This problem practice test:
Abe wants to invest $3000 in a bond paying 2% interest.
A. What will be the amount of the bond after it matures in 8 years.
B. How long will it take to double?
I met to type in school subject.
A. A=P(1+r/n)^n*t
A=$3000(1*0.02/8)^8*8
A=$3000(1+0.0025)^64
A=$3000(1.0025)^64
A=3000(1.173276583)
A=3519.829749=3520.82
B. A=P(1+r/n)^n*t
2000=$3000(1+0.02/8)^8*t
-2000-2000
$1000(1+0.00025/8)^8t
$1000(1.0025)8t
1000(1.020175878)
=1020.175878=102.185
You have the correct formula
A=P(1+r/n)^(n*t)
but "n" is the number of times per year that the interest is compounded. Since it does not specify a shorter compounding period, you should assume annual compounding, so n=1. So, you should proceed with
A. 3000(1+0.02)^8 = 3514.98
B. 3000(1+.02)^t = 2*3000
Not sure where you got that 2000, since it bears no relation to doubling. In fact, the doubling time is unrelated to the amount invested, since the 3000 can be canceled, leaving
(1+.02)^t = 2
t log1.02 = log2
t = log2/log1.02 = 35
To calculate the amount of the bond after it matures in 8 years, 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 = annual interest rate (as a decimal)
n = number of times interest is compounded per year
t = number of years
Let's plug in the values given:
P = $3000
r = 2% = 0.02 (as a decimal)
n = 1 (compounded annually)
t = 8 years
A = 3000(1 + 0.02/1)^(1*8)
A = 3000(1 + 0.02)^8
A = 3000(1.02)^8
A = 3000(1.1716)
A ≈ $3,514.80
So, the amount of the bond after it matures in 8 years will be approximately $3,514.80.
Now, let's move on to calculating how long it will take for the bond to double in value.
To find the time it takes to double an investment with compound interest, we can use the formula:
t = log(2) / log(1 + r/n)
In this case, we want to find t when the principal (P) doubles, so the future value (A) will be 2 times the principal amount.
Let's set up the equation:
2P = P(1 + r/n)^(nt)
Dividing both sides by P:
2 = (1 + r/n)^(nt)
Now solve for t:
log(2) = log((1 + r/n)^(nt))
log(2) = nt * log(1 + r/n)
t = log(2) / (n * log(1 + r/n))
Plugging in the values from the given information:
r = 2% = 0.02 (as a decimal)
n = 1 (compounded annually)
t = log(2) / (1 * log(1 + 0.02/1))
t = log(2) / (log(1.02))
t ≈ 35.00
So, it will take approximately 35 years to double the value of the bond at a 2% interest rate.