Tia is investing $2500 that she would like to grow to $6000 in 10 years. At
what annual interest rate, compounded quarterly, must Tia invest her money?
Round your answer to two decimal places.
2500(1+r/4)^(4*10) = 6000
(1+r/4)^40 = 2.4
1+r/4 = 1.022128
r/4 = 0.022128
r = 0.0885 = 8.85%
Well, Tia sure has some big financial goals! If she wants to turn $2500 into $6000 in 10 years, we need to figure out the interest rate.
To do this, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = the future value (in this case, $6000)
P = the principal (initial investment of $2500)
r = the annual interest rate (what we're trying to find)
n = the number of times the interest is compounded per year (in this case, quarterly)
t = the number of years (in this case, 10)
Plugging in the values, we get:
6000 = 2500 * (1 + r/4)^(4 * 10)
Now, let's solve for r. But since we clowns like to have some fun, let's rearrange the equation a bit to make it a little more exciting:
6000/2500 = (1 + r/4)^(40)
Now, let's take the logarithm of both sides because I certainly don't want to do all that exponentiation manually:
log(6000/2500) = log((1 + r/4)^(40))
Using the logarithmic property of exponents, we can move that pesky 40 down:
log(6000/2500) = 40 * log(1 + r/4)
Now, let's isolate that r/4:
log(6000/2500)/40 = log(1 + r/4)
And finally, let's multiply both sides by 4:
4 * log(6000/2500)/40 = r
Oh no! It looks like I made a miscalculation while juggling those numbers! Let's try that again, but this time, I'll get it right.
r ≈ 2.94%
So Tia needs to invest her money at an annual interest rate of approximately 2.94%, compounded quarterly, to reach her goal of $6000 in 10 years. Good luck, Tia! Maybe she should invest in clown shoes to give her investment a little extra bounce!
To determine the annual interest rate required, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment
P = the principal amount (initial investment)
r = the annual interest rate (as a decimal)
n = the number of times interest is compounded per year
t = the number of years
Given:
P = $2500
A = $6000
n = 4 (compounded quarterly)
t = 10
Plugging in the values into the formula, we can rearrange it to solve for r:
A = P(1 + r/n)^(nt)
6000 = 2500(1 + r/4)^(4*10)
Divide both sides by 2500:
2.4 = (1 + r/4)^(40)
Taking the 40th root:
(1 + r/4) = 2.4^(1/40)
Subtracting 1:
r/4 = 2.4^(1/40) - 1
Multiply both sides by 4:
r = 4 * (2.4^(1/40) - 1)
Using a calculator, we can find the value of r as approximately:
r = 0.0379
Therefore, Tia must invest her money at an annual interest rate of approximately 3.79% when compounded quarterly.
To find the annual interest rate, compounded quarterly, at which Tia must invest her money to grow it to $6000 in 10 years, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = final amount ($6000)
P = principal amount ($2500)
r = annual interest rate (unknown)
n = number of times compounding occurs per year (quarterly, so n = 4)
t = number of years (10 years)
We need to solve for r. Rearranging the formula, we have:
r = (A/P)^(1/(n*t)) - 1
Substituting the values, we get:
r = (6000/2500)^(1/(4*10)) - 1
Calculating this expression, we find the annual interest rate (compounded quarterly) required for Tia's investment to grow to $6000 in 10 years:
r ≈ 0.0754 or 7.54%
Therefore, Tia must invest her money at an annual interest rate of approximately 7.54%, compounded quarterly, to reach her goal.
6000 = 2500 (1 + r/4)^(10 * 4)
2.4 = (1 + r/4)^40
log(2.4) = 40 log(1 + r/4)
.0095053 = log(1 + r/4)
1.002213 = 1 + r/4