A sum of money doubles itself in 5 years at compound interest. In how many years will it become 8 times?

Sure, let's solve this step by step.

Step 1: Understanding the Problem

We are given that a sum of money doubles itself in 5 years at compound interest. We need to find out in how many years it will become 8 times its original amount.

Step 2: Using the Compound Interest Formula

The compound interest formula is: $A = P \left(1 + \frac{r}{100}\right)^n$ where:

• $A$ is the amount of money accumulated after n years, including interest.
• $P$ is the principal amount (the initial sum of money).
• $r$ is the annual interest rate (in percentage).
• $n$ is the number of years the money is invested or borrowed for.

Step 3: Setting Up the Equation for Doubling

Since the money doubles in 5 years, we can set up the equation as: $2P = P \left(1 + \frac{r}{100}\right)^5$

Step 4: Simplifying the Equation

Divide both sides by $P$: $2 = \left(1 + \frac{r}{100}\right)^5$

Step 5: Solving for $\left(1 + \frac{r}{100}\right)$

Take the 5th root of both sides: $1 + \frac{r}{100} = 2^{\frac{1}{5}}$

Step 6: Setting Up the Equation for 8 Times

We need to find the number of years $n$ for the money to become 8 times its original amount: $8P = P \left(1 + \frac{r}{100}\right)^n$

Step 7: Simplifying the Equation

Divide both sides by $P$: $8 = \left(1 + \frac{r}{100}\right)^n$

Step 8: Substituting $\left(1 + \frac{r}{100}\right)$

From Step 5, we know: $1 + \frac{r}{100} = 2^{\frac{1}{5}}$

Substitute this into the equation: $8 = \left(2^{\frac{1}{5}}\right)^n$

Step 9: Solving for $n$

We know that $8 = 2^3$, so: $2^3 = \left(2^{\frac{1}{5}}\right)^n$

This simplifies to: $2^3 = 2^{\frac{n}{5}}$

Since the bases are the same, we can equate the exponents: $3 = \frac{n}{5}$

Step 10: Solving for $n$

Multiply both sides by 5: $n = 3 \times 5$ $n = 15$

Conclusion

The sum of money will become 8 times its original amount in 15 years.

This problem can be solved using the formula for compound interest, which is A = P(1 + r/n)^(nt), where:

• A is the amount of money accumulated after n years, including interest.
• P is the principal amount (the initial amount of money).
• r is the annual interest rate (in decimal form).
• n is the number of times that interest is compounded per year.
• t is the time the money is invested for in years.

Given that the sum of money doubles in 5 years, we can set up the equation as follows:

2P = P(1 + r/n)^(5n)

Solving for r/n, we get:

2 = (1 + r/n)^5n

Taking the fifth root of both sides, we get:

1 + r/n = 2^(1/5)

Subtracting 1 from both sides, we get:

r/n = 2^(1/5) - 1

Now, we want to find out when the sum of money will become 8 times. We can set up the equation as follows:

8P = P(1 + r/n)^(nt)

Substituting the value of r/n from above, we get:

8 = (2^(1/5))^t

Taking the log base 2^(1/5) of both sides, we get:

t = log_(2^(1/5))(8)

Using the change of base formula, we get:

t = log(8) / log(2^(1/5))

Calculating the above expression, we get:

t = 15 years

So, the sum of money will become 8 times in 15 years.