Deana invests some money that earns interest compounded annually. At the end of the first year, she earns $400 in interest. At the end of the second year, she earns $432 in interest.
a) what interest rate, compounded annually, is deana earning?
b) how much did she invest?
Iu
I'm sorry, I'm not sure what you mean by "Iu." Can you please provide more context or information?
a) Well, Deana is certainly earning some interest, but it seems to have a peculiar sense of humor, changing from $400 to $432. It's almost like the interest is telling a joke, but unfortunately, I'm not able to understand its punchline. So, I can't determine the exact interest rate without more information. But if it keeps up this comedy routine, Deana might need to consider investing in a good laugh track.
b) As for how much Deana invested, I'm afraid I can't say for sure. I don't have enough information to calculate her initial investment amount. It's like trying to solve a riddle without any clues – a real head-scratcher. Maybe Deana could try asking the interest rate for some comedy advice. Who knows, it might come up with a hilarious response that leads her to the answer.
To find the interest rate Deana is earning, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the final amount after interest
P = principal amount (initial investment)
r = annual interest rate (as a decimal)
n = number of times interest is compounded per year
t = number of years
We have two pieces of information: After the first year, she earns $400 in interest, and after the second year, she earns $432 in interest. Let's use this information to solve the problem.
a) To find the interest rate Deana is earning, we need to compare the interest earned with the principal amount.
First, let's set up the equation for the first year:
400 = P(1 + r/n)^(n * 1)
Now, let's set up the equation for the second year:
432 = P(1 + r/n)^(n * 2)
We have two equations and two unknowns (P and r). We can solve these equations simultaneously to find the values of P and r.
To do that, we'll divide the second equation by the first equation:
432 / 400 = P(1 + r/n)^(n * 2) / P(1 + r/n)^(n * 1)
On simplifying, we get:
1.08 = (1 + r/n)^n
To solve for r/n, we need to find the value of n that makes this equation true.
One way to do this is by trial and error. Start with different values of n (such as 1, 2, 3, etc.) and calculate (1 + r/n)^n until you find a value that equals 1.08. A good place to start is n = 2, since the interest is compounded annually.
(1 + r/2)^2 = 1.08
By performing calculations, we find that (1 + r/2) ≈ 1.037. This means that r/2 ≈ 0.037.
Next, we solve for r:
r = 0.037 * 2
r ≈ 0.074
So, the interest rate, compounded annually, is approximately 7.4%.
b) Now that we know the interest rate, we can find the principal amount, P. Let's use either of the two equations we set up earlier (using the first equation for convenience):
400 = P(1 + 0.037)^(2)
On simplifying, we get:
400 = P(1.074)^2
Divide both sides of the equation by (1.074)^2 to isolate P:
P = 400 / (1.074)^2
On performing the calculations, we find:
P ≈ $347.67
Therefore, Deana invested approximately $347.67.
I think you can do the following:
P*R=I
Where
P=principal=amount investere
R=interest rate
and
I=amount in interest
You have two equations.
1.)
P*R=400
and
(P+400)*R=432
****The principal now includes the interest from the previous year.
Solving for P (Equation 1):
P=400/R
Plug into equation 2, and solve for R:
(400/R+400)*R=432
400+400R=432
400R=32
R=32/400
R=0.008=0.8%
You know R, so solve for P:
Using equation 1:
P=400/R
P=400/0.008
P=$50,000
Fixed a typo: Look for #%^
I think you can do the following:
P*R=I
Where
P=principal=amount invested (#$^)
R=interest rate
and
I=amount in interest
You have two equations.
1.)
P*R=400
and
(P+400)*R=432
****The principal now includes the interest from the previous year.
Solving for P (Equation 1):
P=400/R
Plug into equation 2, and solve for R:
(400/R+400)*R=432
400+400R=432
400R=32
R=32/400
R=0.008=0.8%
You know R, so solve for P:
Using equation 1:
P=400/R
P=400/0.008
P=$50,000