³⁰) / ((1 + 0.04)³⁰ - 1)
• 2. Calculate (1 + 0.04)³⁰:
• (1 + 0.04)³⁰ = 1.04³⁰ ≈ 3.2434
• 3. Substitute this value back into the formula:
• EMI = (10000 × 0.04 × 3.2434) / (3.2434 - 1)
• 4. Calculate the EMI:
• EMI = (10000 × 0.129736) / 2.2434 ≈ 578.02
• Final Answer:
• The amount of each installment to cover the principal and interest at 4% per annum compound interest is Rs. 578.02.

• Formula:
• V = 1200 / 0.08 × [1 - 1 / (1.08)¹²]
• Step-by-Step Solution:
• 1. Simplify the fraction:
• 1200 / 0.08 = 15000
• 2. Calculate (1.08)¹²
• (1.08)¹² ≈ 2.5182
• 3. Find the term (1 - 1 / (1.08)¹²):
• 1 / 2.5182 ≈ 0.397 So, 1 - 0.397 ≈ 0.603
• 4. Multiply by 15000:
• 15000 × 0.603 ≈ 9045
• Final Answer:
• The value of V is approximately: Rs. 9045

• a: Rs. 100 (Annuity payment)
• i: 5% = 0.05 (Rate of interest per period)
• n: 10 (Number of periods)

Step-by-Step Solution:

1. Substitute the given values into the formula:

FV = 100 * {(1 + 0.05)¹⁰ - 1} / 0.05

2. Calculate (1 + 0.05)¹⁰

(1.05)¹⁰ ≈ 1.62889463

3. Subtract 1 from the result:

1.62889463 - 1 = 0.62889463

4. Divide by 0.05:

0.62889463 / 0.05 = 12.5778926

5. Multiply by the annuity payment (a = 100):

FV = 100 * 12.5778926 ≈ 1257.79

Final Answer:

The Future Value (FV) of the annuity is approximately: Rs. 1257.79 = 1258

Given:

• Future value (amount accumulated): Rs. 50,000
• Interest rate (compound interest rate): 5% per annum (0.05 in decimal)
• Number of years (periods): 25 years