Derrick is investing $1,000 at 5% interest and Anna is investing $750 at 7% interest. Both interest rates are compounded annually. When will they have the same amount saved? (Hint: 1.05t/1.07t =(1.05/1.07)t )
we want
1000*1.05^t = 750*1.07^t
(1.05/1.07)^t = 3/4
.9813^t = .75
t = log(.75)/log(.9813) = 15.24 years
Extra credit: after that point, who gets the larger amount?
calculate the compound interest on an investment of $18,000 at 8%, interest compounded quarterly, for 15 months.
To determine when Derrick and Anna will have the same amount saved, we can set up an equation using the formula for compound interest:
Amount = Principal * (1 + Interest Rate)^Time
Let's denote the time in years as 't'. Now we can set up an equation for Derrick's savings (D) and Anna's savings (A):
D = $1,000 * (1 + 0.05)^t
A = $750 * (1 + 0.07)^t
To find when they will have the same amount saved, we can set D equal to A and solve for t:
$1,000 * (1 + 0.05)^t = $750 * (1 + 0.07)^t
Divide both sides of the equation by $750 * (1 + 0.07)^t:
(1 + 0.05)^t / (1 + 0.07)^t = $1,000 / $750
Simplify the right side:
(1.05/1.07)^t = 4/3
Now we can take the logarithm of both sides of the equation to solve for t:
log[(1.05/1.07)^t] = log(4/3)
Using the logarithmic properties, we can bring down the exponent:
t * log(1.05/1.07) = log(4/3)
Finally, divide both sides by log(1.05/1.07):
t = log(4/3) / log(1.05/1.07)
Use a calculator to find the logarithms and perform the division to get the value for t. This will tell you when Derrick and Anna will have the same amount saved.