I want the solutions for this question in fractions

Certainly! Let's solve the problem using fractions.

We have:
- After 4 years, the amount is Rs. 54,000.
- After 7 years, the amount is Rs. 59,568.

Using the compound interest formula:

\[
A = P \left(1 + \frac{r}{100}\right)^n
\]

We have two equations:

1. \( 54000 = P \left(1 + \frac{r}{100}\right)^4 \)  \(\quad \text{(Equation 1)}\)
2. \( 59568 = P \left(1 + \frac{r}{100}\right)^7 \)  \(\quad \text{(Equation 2)}\)

### Step 1: Divide Equation 2 by Equation 1
\[
\frac{59568}{54000} = \frac{\left(1 + \frac{r}{100}\right)^7}{\left(1 + \frac{r}{100}\right)^4}
\]

Simplifying this, we get:

\[
\frac{59568}{54000} = \left(1 + \frac{r}{100}\right)^3
\]

### Step 2: Simplify the Fraction
First, simplify \(\frac{59568}{54000}\):

\[
\frac{59568}{54000} = \frac{59568 \div 72}{54000 \div 72} = \frac{828}{750}
\]

Further simplifying:

\[
\frac{828}{750} = \frac{276}{250} = \frac{138}{125}
\]

Now the equation becomes:

\[
\left(1 + \frac{r}{100}\right)^3 = \frac{138}{125}
\]

### Step 3: Solve for \(r\)
Taking the cube root on both sides:

\[
1 + \frac{r}{100} = \left(\frac{138}{125}\right)^{\frac{1}{3}}
\]

Now, find the cube root of \(\frac{138}{125}\):

\[
1 + \frac{r}{100} = \frac{\sqrt[3]{138}}{\sqrt[3]{125}}
\]

Since \(\sqrt[3]{125} = 5\), the equation simplifies to:

\[
1 + \frac{r}{100} = \frac{\sqrt[3]{138}}{5}
\]

Therefore:

\[
\frac{r}{100} = \frac{\sqrt[3]{138}}{5} - 1 = \frac{\sqrt[3]{138} - 5}{5}
\]

To express \( r \) in terms of fractions:

\[
r = 100 \times \frac{\sqrt[3]{138} - 5}{5} = 20 \times \left(\sqrt[3]{138} - 5\right)
\]

This gives the exact expression for \( r \) in terms of a fraction.

If we approximate \(\sqrt[3]{138}\) to a fractional value:

\[
\sqrt[3]{138} \approx 5.14 \quad (\text{approximate value})
\]

Thus,

\[
r \approx 20 \times (5.14 - 5) = 20 \times 0.14 = 2.8\%
\]

This is a more precise fractional solution before approximation.

Would you like further details or have any additional questions?

Here are five more related questions:
1. How would the solution change if we used a different principal value?
2. What is the relationship between the principal and the rate of interest in compound interest problems?
3. Can we express the interest rate as a ratio instead of a percentage?
4. How does the compound interest formula change if the interest is compounded semi-annually?
5. Can the derived fractional formula be simplified further for a particular range of values?

**Tip:** Using fractions in financial calculations can help avoid rounding errors, leading to more accurate results.

Learn how to solve compound interest problems using fractions with a detailed step-by-step guide. This problem involves calculating compound interest over different periods and finding the interest rate in fractional terms. Suitable for undergraduate-level students and financial analysts.

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