Calculating Time in Simple Interest

The problem asks us to find the time it takes for a principal amount to grow to a specific amount under simple interest at a given rate. We are given the principal amount, the final amount, and the simple interest rate per annum. We need to calculate the time in years.

Understanding the Given Information

• Principal amount ($\(P\)$) = ₹90
• Final Amount ($\(A\)$) = ₹108
• Simple Interest Rate ($\(R\)$) = 6¼% per annum

Calculating the Simple Interest Earned

Simple Interest ($\(SI\)$) is the difference between the final amount and the principal amount.

$\(SI = A - P\)$

Substitute the given values:

$\(SI = ₹108 - ₹90\)$

$\(SI = ₹18\)$

So, the simple interest earned is ₹18.

Converting the Interest Rate

The interest rate is given as 6¼%. We can convert this mixed fraction percentage into a simple fraction or a decimal.

6¼% = 6 + ¼% = 6.25%

As a fraction, 6.25% is $\(\frac{6.25}{100}\)$ or $\(\frac{6\frac{1}{4}}{100} = \frac{\frac{25}{4}}{100} = \frac{25}{4 \times 100} = \frac{25}{400}\)$ which simplifies to $\(\frac{1}{16}\)$. Using the fraction $\(\frac{25}{4}\)$ in the formula with the division by 100 is usually easier.

Using the Simple Interest Formula to Find Time

The formula for simple interest is:

$\(SI = \frac{P \times R \times T}{100}\)$

Where:

• $\(SI\)$ is the Simple Interest
• $\(P\)$ is the Principal amount
• $\(R\)$ is the Rate of interest per annum
• $\(T\)$ is the Time in years

We need to find $\(T\)$. We can rearrange the formula to solve for $\(T\)$:

$\(T = \frac{SI \times 100}{P \times R}\)$

Substituting Values and Calculating Time

Substitute the values we know into the rearranged formula:

$\(T = \frac{₹18 \times 100}{₹90 \times 6\frac{1}{4}\%}\)$

Using $\(R = \frac{25}{4}\% \)$: Note that the rate $\(R\)$ in the formula $\(\frac{P \times R \times T}{100}\)$ is the numerical value of the percentage rate. So, R = $\(\frac{25}{4}\)$.

$\(T = \frac{18 \times 100}{90 \times \frac{25}{4}}\)$

$\(T = \frac{1800}{\frac{90 \times 25}{4}}\)$

$\(T = \frac{1800}{\frac{2250}{4}}\)$

To divide by a fraction, multiply by its reciprocal:

$\(T = 1800 \times \frac{4}{2250}\)$

$\(T = \frac{1800 \times 4}{2250}\)$

$\(T = \frac{7200}{2250}\)$

Now, simplify the fraction by dividing the numerator and denominator by common factors. We can start by cancelling a 0 from both:

$\(T = \frac{720}{225}\)$

Both numbers are divisible by 5:

$\(T = \frac{720 \div 5}{225 \div 5} = \frac{144}{45}\)$

Both numbers are divisible by 9:

$\(T = \frac{144 \div 9}{45 \div 9} = \frac{16}{5}\)$

The time is $\(\frac{16}{5}\)$ years. Convert this improper fraction to a mixed fraction:

$\(\frac{16}{5} = 3\frac{1}{5}\)$ years.

So, it will take 3⅕ years for the sum of ₹90 to become ₹108 at 6¼% simple interest.

Revision Table: Simple Interest Concepts

Additional Information on Simple Interest Calculations

Simple interest is a basic concept in finance. It is the easiest type of interest to calculate because it is based solely on the original principal amount. This means the interest earned in each period (e.g., each year) is constant, unlike compound interest where interest is calculated on the principal plus accumulated interest.

Understanding simple interest is important for calculating earnings on simple deposit schemes or interest on simple loans. The formula $\(SI = \frac{PRT}{100}\)$ can be rearranged to find any of the variables ($\(P\)$, $\(R\)$, or $\(T\)$) if the other three and the Simple Interest ($\(SI\)$) are known. Remember that the rate $\(R\)$ is typically an annual rate, and the time $\(T\)$ must also be in years for the formula to work correctly. If the time is given in months, divide by 12; if in days, divide by 365 (or 366 for a leap year).