Simple Interest All Competence Aptitude

Let's calculate the interest earned by the man and determine whether it results in a profit or loss.

The formula for simple interest is given by:

\[ I = P \cdot R \cdot T \]

where:
- \( I \) is the interest earned,
- \( P \) is the principal amount,
- \( R \) is the rate of interest per annum, and
- \( T \) is the time in years.

Given:
- Principal amount (\( P \)) = $1950
- Selling price (\( S \)) = $2200
- Rate of interest (\( R \)) = 10%
- Time (\( T \)) = 1 year

First, let's calculate the interest earned (\( I \)):

\[ I = P \cdot R \cdot T \]

\[ I = 1950 \cdot 0.10 \cdot 1 \]

\[ I = 195 \]

Now, let's determine whether it's a profit or loss:

\[ \text{Profit or Loss} = \text{Selling Price} - \text{Cost Price} \]

\[ \text{Profit or Loss} = 2200 - 1950 \]

\[ \text{Profit or Loss} = 250 \]

Since the calculated interest (\( I \)) is $195 and the profit or loss is $250, it means there is a profit of $55.

Therefore, the correct answer is:

A. $55 gain

The formula for calculating simple interest is given by:

\[ I = P \cdot R \cdot T \]

where:
- \( I \) is the interest,
- \( P \) is the principal amount,
- \( R \) is the rate of interest per annum, and
- \( T \) is the time in years.

Given:
- Principal amount (\( P \)) = $1500
- Time (\( T \)) = 3 years
- Difference in simple interest (\( \Delta I \)) = $13.50

The formula for simple interest is \( I = P \cdot R \cdot T \), and the difference in simple interest between two sources is \( \Delta I = P \cdot \Delta R \cdot T \), where \( \Delta R \) is the difference in rates of interest.

Now, we can solve for \( \Delta R \):

\[ \Delta R = \dfrac{\Delta I}{P \cdot T} \]

Substitute the given values:

\[ \Delta R = \dfrac{13.50}{1500 \cdot 3} \]

\[ \Delta R = \dfrac{13.50}{4500} \]

\[ \Delta R = 0.003 \]

To express this as a percentage, multiply by 100:

\[ \Delta R = 0.3\% \]

Therefore, the correct answer is:

C. 0.3%

The formula for calculating simple interest is given by:

\[ I = P \cdot R \cdot T \]

where:
- \( I \) is the interest,
- \( P \) is the principal amount,
- \( R \) is the rate of interest per annum, and
- \( T \) is the time in years.

Given:
- Principal amount (\( P \)) = $4000
- Rate of interest (\( R \)) = 12% per annum
- Time (\( T \)) = 18 months

To calculate the interest, we need to convert the time from months to years by dividing it by 12:

\[ T = \dfrac{18}{12} = 1.5 \]

Now, substitute the values into the formula:

\[ I = 4000 \cdot 0.12 \cdot 1.5 \]

\[ I = 720 \]

Therefore, the simple interest earned will be $720.

So, the correct answer is:

C. $720

Let's denote the time for the first investment ($8,000 at 3% per annum) as $T_1$ and the time for the second investment ($6,000 at 4% per annum) as $T_2$.

The formula for calculating simple interest is given by:

$$I=P\cdot R\cdot T$$

where:
- $I$ is the interest,
- $P$ is the principal amount,
- $R$ is the rate of interest per annum, and
- $T$ is the time in years.

Given:
1. For the first investment:
   - Principal amount ($P_1$) = $8,000
   - Rate of interest ($R_1$) = 3%
   - Time ($T_1$) = unknown

2. For the second investment:
   - Principal amount ($P_2$) = $6,000
   - Rate of interest ($R_2$) = 4%
   - Time ($T_2$) = 5 years

We know that the interests earned by both investments are equal:

$$P_1\cdot R_1\cdot T_1=P_2\cdot R_2\cdot T_2$$

Now, let's substitute the given values:

$$8000\cdot 0.03\cdot T_1=6000\cdot 0.04\cdot 5$$

Solve for $T_1$:

$$240T_1=1200$$

$$T_1=\frac{1200}{240}$$

$$T_1=5\text{ years}$$

Therefore, the time required for the $8,000 investment at 3% per annum to produce the same interest as the $6,000 investment at 4% per annum in 5 years is 5 years.

So, the correct answer is:

A. 5 years

To find the present population, considering a 20% decrease per annum, we can use the formula for exponential decay:

\[ \text{Population} = \text{Initial Population} \times (1 - \text{Rate of Decrease})^{\text{Time}} \]

Given:
- Initial Population = 10,000
- Rate of Decrease = 20% or 0.2 (as a decimal)
- Time = 2 years

Substitute the values into the formula:

\[ \text{Population} = 10,000 \times (1 - 0.2)^2 \]

Calculate the result:

\[ \text{Population} = 10,000 \times (0.8)^2 \]

\[ \text{Population} = 10,000 \times 0.64 \]

\[ \text{Population} = 6400 \]

Therefore, the present population is 6400.

So, the correct answer is:

B. 6400

Let the initial capital (principal) be \( P \) and the initial rate of interest be \( R_1 \), and the diminished capital after the fall in the interest rate be \( P' \) with a rate of interest \( R_2 \).

The formula for calculating simple interest is \( I = P \cdot R \cdot T \), where \( I \) is the interest, \( P \) is the principal amount, \( R \) is the rate of interest per annum, and \( T \) is the time in years.

Given:
- Initial rate of interest (\( R_1 \)) = 8%
- Diminished rate of interest (\( R_2 \)) = 7 and \( \dfrac{3}{4} \)%
- Decrease in income (\( \Delta I \)) = $61.50

Now, the difference in the interest (\( \Delta I \)) is given by:\[ \Delta I = P \cdot (R_1 - R_2) \cdot T \]

Since the fall in the rate of interest occurs yearly, we can assume \( T = 1 \).

Substitute the given values:\[ 61.50 = P \cdot (8\% - 7.75\%) \]

Simplify the expression:\[ 61.50 = P \cdot 0.0025 \]

Now, solve for \( P \):\[ P = \dfrac{61.50}{0.0025} \]

\[ P = 24,600 \]

Therefore, the initial capital (principal) is $24,600.

So, the correct answer is:

C. $24,600

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

\[ I = P \cdot R \cdot T \]

where:
- \( I \) is the interest,
- \( P \) is the principal amount,
- \( R \) is the rate of interest per annum, and
- \( T \) is the time in years.

Given:
- Principal amount (\( P \)) = $1600
- Simple interest (\( I \)) = $252
- Time (\( T \)) = 2 years and 3 months

First, convert the time to years:\[ T = 2 + \dfrac{3}{12} = 2.25 \text{ years} \]

Now, we can use the formula to find the rate of interest (\( R \)):\[ 252 = 1600 \cdot R \cdot 2.25 \]

Solve for \( R \):\[ R = \dfrac{252}{1600 \cdot 2.25} \]

\[ R = \dfrac{252}{3600} \]

\[ R = \dfrac{7}{100} \]

Now, express \( R \) as a percentage:\[ R = 7\% \]

Therefore, the rate of interest per annum is 7%.

So, the correct answer is:

C. 7%

Let's denote the principal amount as \( P \), the rate of interest as \( R \), and the time as \( T \).

The formula for simple interest is \( I = P \cdot R \cdot T \).

Given:
1. Simple interest for 3 years: \( I_1 = P \cdot 0.05 \cdot 3 \)
2. Simple interest for 4 years: \( I_2 = P \cdot 0.05 \cdot 4 \)
3. The difference in simple interest: \( \Delta I = I_2 - I_1 = P \cdot 0.05 \cdot (4 - 3) = P \cdot 0.05 \)

Given that \( \Delta I = 42 \):\[ P \cdot 0.05 = 42 \]\[ P = \dfrac{42}{0.05} \]

\[ P = 840 \]

Therefore, the sum (principal) is $840.

So, the correct answer is:

D. 840

Let's denote the first part of the sum as \(x\) and the second part as \(1750 - x\).

According to the problem, the interest earned on the first part at 8% per annum and the interest earned on the second part at 6% per annum are equal.

So, we can set up the equation:

\[0.08x = 0.06(1750 - x)\]

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

\[0.08x = 0.06(1750 - x)\]\[0.08x = 105 - 0.06x\]\[0.08x + 0.06x = 105\]\[0.14x = 105\]

Now, divide both sides by 0.14:

\[x = \dfrac{105}{0.14}\]\[x = 750\]

So, the first part is \(750\) rupees, and the second part is \(1750 - 750 = 1000\) rupees.

Now, let's calculate the interest earned on each part:

Interest on the first part at 8% per annum:\[0.08 \times 750 = 60\] rupees

Interest on the second part at 6% per annum:\[0.06 \times 1000 = 60\] rupees

Thus, the interest on each part is \(\text{Rs} \ 60\). 

Therefore, the correct answer is \(\text{Rs} \ 60\).

Let's denote the amount lent at 5% as \(x\) and the amount lent at 8% as \(1550 - x\).

We know that the total interest received after 3 years is \(300\) rupees.

The interest earned from the amount lent at 5% after 3 years is \(3 \times 0.05 \times x\) rupees.

The interest earned from the amount lent at 8% after 3 years is \(3 \times 0.08 \times (1550 - x)\) rupees.

According to the problem, the total interest received after 3 years is \(300\) rupees. So, we can set up the equation:

\[3 \times 0.05 \times x + 3 \times 0.08 \times (1550 - x) = 300\]

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

\[0.15x + 0.24(1550 - x) = 300\]\[0.15x + 372 - 0.24x = 300\]\[0.15x - 0.24x = 300 - 372\]\[-0.09x = -72\]

Now, divide both sides by \(-0.09\):

\[x = \dfrac{-72}{-0.09}\]\[x = 800\]

So, the amount lent at 5% is \(800\) rupees, and the amount lent at 8% is \(1550 - 800 = 750\) rupees.

Now, let's find the ratio of money lent at 5% to that at 8%:

\[\text{Ratio} = \dfrac{\text{Amount lent at 5%}}{\text{Amount lent at 8%}}\]\[\text{Ratio} = \dfrac{800}{750}\]\[\text{Ratio} = \dfrac{16}{15}\]

So, the ratio of money lent at 5% to that at 8% is \(16 : 15\).

Therefore, the correct answer is \(16 : 15\).