Simple Interest All Competence Aptitude

To find the rate of interest, we can use the formula for simple interest:

\[ \text{Simple Interest} = \dfrac{P \times R \times T}{100} \]

Where:
- \( P \) is the principal amount (Rs. 12,500 in this case).
- \( R \) is the rate of interest per annum (what we're trying to find).
- \( T \) is the time in years (4 years in this case).
- The amount after \( T \) years is \( P + \text{Simple Interest} \) (Rs. 15,500 in this case).

We can rewrite the formula to solve for \( R \):

\[ R = \dfrac{100 \times \text{Simple Interest}}{P \times T} \]

Given that the principal amount (\( P \)) is Rs. 12,500, the amount after 4 years (\( P + \text{Simple Interest} \)) is Rs. 15,500. So, the simple interest earned (\( \text{Simple Interest} \)) is:

\[ \text{Simple Interest} = \text{Amount after 4 years} - \text{Principal} \]\[ \text{Simple Interest} = 15500 - 12500 \]\[ \text{Simple Interest} = 3000 \]

Now, we can substitute the values into the formula to find the rate of interest (\( R \)):

\[ R = \dfrac{100 \times 3000}{12500 \times 4} \]\[ R = \dfrac{300000}{50000} \]\[ R = 6 \]

So, the rate of interest is 6%.

Therefore, the correct answer is: 6%.

To find the principal amount (the sum of money lent), let's denote it as \( P \).

We are given two scenarios:

1. Amounts to 880 in 2 years
2. Amounts to 920 in 3 years

Using the formula for simple interest:\[ \text{Simple Interest} = \dfrac{P \times R \times T}{100} \]

We can set up two equations based on the information given:

1. For the scenario where the amount is 880 in 2 years:\[ \text{Simple Interest} = \dfrac{P \times R \times 2}{100} \]\[ 880 = \dfrac{P \times R \times 2}{100} \]

2. For the scenario where the amount is 920 in 3 years:\[ \text{Simple Interest} = \dfrac{P \times R \times 3}{100} \]\[ 920 = \dfrac{P \times R \times 3}{100} \]

Now, we have a system of equations:

1. \( 880 = \dfrac{2PR}{100} \)
2. \( 920 = \dfrac{3PR}{100} \)

To solve this system, we can divide the second equation by the first:

\[ \dfrac{920}{880} = \dfrac{\dfrac{3PR}{100}}{\dfrac{2PR}{100}} \]

\[ \dfrac{23}{22} = \dfrac{3}{2} \]

\[ 23 \times 2 = 3 \times 22 \]

\[ 46 = 66 \]

This equation is not true. It seems we made an error in our calculations. Let's go back and correct it.

From the first equation:

\[ 880 = \dfrac{2PR}{100} \]

\[ 880 \times 100 = 2PR \]

\[ 88000 = 2PR \]

\[ PR = \dfrac{88000}{2R} \]

\[ PR = 44000 \div R \]

Similarly, from the second equation:

\[ 920 = \dfrac{3PR}{100} \]

\[ 920 \times 100 = 3PR \]

\[ 92000 = 3PR \]

\[ PR = \dfrac{92000}{3R} \]

\[ PR = \dfrac{92000}{3R} \]

Since \( PR \) must be equal in both equations, we have:

\[ \dfrac{44000}{R} = \dfrac{92000}{3R} \]

\[ 3 \times 44000 = 92000 \]

\[ 132000 = 92000 \]

This equation is also not true. It seems we made an error in our setup. Let's reassess the problem. 

Given that the interest is simple interest, if we look at the difference between the amounts in 2 years and 3 years, we see that it's the interest earned in the third year. 

\[ 920 - 880 = 40 \]

This is the interest earned in the third year.

Now, let's find the interest earned per year:

\[ \text{Interest per year} = \dfrac{40}{1} = 40 \]

So, the interest earned per year is 40.

Using this information, we can find the principal amount:

\[ \text{Principal} = \text{Amount after 2 years} - \text{Interest per year} \times \text{Time in years} \]\[ \text{Principal} = 880 - 40 \times 2 \]\[ \text{Principal} = 800 \]

So, the sum of money lent (the principal amount) is 800 rupees.

Therefore, the correct answer is: 800.

Amount after 2 years = Rs. 756

Amount after 3.5 years = Rs. 873

Formula Used:

Simple interest = (Principal × Rate × Time)/100

Concept Used:

Simple interest for every year is always equal in amount.

Calculation:

Interest of Three and half year = 873 - 756 = Rs. 117

Interest earned in one year = (117/3) × 2 = Rs. 78

so, 

Interest earned during initial two years = 2 × 78 = Rs. 156

Principal = Rs. 756 - Rs. 156 = Rs. 600

78 = (600 × R × 1)/100

⇒ R = 78/6 = 13%

Thus, the Rate of Interest on the sum is 13% per annum. 

Given:
Time \( T_1 = \dfrac{15}{12} \) years
Rate \( R_1 = 7.5\% = \dfrac{7.5}{100} \)
Time \( T_2 = \dfrac{8}{12} \) years
Rate \( R_2 = 12.5\% = \dfrac{12.5}{100} \)

Using the formula for simple interest:\[ \text{Simple Interest} = \dfrac{P \times R \times T}{100} \]

We can set up the equation based on the information given:

\[ \dfrac{P \times 7.5 \times \dfrac{15}{12}}{12 \times 100} - \dfrac{P \times 12.5 \times \dfrac{8}{12}}{12 \times 100} = 32.50 \]

\[ \dfrac{112.5P - 100P}{1200} = 32.50 \times 12 \]

\[ \dfrac{12.5P}{1200} = 32.50 \times 12 \]

\[ 12.5P = 32.50 \times 12 \times 1200 \]

\[ P = \dfrac{32.50 \times 12 \times 1200}{12.5} \]

\[ P = \dfrac{390000}{12.5} \]

\[ P = 3120 \]

So, the sum of money is Rs. 3120.

Your calculation and approach are indeed correct. The sum of money is Rs. 3120. Thank you for providing the verification!

Let's first calculate the amount of profit Arun wants to make on the whole sum of money.

Arun lends Rs. 20,000 in total.

He lends Rs. 12,000 to the first friend at 8% per annum simple interest.

The simple interest earned on this amount in one year is:

\[ \text{Interest}_1 = \dfrac{P \times R \times T}{100} = \dfrac{12000 \times 8 \times 1}{100} = 960 \]

Now, Arun wants to make a profit of 10% on the whole sum of money. So, his profit should be 10% of Rs. 20,000:

\[ \text{Profit} = \dfrac{10}{100} \times 20000 = 2000 \]

To achieve this profit, the total amount Arun needs to receive (which includes the principal plus the profit) is:

\[ \text{Total amount to receive} = \text{Principal} + \text{Profit} = 20000 + 2000 = 22000 \]

Now, Arun has already lent Rs. 12,000 to the first friend. So, the remaining amount he needs to lend to the second friend is:

\[ \text{Remaining amount} = 20000 - 12000 = 8000 \]

Now, let's find the rate at which he should lend this remaining amount to the second friend.

Let \( R_2 \) be the rate at which he lends Rs. 8000 to the second friend.

We can use the formula for simple interest to find \( R_2 \):

\[ \text{Interest}_2 = \dfrac{P \times R_2 \times T}{100} \]

We know that the interest earned on this amount should be 10% of the total amount Arun wants to receive (Rs. 22000), which is Rs. 2000.

So, we have:

\[ \dfrac{8000 \times R_2 \times 1}{100} = 2000 \]

Solving for \( R_2 \):

\[ 8000 \times R_2 = 200000 \]

\[ R_2 = \dfrac{200000}{8000} = 25 \]

But this is the interest rate for the full year, however, the interest rate per annum for 12 months is required.

Let's convert 25% annual interest to a monthly rate:

\[ \text{Monthly interest rate} = \dfrac{25}{12} \]

\[ \text{Monthly interest rate} \approx 2.08\% \]

So, Arun should lend the remaining sum of money to the second friend at a simple interest rate of approximately 2.08% per month to achieve a 10% profit on the whole amount.

Therefore, the correct answer is not listed among the options provided. Let's recalculate and choose the closest option. 

Given that Arun wants to make a profit of 10%, the total amount he wants to receive from both friends is Rs. 22,000. 

He has already lent Rs. 12,000 at 8% interest, earning an interest of:

\[ \text{Interest}_1 = \dfrac{12000 \times 8 \times 1}{100} = 960 \]

So, the remaining profit he needs to make from the second friend's loan is Rs. \( 2000 - 960 = 1040 \).

Now, let's find the interest rate \( R_2 \) for the remaining Rs. 8,000:

\[ \text{Interest}_2 = \dfrac{8000 \times R_2 \times 1}{100} \]

\[ 1040 = 80R_2 \]

\[ R_2 = \dfrac{1040}{80} = 13 \]

So, Arun should lend the remaining sum of money to the second friend at a simple interest rate of 13% per annum to achieve a 10% profit on the whole amount.

Therefore, the closest option among the given ones is 13%.

Let's break down the problem:

1. Ramesh borrowed a sum from Rahul at 5% per annum simple interest.
2. He returns the amount after 5 years.
3. Rahul returns 2% of the total amount received, which is Rs. 5.

Let's denote the principal amount (the sum Ramesh borrowed) as \( P \).

The simple interest earned by Ramesh in 5 years is given by:

\[ \text{Simple Interest} = \dfrac{P \times R \times T}{100} \]

Given that the rate of interest \( R \) is 5% per annum and the time \( T \) is 5 years:

\[ \text{Simple Interest} = \dfrac{P \times 5 \times 5}{100} \]

\[ \text{Simple Interest} = \dfrac{25P}{100} \]

\[ \text{Simple Interest} = 0.25P \]

Now, the total amount Ramesh receives after 5 years includes the principal amount and the interest earned:

\[ \text{Total amount} = P + \text{Simple Interest} \]

Given that Rahul returns 2% of the total amount received, which is Rs. 5:

\[ 0.02 \times (\text{Total amount}) = 5 \]

\[ 0.02 \times (P + 0.25P) = 5 \]

\[ 0.02 \times 1.25P = 5 \]

\[ 0.025P = 5 \]

\[ P = \dfrac{5}{0.025} \]

\[ P = 200 \]

So, Ramesh borrowed Rs. 200.

Therefore, the correct answer is Rs. 200.

Let's denote the sum lent at 4% interest as \( P_1 \) and the sum lent at 5% interest as \( P_2 \).

We know that the total sum lent is Rs. 5,000, so we have:

\[ P_1 + P_2 = 5000 \]

Now, we'll calculate the interest earned on each part.

For the sum lent at 4% interest:\[ \text{Interest}_1 = \dfrac{P_1 \times 4 \times 2}{100} = \dfrac{8P_1}{100} = 0.08P_1 \]

For the sum lent at 5% interest:\[ \text{Interest}_2 = \dfrac{P_2 \times 5 \times 2}{100} = \dfrac{10P_2}{100} = 0.10P_2 \]

Given that the total interest after 2 years is Rs. 440:\[ \text{Total interest} = \text{Interest}_1 + \text{Interest}_2 = 440 \]

\[ 0.08P_1 + 0.10P_2 = 440 \]

Now, we need to express one of the variables in terms of the other using the relationship \( P_1 + P_2 = 5000 \).

Let's express \( P_2 \) in terms of \( P_1 \):

\[ P_2 = 5000 - P_1 \]

Now, we can substitute this into the equation for total interest:

\[ 0.08P_1 + 0.10(5000 - P_1) = 440 \]

\[ 0.08P_1 + 500 - 0.10P_1 = 440 \]

\[ -0.02P_1 = -60 \]

\[ P_1 = \dfrac{-60}{-0.02} \]

\[ P_1 = 3000 \]

Now that we have found \( P_1 \), we can find \( P_2 \):

\[ P_2 = 5000 - 3000 \]

\[ P_2 = 2000 \]

So, the sum of money lent at 4% interest is Rs. 3000, and the sum of money lent at 5% interest is Rs. 2000.

To find the ratio in which the total sum is divided, we divide both amounts by the least common multiple of 3000 and 2000, which is 2000.

\[ \dfrac{3000}{2000} = \dfrac{3}{2} \]\[ \dfrac{2000}{2000} = 1 \]

So, the total sum of money lent at each rate is divided in the ratio \( 3 : 2 \).

Therefore, the correct answer is \( 3 : 2 \).

To solve this problem, we can use the formula for simple interest:

\[ \text{Simple Interest (S.I)} = \frac{P \times R \times T}{100} \]

Where:
- P is the principal amount (the initial amount lent or borrowed).
- R is the rate of interest per annum.
- T is the time in years.

Given:
P = 1860
A = 2641.20
R = 12%
\( \displaystyle R = 12\% = \frac{12}{100} \)

We need to find the time T it takes for the amount to reach 2641.20 at the given rate of interest.

First, let's find the simple interest earned:
S.I = A - P
S.I = 2641.20 - 1860
S.I = 781.2

Now, let's use the formula for simple interest to find the time T:

\[ \text{S.I} = \frac{P \times R \times T}{100} \]

Substituting the known values:

\[ 781.20 = \frac{1860 \times 12 \times T}{100} \]

Now, let's solve for T:

\[ 781.20 = \frac{22320T}{100} \] \[ 78120 = 22320T \] \[ \displaystyle T = \frac{78120}{22320} = 3.5 \]

Given the provided calculations:

\[ \text{Amount in 2.5 years} = 1012 \]

\[ \text{Amount in 4 years} = 1067.20 \]

We found the interest for 1.5 years:

\[ \text{Interest for 1.5 years} = 1067.20 - 1012 = 55.20 \]

\[ \text{Interest for 1 year} = \dfrac{55.20}{1.5} = 36.8 \]

\[ \text{Interest for 2.5 years} = 36.8 \times 2.5 = 92 \]

Then we found the principal:

\[ \text{Principal} = \text{Amount} - \text{Interest} \]

\[ \text{Principal} = 1012 - 92 = 920 \]

Finally, we calculated the rate:

\[ \text{Rate} = \dfrac{100 \times \text{Interest}}{\text{Principal} \times \text{Time}} \]

\[ \text{Rate} = \dfrac{100 \times 92}{920 \times \left(\dfrac{5}{2}\right)} = 4\% \]

So, the rate of interest is 4%. 

Thank you for providing the detailed calculations!

To find the principal amount (the sum of money), we'll use the simple interest formula:

\[ \text{Simple Interest} = \dfrac{P \times R \times T}{100} \]

Where:
- \( P \) is the principal amount (what we want to find).
- \( R \) is the rate of interest per annum (4% in this case).
- \( T \) is the time in years (6 months or \( \dfrac{1}{2} \) years).

Given:
- Rate of interest \( R = 4\% = \dfrac{4}{100} \)
- Time \( T = \dfrac{1}{2} \) years
- Simple Interest \( SI = 150 \)

Let's substitute the given values into the formula and solve for \( P \):

\[ 150 = \dfrac{P \times 4 \times \dfrac{1}{2}}{100} \]

\[ 150 = \dfrac{P \times 2}{100} \]

\[ 150 \times 100 = P \times 2 \]

\[ P = \dfrac{150 \times 100}{2} \]

\[ P = \dfrac{15000}{2} \]

\[ P = 7500 \]

So, the sum of money that must be given as simple interest for six months at 4% per annum in order to earn Rs. 150 interest is Rs. 7500. 

Therefore, the correct answer is 7500.