Simple Interest All Competence Aptitude

Let's solve this problem step by step.

Let's denote the principal amount as \( P \) and the rate of interest as \( R \). 

Given that after 2 years the amount becomes 720, we can write the equation:

\[ A = P + P \cdot R \cdot 2 = 720 \]

After simplifying, we get:

\[ P(1 + 2R) = 720 \]  (Equation 1)

Similarly, after a further period of 5 years (a total of 7 years), the amount becomes 1020:

\[ A = P + P \cdot R \cdot 7 = 1020 \]

Simplifying this equation, we get:

\[ P(1 + 7R) = 1020 \]  (Equation 2)

Now, we can solve these two equations to find the values of \( P \) and \( R \).

Let's divide Equation 2 by Equation 1:

\[ \dfrac{1 + 7R}{1 + 2R} = \dfrac{1020}{720} \]

Solving this, we find:

\[ \dfrac{1 + 7R}{1 + 2R} = \dfrac{17}{12} \]

Cross-multiplying:

\[ 12(1 + 7R) = 17(1 + 2R) \]

Expanding and solving for \( R \):

\[ 12 + 84R = 17 + 34R \]\[ 50R = 5 \]\[ R = \dfrac{5}{50} = \dfrac{1}{10} \]

Now that we have the value of \( R \), we can substitute it into Equation 1 to find \( P \):

\[ P(1 + 2 \times \dfrac{1}{10}) = 720 \]\[ P(1 + \dfrac{1}{5}) = 720 \]\[ P(\dfrac{6}{5}) = 720 \]\[ P = \dfrac{720 \times 5}{6} = 600 \]

So, the principal amount \( P \) is $600. 

Therefore, the correct answer is \( \boxed{600} \).

It appears there's a calculation error in your approach. Let's correct it:

Given:

- Principal amount \( P = \$7300 \)
- Rate of interest \( R = 5\% \) per annum

You have correctly calculated the total number of days as 123 days.

Now, using the formula for simple interest:

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

Where:
- \( P \) is the principal amount
- \( R \) is the rate of interest per annum
- \( T \) is the time period in years

Let's plug in the values:

\[ \text{Simple Interest} = \dfrac{7300 \times 5 \times \dfrac{123}{365}}{100} \]

\[ \text{Simple Interest} = \dfrac{7300 \times 5 \times 123}{100 \times 365} \]

\[ \text{Simple Interest} = \dfrac{36500 \times 123}{36500} \]

\[ \text{Simple Interest} = 123 \]

Therefore, the correct answer is indeed \( \boxed{123} \). Your initial calculation was correct; however, the subsequent explanation appears to have contained a mistake.

To find out how many years it takes for a sum of money to double itself at a given rate of simple interest, we can use the formula for simple interest:

\[ A = P + P \times R \times T \]

Where:
- \( A \) is the total amount after interest
- \( P \) is the principal amount
- \( R \) is the rate of interest per annum
- \( T \) is the time period in years

Given that the sum doubles itself, \( A = 2P \).

\[ 2P = P + P \times R \times T \]

\[ 2 = 1 + R \times T \]

\[ T = \dfrac{1}{R} \times 100 \]

Given that the rate of interest \( R = 6\dfrac{1}{4}\% = 6.25\% = \dfrac{25}{4}\% \), we can plug this value into the equation:

\[ T = \dfrac{1}{\dfrac{25}{4}} \times 100 \]

\[ T = \dfrac{4}{25} \times 100 \]

\[ T = \dfrac{400}{25} \]

\[ T = 16 \]

So, it takes 16 years for the sum of money to double itself at a simple interest rate of \( 6\dfrac{1}{4}\% \) per annum.

Therefore, the correct answer is \( \boxed{16} \) years.

Let's denote the principal amount as \( P \) and the rate of interest as \( R \).

Given that the simple interest on a sum after 4 years is \( \dfrac{1}{5} \) of the sum, we can write the equation:

\[ SI = \dfrac{1}{5}P \]

We know that simple interest is calculated using the formula:

\[ SI = P \times R \times T \]

Given \( T = 4 \) years, we can rewrite the equation for simple interest:

\[ P \times R \times 4 = \dfrac{1}{5}P \]

Now, we can solve this equation to find the rate of interest \( R \).

\[ 4R = \dfrac{1}{5} \]

\[ R = \dfrac{1}{5 \times 4} \]

\[ R = \dfrac{1}{20} \]

Converting this to a percentage:

\[ R = \dfrac{1}{20} \times 100\% \]

\[ R = 5\% \]

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

Let's denote:
- \( SI_1 \) as the simple interest received from the first bank,
- \( SI_2 \) as the simple interest received from the second bank.

Given:
1. Principal amount \( P = \$500 \)
2. Time \( T = 2 \) years

We know that the formula for simple interest is \( SI = P \times R \times T \), where \( R \) is the rate of interest.

So, the difference in simple interest received from the two banks is:

\[ SI_2 - SI_1 = P \times R_2 \times T - P \times R_1 \times T \]\[ SI_2 - SI_1 = P \times T \times (R_2 - R_1) \]

Given that this difference is \( \$2.50 \), we can write:

\[ 2.50 = 500 \times 2 \times (R_2 - R_1) \]\[ 2.50 = 1000 \times (R_2 - R_1) \]

To find the difference in their rates of interest \( R_2 - R_1 \), we need to divide both sides by 1000:

\[ R_2 - R_1 = \dfrac{2.50}{1000} \]\[ R_2 - R_1 = 0.0025 \]

Converting this to a percentage:

\[ R_2 - R_1 = 0.25\% \]

So, the difference between their (per annum) rates of interest is \( \boxed{0.25\%} \).

Let's denote:
- \( SI_1 \) as the simple interest received from the first scenario (on $4000 for 3 years at the rate of \(x\)% per annum),
- \( SI_2 \) as the simple interest received from the second scenario (on $5000 for 2 years at the rate of 12% per annum).

Given:
1. Principal amount for the first scenario \( P_1 = \$4000 \), Time \( T_1 = 3 \) years
2. Principal amount for the second scenario \( P_2 = \$5000 \), Time \( T_2 = 2 \) years
3. Rate of interest for the second scenario \( R_2 = 12\% \)

We are asked to find the rate of interest \( x \) for the first scenario.

The formula for simple interest is \( SI = P \times R \times T \), where \( R \) is the rate of interest.

So, for the first scenario:

\[ SI_1 = P_1 \times R_1 \times T_1 \]

For the second scenario:

\[ SI_2 = P_2 \times R_2 \times T_2 \]

Given that \( SI_1 = SI_2 \), we can equate them:

\[ P_1 \times R_1 \times T_1 = P_2 \times R_2 \times T_2 \]

\[ 4000 \times R_1 \times 3 = 5000 \times 12\% \times 2 \]

\[ 12000 \times R_1 = 5000 \times 0.12 \times 2 \]

\[ 12000 \times R_1 = 5000 \times 0.24 \]

\[ 12000 \times R_1 = 1200 \]

Now, solving for \( R_1 \):

\[ R_1 = \dfrac{1200}{12000} = 0.1 \]

Converting this to a percentage:

\[ R_1 = 10\% \]

So, the value of \( x \) is \( \boxed{10\%} \).

Initially, the ratio of principal to amount is 4 : 5.
After 3 years, the ratio becomes 5 : 7.

Let's denote the principal as \( P \) and the amount as \( A \).

1. Initially, the ratio \( \dfrac{P}{A} = \dfrac{4}{5} \).
2. After 3 years, the ratio \( \dfrac{P}{A} = \dfrac{5}{7} \).

To make the initial ratios equal, let's scale them by a common factor:
- Multiply the first ratio by 25: \( \dfrac{4}{5} \times 25 = \dfrac{100}{125} \).
- Multiply the second ratio by 20: \( \dfrac{5}{7} \times 20 = \dfrac{100}{140} \).

Now, the ratios become:
1. Initially, \( \dfrac{P}{A} = \dfrac{100}{125} \).
2. After 3 years, \( \dfrac{P}{A} = \dfrac{100}{140} \).

The difference in the amounts after 3 years compared to initially represents the interest earned over those 3 years.

\[ \text{Difference in Amount} = 140 - 125 = 15 \]

Given that this difference occurred over 3 years, we can find the interest earned per year:

\[ \text{Interest per Year} = \dfrac{\text{Difference in Amount}}{3} = \dfrac{15}{3} = 5 \]

So, the annual interest earned is $5.

Now, let's calculate the rate of interest using the formula:

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

\[ \text{Rate} = \dfrac{5 \times 100}{100 \times 1} = \dfrac{5}{1} = 5\% \]

Therefore, the required rate of interest is \( \boxed{5\%} \), as per your solution.

To find the time it takes for the simple interest to become \( \dfrac{2}{5} \) of the principal amount at 8% per annum, we can use the formula for simple interest:

\[ SI = \dfrac{P \times R \times T}{100} \]

Where:
- \( SI \) is the simple interest,
- \( P \) is the principal amount,
- \( R \) is the rate of interest per annum,
- \( T \) is the time period in years.

Given that the simple interest is \( \dfrac{2}{5} \) of the principal amount, we have:

\[ \dfrac{2}{5}P = \dfrac{P \times 8 \times T}{100} \]

Now, we can solve this equation for \( T \), the time period.

First, let's cancel out \( P \) from both sides:

\[ \dfrac{2}{5} = \dfrac{8 \times T}{100} \]

Now, let's solve for \( T \):

\[ T = \dfrac{\dfrac{2}{5} \times 100}{8} \]

\[ T = \dfrac{40}{8} \]

\[ T = 5 \]

So, it will take \( \boxed{5} \) years for the simple interest to be \( \dfrac{2}{5} \) of the principal amount at 8% per annum.

To find the rate of simple interest for which a sum of money becomes 5 times of itself in 8 years, we can use the formula for simple interest:

\[ A = P + P \times R \times T \]

Where:
- \( A \) is the total amount after interest
- \( P \) is the principal amount
- \( R \) is the rate of interest per annum
- \( T \) is the time period in years

Given that the sum becomes 5 times itself, we have:

\[ A = 5P \]

We are also given that the time period is 8 years.

So, substituting the values into the formula, we get:

\[ 5P = P + P \times R \times 8 \]

\[ 5P = P(1 + 8R) \]

To find the rate of interest \( R \), we divide both sides by \( P \):

\[ 5 = 1 + 8R \]

\[ 8R = 5 - 1 \]

\[ 8R = 4 \]

\[ R = \dfrac{4}{8} \]

\[ R = \dfrac{1}{2} \]

\[ R = 0.5 \]

Converting this to a percentage:

\[ R = 0.5 \times 100\% \]

\[ R = 50\% \]

So, the rate of simple interest for which a sum of money becomes 5 times of itself in 8 years is \( \boxed{50\%} \).

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

\[ A = P + P \times R \times T \]

Where:
- \( A \) is the total amount after interest
- \( P \) is the principal amount
- \( R \) is the rate of interest per annum
- \( T \) is the time period in years

Given that the sum amounts to double in 8 years, we have:

\[ A = 2P \]

We are also given that the time period is 8 years.

So, substituting the values into the formula, we get:

\[ 2P = P + P \times R \times 8 \]

\[ 2P = P(1 + 8R) \]

To find the rate of interest \( R \), we divide both sides by \( P \):

\[ 2 = 1 + 8R \]

\[ 8R = 2 - 1 \]

\[ 8R = 1 \]

\[ R = \dfrac{1}{8} \]

\[ R = 0.125 \]

Converting this to a percentage:

\[ R = 0.125 \times 100\% \]

\[ R = 12.5\% \]

So, the rate of simple interest per annum is \( \boxed{12.5\%} \).