Question

At compound interest, if a certain sum of money doubles in \(n\) years, then the amount will be four fold in : (a) \(n^{2}\) years (b) \(2 n^{2}\) years (c) \(2 n\) years (d) \(4 \pi\) years

Short answer

(c) 2n years

Step-by-step solution

Recall the compound interest formula

The formula for compound interest is given by: \(A = P(1 + \frac{r}{100})^{t}\) where, \(A\) = final amount \(P\) = principal amount \(r\) = rate of interest \(t\) = number of years

Setup the given condition that money doubles in n years

According to the given condition, the amount doubles in \(n\) years which means: \(2P = P(1 + \frac{r}{100})^{n}\)

Solve for r

To solve for the rate of interest (r), we first divide both sides of the equation by P, giving us: \(2 = (1 + \frac{r}{100})^{n}\) Now, taking the nth root of both sides, we get: \((1 + \frac{r}{100}) = \sqrt[n]{2}\) Now, solve for r: \(r = 100(\sqrt[n]{2} - 1)\)

Find the condition for the amount to become four times the original amount

We need to find the time it takes for the amount to become four times the original amount, so: \(4P = P(1 + \frac{r}{100})^{t}\)

Substitute r from Step 3 and solve for t

Substitute the expression we found in Step 3 for r: \(4P = P(1 + \frac{100(\sqrt[n]{2} - 1)}{100})^{t}\) Divide both sides by P, and simplify the equation: \(4 = (\sqrt[n]{2})^{t}\) Taking the logarithm of both sides, we have: \(\log 4 = t\log(\sqrt[n]{2})\) Now, solve for t: \(t = \frac{\log 4}{\log(\sqrt[n]{2})}\) Since \(4 = 2^{2}\), we can rewrite the equation as: \(t = \frac{\log 2^{2}}{\log(\sqrt[n]{2})}\) Applying the rule of logarithms \(\log a^{b} = b\log a\), we have: \(t = \frac{2\log 2}{\frac{1}{n}\log 2}\) Finally, simplify to find the value of t: \(t = 2n\)

Choose the correct answer

From our calculations, we found that the amount will be four times the original amount in \(2n\) years. Therefore, the correct answer is (c) \(2n\) years.

Key concepts

Doubling Time

Doubling time in the context of compound interest is the period it takes for an investment to grow to twice its size. It is an important concept when determining how quickly you can expect your investment to grow. Understanding this concept begins with the compound interest formula:

• \( A = P(1 + \frac{r}{100})^t \)

- Here, \( A \) is the final amount, \( P \) is the principal, \( r \) is the rate of interest, and \( t \) is the time in years.If a certain amount doubles in \( n \) years, then we can set up the equation:

• \( 2P = P(1 + \frac{r}{100})^n \)

This equation indicates that the final amount \( 2P \) is twice the principal, showing the doubling effect. Handling these calculations provides insight into how quick growth can be attained given a fixed rate of interest. Recognizing doubling time helps in planning investment goals effectively.

Logarithms

Logarithms are mathematical tools used to simplify the process of solving exponential equations. They turn multiplications into additions, which are easier to solve. When dealing with compound interest problems, logarithms can be invaluable to determine time or interest rates where variables are involved in an exponent.In our problem of determining when an amount quadruples its original value, we use logarithms as follows:

• We derived \( 4 = (\sqrt[n]{2})^t \)

Taking logarithms on both sides helps us simplify this equation for \( t \):

• \( \log 4 = t \log (\sqrt[n]{2}) \)

By using the property \( \log a^b = b \log a \), further simplification leads to:

• \( t = \frac{\log 2^2}{\frac{1}{n} \log 2} \)

Logarithms thus help unravel the complexity of working with exponents, aiding in calculating doubling time or any variations of growth related inquiries.

Rate of Interest

The rate of interest \( r \) is a crucial component of compound interest. It represents how much the principal grows each period. This problem demonstrates how to determine the rate if we know the doubling time:Using the given equation:

• \( 2 = (1 + \frac{r}{100})^n \)

- It indicates that a sum grows to double its original amount in \( n \) years.Solving for \( r \), we first rearrange:

• \( 1 + \frac{r}{100} = \sqrt[n]{2} \)

Subtracting 1 from both sides gives us:

• \( \frac{r}{100} = \sqrt[n]{2} - 1 \)

And multiplying by 100 to clear the fraction:

• \( r = 100(\sqrt[n]{2} - 1) \)

This expression allows you to calculate the interest rate associated with a specific doubling period, letting you adjust investment strategies based on predicted returns.

Growth Calculation

Calculating growth in terms of compound interest involves understanding the formula that defines how our investment increases over time. Key elements of calculation include initial principal, interest rate, and the compounding period. By piecing these together, we can determine future investment value:The formula used is:

• \( A = P (1 + \frac{r}{100})^t \)

In the context of the exercise, once the interest rate and time for doubling (\( n \) years) are known, determining when the complete amount grows four-fold is a matter of applying these principles.We establish that:

• \( 4P = P (1 + \frac{r}{100})^t \)

Dividing and simplifying:

• \( 4 = (\sqrt[n]{2})^t \)

Finally using logarithms to solve for \( t \) indicated quadruple growth occurs over \( 2n \) years. These steps reveal how exponential growth through compound interest can be comprehended and calculated, essential for investment forecasting.