Jasmine invested $73,000 in a fixed rate account offering 7.2% interest rate compounded semiannually. How long will it take her to grow it to $100,000?
100,000=(73,000)(0.072)^2)t
P = Po(1+r)n = 100,000.
r = 0.072/2 = 0.036 = Semi-annual % rate expressed as a decimal.
n = 2Comp./yr. * t yrs. = 2t Compounding periods.
73,000(1.036)^2t = 100,000.
(1.036)^2t = 100,000/73,000 = 1.370, 2t*Log1.036 = Log1.370,
2t = Log1.370/Log1.036 = 8.90, t = 4.45 Years.
To find out how long it will take Jasmine to grow her investment to $100,000, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the final amount (in this case, $100,000)
P = the initial principal (in this case, $73,000)
r = the annual interest rate (in this case, 7.2% or 0.072)
n = the number of times the interest is compounded per year (in this case, semiannually, so 2)
t = the time in years
Plugging in the given values, the equation becomes:
100,000 = 73,000(1 + 0.072/2)^(2t)
Now, we can solve for t. Let's simplify the equation further:
100,000 = 73,000(1.036)^2t
Divide both sides of the equation by 73,000:
1.3699 = (1.036)^2t
Next, take the natural logarithm (ln) of both sides of the equation:
ln(1.3699) = ln[(1.036)^2t]
Using the logarithm property ln(a^b) = b * ln(a):
ln(1.3699) = 2t * ln(1.036)
Now, divide both sides of the equation by 2 * ln(1.036):
t = ln(1.3699) / (2 * ln(1.036))
Using a calculator, we can find that ln(1.3699) ≈ 0.3156 and ln(1.036) ≈ 0.0351:
t = 0.3156 / (2 * 0.0351)
Simplifying further:
t ≈ 4.51
Therefore, it will take approximately 4.51 years for Jasmine to grow her investment to $100,000.
To find out how long it will take for Jasmine's investment to grow to $100,000, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
In this case:
A = $100,000 (the desired amount)
P = $73,000 (the initial investment)
r = 7.2% (the interest rate expressed in decimal form: 0.072)
n = 2 (compounded semiannually, so twice a year)
t = the time in years
Substituting these values into the formula, we get:
100,000 = 73,000(1 + 0.072/2)^(2t)
Next, we can simplify the equation by dividing both sides by 73,000:
100,000 / 73,000 = (1 + 0.072/2)^(2t)
1.3699 = (1.036)^2t
To isolate the variable, we can take the natural logarithm of both sides:
ln(1.3699) = ln[(1.036)^2t]
Using the property of logarithms, we can bring the exponent down:
ln(1.3699) = 2t * ln(1.036)
Now, divide both sides by 2 * ln(1.036) to solve for t:
t = ln(1.3699) / (2 * ln(1.036))
Using a calculator, you can find the natural logarithm of 1.3699 and 1.036 and complete the division to find the value of t. The result of this calculation will give you the number of years it will take for Jasmine's investment to grow to $100,000.