Joe invests $10 000 in a savings account with compound interest x % per annum.
After 5 years, the balance in the account is $11 000. Find the value of x.
To find the value of x, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = final amount
P = principal amount (initial investment)
r = annual interest rate (as a decimal)
n = number of times the interest is compounded per year
t = number of years
Given:
P = $10,000
A = $11,000
t = 5 years
Plugging in the values, we get:
11,000 = 10,000(1 + r/n)^(n*5)
Simplifying this equation:
11/10 = (1 + r/n)^(5n)
Taking the natural log (ln) of both sides:
ln(11/10) = ln((1 + r/n)^(5n))
Using the property of logarithms (ln(a^b) = b * ln(a)):
ln(11/10) = 5n * ln(1 + r/n)
Dividing both sides by 5:
ln(11/10) / 5 = n * ln(1 + r/n)
Let's define a variable, k, as the left side of the equation:
k = ln(11/10) / 5
Substituting this back into the equation:
k = n * ln(1 + r/n)
Now we can solve for x, the interest rate:
x/n = e^k - 1
Where e is Euler's number (approximately 2.71828).
Substituting k and simplifying:
x/n = e^(ln(11/10) / 5) - 1
x/n = (11/10)^(1/5) - 1
Since the interest rate (x) is given as a percentage, we multiply x/n by 100:
100(x/n) = (11/10)^(1/5) - 1
Multiply both sides by n:
100x = n * ((11/10)^(1/5) - 1)
To find the value of x, we need to know the value of n (number of times the interest is compounded per year).
Without knowing the value of n, we cannot determine the exact value of x.
To find the value of x, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = final balance = $11,000
P = principal amount = $10,000
r = annual interest rate (in decimal form)
n = number of times interest is compounded per year
t = number of years
In this case, the principal amount is $10,000, the final balance is $11,000, and the time is 5 years. We need to find the value of the annual interest rate, represented as x%.
Let's substitute the given values into the formula:
$11,000 = $10,000(1 + x/100)^5
Dividing both sides of the equation by $10,000:
1.1 = (1 + x/100)^5
To isolate the base (1 + x/100), we take the fifth root of both sides:
(1 + x/100) = √(1.1)
Next, we solve for x:
1 + x/100 = √(1.1)
Subtracting 1 from both sides:
x/100 = √(1.1) - 1
Multiplying both sides by 100:
x = 100 * (√(1.1) - 1)
Evaluating the expression on the right side using a calculator, we find:
x ≈ 4.5
Therefore, the value of x is approximately 4.5%.