Question

A sum of money becomes \(13 / 5\) times of itself in 32 years at \(r \%\) of SI. What is the value of \(r ?\) (a) \(6 \%\) (b) \(7 \%\) (c) \(5 \%\) (d) \(18 \%\)

Short answer

Answer: 5%

Step-by-step solution

Write down the given information

It is given that the sum of money becomes \(13 / 5\) times of itself in 32 years and we need to find the rate of interest \(r\%\).

Set up an equation for Simple Interest

Using the Simple Interest formula, we can write the equation as \(I = P * R * T/100\). We also know that the final amount \(A = P + I\) and we are given that the sum of money becomes \(13/5\) times of itself, so \(A = \frac{13}{5}P\). Substitute \(I = A - P\) in the Simple Interest equation: \((A - P) = P * R * T/100\).

Substitute the value of A

Since \(A = \frac{13}{5}P\), substitute this into the previous equation: \((\frac{13}{5}P - P) = P * R * T / 100\).

Simplify and substitute the given value of T

The time given is 32 years. So, substitute \(T=32\): \((\frac{13}{5}P - P) = P * R * 32 / 100\). Simplify the equation: \((\frac{8}{5}P) = P * R * 32 / 100\).

Solve for R

To solve for \(R\), first, we can cancel out \(P\): \(\frac{8}{5} = R * 32 / 100\). Now, multiply both sides by \(100 / 32\): \(R = \frac{8}{5} * \frac{100}{32} = \frac{8 * 100}{5 * 32}\). Simplify: \(R = \frac{800}{160} = 5\). So, the value of \(r\) is \(5\%\). The correct answer is (c) \(5\%\).

Key concepts

Rate of Interest Calculation

When dealing with financial mathematics, calculating the rate of interest is crucial, particularly when considering simple interest. Simple interest is calculated using the formula \( I = P \times R \times T / 100 \), where:

• \(I\) is the interest.
• \(P\) is the principal amount.
• \(R\) is the rate of interest per annum.
• \(T\) is the time in years.

In this exercise, you're provided information that helps in finding the rate of interest. The principal becomes \(\frac{13}{5}\) times itself over 32 years. By rearranging the simple interest formula, you can easily solve for \(R\), expressing it in terms of the other known values. This allows you to find the rate of interest that makes the principal grow at this rate.

Equation Simplification

Let's simplify equations step by step to solve complex problems. You begin with the fundamental equations and alter them to find the unknown variable. Here we're dealing with the equation \( (\frac{13}{5}P - P) = P \times R \times 32 / 100 \). By switching terms around and eliminating common factors like \(P\), it reduces the complexity drastically.Simplification:

• The term \(\frac{13}{5}P - P\) simplifies to \(\frac{8}{5}P\).
• You can cancel the \(P\) from both sides to further simplify. This result is \(\frac{8}{5} = R \times 32 / 100\).

This simplification clears the pathway to solving for \(R\) by using perfectly matched coefficients and then multiplying by reciprocal factors, such as \(100/32\) in this context, to isolate \(R\). This process unravels the complex math into simple steps.

Financial Mathematics

Financial mathematics is the branch that deals with calculations related to finances, including loans and interests. It equips you to understand and manipulate equations to make sense of financial decisions and projections. Using the knowledge of financial mathematics:

• You understand how principal amounts transform over time and gain insight into how different parameters affect this change.
• The management of equations such as simple interest is foundational, as shown by needing to understand how the principal and interest relate to terms like interest rate and time.

Practicing financial mathematics ensures you have the skills to approach real-world financial problems with confidence. By mastering the calculations and simplifications involved, like in the given exercise, you enhance your ability to make informed financial decisions regarding investments or loans.