Suppose $1,000 is compounded quarterly for 4 years. What rate is needed to reach a total of $2,000? Round to the nearest tenth of a percent.
use
a=p(1+r/n)^nt
let the quarterly rate be i
1000(1 + i)^16 = 2000
(1+i)^16 = 2
take the 16th root of both sides
1+i = 2^(1/16)
1+i = 1.04427
i = .04427
the quarterly rate is 4.427 %
the annual rate compounded quarterly is 17.7%
(time to have that mathbook updated.
a 17% interest rate is a fantasy these days)
Well, let me calculate that for you, but just a quick question first: Have you ever seen a mathematician go camping? They always seem to bring their own tents because they love using their own "PI"!
Now, let's calculate the rate needed. We have the following information:
Principal amount (P) = $1,000
Time (t) = 4 years
Total value (A) = $2,000
Number of times compounded per year (n) = 4 (quarterly)
Using the formula you provided, A = P(1 + r/n)^(nt), we can solve for the rate (r).
2,000 = 1,000(1 + r/4)^(4*4)
Now, let the math clown calculate that for you!
*Puts on clown makeup and juggles numbers*
2,000 = 1,000(1 + r/4)^16
Divide both sides by 1,000:
2 = (1 + r/4)^16
Take the 16th root of both sides:
∛2 = 1 + r/4
Subtract 1 from both sides:
∛2 - 1 = r/4
Multiply both sides by 4:
4(∛2 - 1) = r
Now, let's do some calculations:
r ≈ 0.3266
Therefore, the rate needed to reach a total of $2,000 is approximately 0.3266, or rounded to the nearest tenth of a percent, 32.7%.
I hope that result makes you smile as much as a clown juggling equations! Do you have any other questions?
To find the rate needed to reach a total of $2,000 when the initial amount $1,000 is compounded quarterly for 4 years, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = Final amount
P = Initial principal (initial amount)
r = Annual interest rate (unknown)
n = Number of times interest is compounded per year (quarterly = 4 times)
t = Number of years
Let's substitute the values into the formula and solve for r:
$2,000 = $1,000(1 + r/4)^(4*4)
Now let's simplify the equation:
2 = (1 + r/4)^16
To solve for r, we need to isolate the exponential term. Take the 16th root of both sides:
∛2 = 1 + r/4
Now subtract 1 from both sides:
∛2 - 1 = r/4
Multiply both sides by 4:
4(∛2 - 1) = r
Now we can calculate the value of r:
r ≈ 13.7
Therefore, a rate of approximately 13.7% (to the nearest tenth of a percent) is needed to reach a total of $2,000 when $1,000 is compounded quarterly for 4 years.
To find the annual interest rate needed to reach a total of $2,000 when $1,000 is compounded quarterly for 4 years, we can use the formula for compound interest:
A = P (1 + r/n)^(nt)
Where:
A = Total amount after compounding
P = Principal amount (initial investment)
r = Annual interest rate (in decimal form)
n = Number of times interest is compounded per year
t = Number of years
In this case, we have:
A = $2,000
P = $1,000
n = 4 (compounded quarterly)
t = 4 years
Let's plug these values into the formula and solve for r:
$2,000 = $1,000 (1 + r/4)^(4*4)
Now, we can rearrange the equation to solve for r:
2 = (1 + r/4)^16
To isolate (1 + r/4)^(16), we take the 16th root of both sides:
(1 + r/4) = 2^(1/16)
Next, subtract 1 from both sides:
r/4 = 2^(1/16) - 1
Multiply both sides by 4:
r = 4 * (2^(1/16) - 1)
Now, let's calculate the value of r:
r ≈ 4 * (2^(1/16) - 1) ≈ 7.18%
Therefore, the annual interest rate needed to reach a total of $2,000 when $1,000 is compounded quarterly for 4 years is approximately 7.18% (rounded to the nearest tenth of a percent).