30.1k views
3 votes
$\$5000$ compounded annually at an $x\%$ interest rate takes $6$ years to double. at the same interest rate, how many years will it take $\$300$ to grow to $\$9600$?

User GeNia
by
8.5k points

1 Answer

3 votes

Answer:

10.058 years for $300 to grow to $9600 at the same interest rate.Explanation:

To solve this problem, we can use the compound interest formula:

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

Where:

A is the final amount

P is the principal amount (initial amount)

r is the annual interest rate (as a decimal)

n is the number of times the interest is compounded per year

t is the number of years

For the first scenario:

P = $5000

A = 2P = $10000

t = 6 years

We need to find the interest rate, r, that will make the amount double in 6 years. Let's substitute the values into the formula and solve for r:

$10000 = $5000(1 + r/1)^(1*6)

2 = (1 + r)^6

Taking the sixth root of both sides:

1 + r = √2

r = √2 - 1

Now we can use this interest rate to solve for the time it will take for $300 to grow to $9600.

P = $300

A = $9600

r = √2 - 1

Let's substitute the values into the formula and solve for t:

$9600 = $300(1 + (√2 - 1)/1)^(1*t)

32 = (√2)^(t)

Taking the logarithm of both sides (with base √2):

log(32) = log(√2)^t

log(32) = t * log(√2)

Using a calculator to evaluate the logarithms:

t ≈ log(32) / log(√2)

t ≈ 10.058

Therefore, it will take approximately 10.058 years for $300 to grow to $9600 at the same interest rate.

User Johlo
by
8.4k points