Simple Interest All Competence Aptitude

To find the difference between simple interest and the true discount on Rs. 2400 due 4 years hence at 5% per annum simple interest, we need to calculate both the simple interest and the true discount.

Given:
- Principal amount (\(P\)) = Rs. 2400
- Time (\(T\)) = 4 years
- Rate of interest (\(R\)) = 5%

First, let's calculate the simple interest (SI) using the formula \(SI = \dfrac{P \times R \times T}{100}\):

\[ SI = \dfrac{2400 \times 5 \times 4}{100} = \dfrac{24000}{5} = 480 \]

Now, let's calculate the true discount (TD) using the formula \(TD = \dfrac{SI}{1 + \dfrac{R \times T}{100}}\):

\[ TD = \dfrac{480}{1 + \dfrac{5 \times 4}{100}} = \dfrac{480}{1 + \dfrac{20}{100}} = \dfrac{480}{1 + 0.20} = \dfrac{480}{1.20} = 400 \]

Now, let's find the difference between the simple interest and the true discount:

\[ Difference = SI - TD = 480 - 400 = 80 \]

So, the difference between simple interest and the true discount on Rs. 2400 due 4 years hence at 5% per annum simple interest is Rs. 80.

Therefore, the correct answer is \(\boxed{\text{Rs. 80}}\).

Let's denote the initial rate of interest as \( R \).

Given:
- Principal amount (\( P \)) = Rs. 2400
- Amount after 4 years (\( A \)) = Rs. 3264
- Time (\( T \)) = 4 years

We can use the formula for simple interest to find the initial rate of interest:\[ A = P + \text{Simple Interest} = P + \dfrac{P \times R \times T}{100} \]

For the initial rate of interest:\[ 3264 = 2400 + \dfrac{2400 \times R \times 4}{100} \]\[ 3264 = 2400 + 96R \]\[ 96R = 3264 - 2400 \]\[ 96R = 864 \]\[ R = \dfrac{864}{96} = 9 \]

So, the initial rate of interest is 9%.

Now, if the rate of interest is increased by 1%, the new rate of interest would be \( R + 1 = 9 + 1 = 10 \) percent.

We can now calculate the new amount using the same formula:\[ A' = P + \dfrac{P \times R' \times T}{100} \]\[ A' = 2400 + \dfrac{2400 \times 10 \times 4}{100} \]\[ A' = 2400 + \dfrac{9600}{10} \]\[ A' = 2400 + 960 \]\[ A' = 3360 \]

So, the amount after 4 years, if the rate of interest is increased by 1%, would be Rs. 3360.

Therefore, the correct answer is \(\boxed{3,360}\).

Let's denote the initial rate of interest as \( R \).

Given:
- Principal amount (\( P \)) = Rs. 800
- Amount after 3 years (\( A \)) = Rs. 956
- Time (\( T \)) = 3 years

We can use the formula for simple interest to find the initial rate of interest:\[ A = P + \text{Simple Interest} = P + \dfrac{P \times R \times T}{100} \]

For the initial rate of interest:\[ 956 = 800 + \dfrac{800 \times R \times 3}{100} \]\[ 956 = 800 + 24R \]\[ 24R = 956 - 800 \]\[ 24R = 156 \]\[ R = \dfrac{156}{24} = 6.5 \]

So, the initial rate of interest is 6.5%.

Now, if the rate of interest is increased by 4%, the new rate of interest would be \( R + 4 = 6.5 + 4 = 10.5 \) percent.

We can now calculate the new amount using the same formula:\[ A' = P + \dfrac{P \times R' \times T}{100} \]\[ A' = 800 + \dfrac{800 \times 10.5 \times 3}{100} \]\[ A' = 800 + \dfrac{2520}{10} \]\[ A' = 800 + 252 \]\[ A' = 1052 \]

So, the amount after 3 years, if the rate of interest is increased by 4%, would be Rs. 1052.

Therefore, the correct answer is \(\boxed{Rs. 1052}\).

Let's denote the amount lent at 8% per annum as \( x \) and the amount lent at 10% per annum as \( 10000 - x \).

The average interest is given as 9.2%. Using the concept of weighted averages, we can set up the equation:

\[ \text{Average Interest Rate} = \dfrac{(Amount_1 \times Rate_1) + (Amount_2 \times Rate_2)}{Total Amount} \]

Given:
- \( Rate_1 = 8\% = 0.08 \)
- \( Rate_2 = 10\% = 0.10 \)
- \( \text{Average Interest Rate} = 9.2\% = 0.092 \)
- \( Total Amount = 10000 \)

Plugging these values into the equation:

\[ 0.092 = \dfrac{(x \times 0.08) + ((10000 - x) \times 0.10)}{10000} \]

Now, let's solve for \( x \):

\[ 0.092 \times 10000 = 0.08x + 1000 - 0.10x \]\[ 920 = 0.02x + 1000 - 0.10x \]\[ 0.10x - 0.08x = 1000 - 920 \]\[ 0.02x = 80 \]\[ x = \dfrac{80}{0.02} \]\[ x = 4000 \]

So, the amount lent at 8% per annum is Rs. 4000, and the amount lent at 10% per annum is Rs. 10000 - Rs. 4000 = Rs. 6000.

Therefore, the two parts are \( \boxed{4000, 6000} \).

Thank you for providing the breakdown according to the question. Let's confirm the calculation:

For the first two years:\[ SI_1 = \dfrac{15600 \times 10 \times 2}{100} = 3120 \]

Now, the principal for the next two years becomes:\[ P_2 = 15600 + 3120 = 18720 \]

For the next two years:\[ SI_2 = \dfrac{18720 \times 10 \times 2}{100} = 3744 \]

Now, the total interest earned at the end of the fourth year is:\[ SI_1 + SI_2 = 3120 + 3744 = 6864 \]

However, according to your provided calculation, the interest at the end of the fourth year is calculated as Rs. 1872.

Let's recalculate the interest at the end of the fourth year according to your provided method:

\[ SI_3 = \dfrac{18720 \times 10 \times 1}{100} = 1872 \]

So, the correct interest earned at the end of the fourth year is Rs. 1872.

Therefore, the correct answer is \(\boxed{1872}\). Thank you for providing the corrected calculation.

Let's denote the rate of interest per annum as \( R \).

For the first person:
- Principal (\( P_1 \)) = Rs. 400
- Time (\( T_1 \)) = 3 years

For the second person:
- Principal (\( P_2 \)) = Rs. 500
- Time (\( T_2 \)) = 4 years

Using the formula for simple interest (\( SI = \dfrac{P \times R \times T}{100} \)), we can calculate the total interest received by the money lender:

For the first person:\[ SI_1 = \dfrac{400 \times R \times 3}{100} = 12R \]

For the second person:\[ SI_2 = \dfrac{500 \times R \times 4}{100} = 20R \]

The total interest received is the sum of interest from both transactions:\[ SI_1 + SI_2 = 12R + 20R = 32R \]

Given that the total interest received is Rs. 160, we have the equation:\[ 32R = 160 \]

Solving for \( R \):\[ R = \dfrac{160}{32} = 5 \]

So, the rate of interest per annum is 5%.

Therefore, the correct answer is \(\boxed{5\%}\).

Let's denote the person's capital as \( C \).

Given:
- Income tax rate: 4 paise per rupee, which is equivalent to 4%.
- Fall of interest rate: from 4% to 3.75%.
- Diminishing net yearly income: Rs. 48.

The net yearly income is the difference between the interest earned and the tax paid on the interest.

Initially, when the interest rate is 4%:\[ \text{Net Income} = \text{Interest Earned} - \text{Tax Paid} \]

When the interest rate is 4%, the interest earned is \( 0.04 \times C \) and the tax paid is \( 0.04 \times 0.04 \times C \).

So, initially:\[ \text{Net Income} = 0.04C - (0.04 \times 0.04C) \]

When the interest rate falls to 3.75%:\[ \text{Net Income} = 0.0375C - (0.04 \times 0.0375C) \]

Given that the decrease in net income is Rs. 48, we can set up the equation:

\[ (0.04C - (0.04 \times 0.04C)) - (0.0375C - (0.04 \times 0.0375C)) = 48 \]

Now, let's solve for \( C \):

\[ (0.04C - 0.0016C) - (0.0375C - 0.0015C) = 48 \]\[ (0.0384C) - (0.036C) = 48 \]\[ 0.0024C = 48 \]\[ C = \dfrac{48}{0.0024} = 20000 \]

So, the person's capital is Rs. 20,000.

Therefore, the correct answer is \(\boxed{20,000}\).

Let's denote the rate of interest per annum as \( R \).

Given:
- The amount becomes \( \dfrac{7}{6} \) of itself in 3 years.

We can use the simple interest formula to set up the equation:

\[ \text{Amount} = \text{Principal} + \text{Simple Interest} \]

After 3 years, the amount is \( \dfrac{7}{6} \) of the principal:\[ \text{Amount} = \text{Principal} + \left( \dfrac{Principal \times R \times T}{100} \right) \]

\[ \dfrac{7}{6} \times \text{Principal} = \text{Principal} + \left( \dfrac{Principal \times R \times 3}{100} \right) \]

Now, let's solve for \( R \):

\[ \dfrac{7}{6} = 1 + \dfrac{3R}{100} \]

\[ \dfrac{7}{6} - 1 = \dfrac{3R}{100} \]

\[ \dfrac{1}{6} = \dfrac{3R}{100} \]

\[ R = \dfrac{1}{6} \times \dfrac{100}{3} \]

\[ R = \dfrac{100}{18} \]

\[ R \approx 5.56 \]

So, the rate of interest per annum is approximately \( \boxed{5 \dfrac{5}{9}\%} \).

Let's denote the first part of the division as \( P_1 \) and the second part as \( P_2 \).

Given:
- Total amount (\( P_1 + P_2 \)) = Rs. 12,000
- Simple interest rate for the first part = 12% per annum
- Simple interest rate for the second part = 16% per annum
- Time for the first part = 3 years
- Time for the second part = 4.5 years

We can use the formula for simple interest to set up equations for both parts.

For the first part:\[ \text{Simple Interest}_{\text{first part}} = \dfrac{P_1 \times R_1 \times T_1}{100} = \dfrac{P_1 \times 12 \times 3}{100} = \dfrac{36P_1}{100} \]

For the second part:\[ \text{Simple Interest}_{\text{second part}} = \dfrac{P_2 \times R_2 \times T_2}{100} = \dfrac{P_2 \times 16 \times 4.5}{100} = \dfrac{72P_2}{100} \]

Given that the simple interest on the first part is equal to the simple interest on the second part, we have:\[ \dfrac{36P_1}{100} = \dfrac{72P_2}{100} \]

This simplifies to:\[ 36P_1 = 72P_2 \]

Now, we know that \( P_1 + P_2 = 12,000 \), so we can express one of the variables in terms of the other.

Let's express \( P_2 \) in terms of \( P_1 \):\[ P_2 = \dfrac{1}{2} P_1 \]

Now, we can substitute this expression for \( P_2 \) into the equation \( P_1 + P_2 = 12,000 \):\[ P_1 + \dfrac{1}{2} P_1 = 12,000 \]\[ \dfrac{3}{2} P_1 = 12,000 \]\[ P_1 = \dfrac{12,000 \times 2}{3} = 8,000 \]

So, the first part (\( P_1 \)) is Rs. 8,000, and the second part (\( P_2 \)) is Rs. 4,000.

Therefore, the ratio of the first part to the second part is \( \boxed{2 : 1} \).

Let's denote the principal amount as \( P \) and the simple interest as \( SI \).

Given:
- The ratio of principal to simple interest for 5 years is 10:3.

Using the formula for simple interest (\( SI = \dfrac{P \times R \times T}{100} \)), we can express the given ratio mathematically as:

\[ \dfrac{P}{SI} = \dfrac{10}{3} \]

For 5 years, the time \( T = 5 \) years.

Substitute the formula for simple interest into the ratio:

\[ \dfrac{P}{\left(\dfrac{P \times R \times 5}{100}\right)} = \dfrac{10}{3} \]

Now, simplify the equation:

\[ \dfrac{100}{R \times 5} = \dfrac{10}{3} \]

Cross multiply:

\[ 100 \times 3 = 10 \times R \times 5 \]

\[ 300 = 50R \]

\[ R = \dfrac{300}{50} \]

\[ R = 6 \]

So, the rate of interest is \( \boxed{6\%} \).