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.
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.
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