What rate of interest, to the nearest tenth of a percent, compounded quarterly is needed for an investment of $1700 to grow to $2600 in 12 years?
I am given the formula A=P(1+r/n)^nt
so, solve for r:
2600 = 1700(1+r/4)^(4*12)
26/17 = (1+r/4)^48
(26/17)^(1/48) = 1+r/4
r = 4[(26/17)^(1/48) - 1]
r = 0.0355
= 3.55%
actually 3.55641% yields 2599.9999
which is pretty darn close
To find the rate of interest compounded quarterly, we can use the formula A = P(1 + r/n)^(nt), where:
A = the final amount ($2600)
P = the principal amount ($1700)
r = the interest rate (unknown)
n = the number of times the interest is compounded per year (4 quarters in a year)
t = the number of years (12 years)
To solve for r, we need to rearrange the formula:
A = P(1 + r/n)^(nt)
Divide both sides by P:
A/P = (1 + r/n)^(nt)
Now, substitute the known values:
2600/1700 = (1 + r/4)^(4*12)
Simplify the expression:
1.52941 = (1 + r/4)^48
Next, isolate the exponential term:
Take the 48th root of both sides:
(1 + r/4) = (1.52941)^(1/48)
Now, solve for r:
Subtract 1 from both sides:
r/4 = (1.52941)^(1/48) - 1
Multiply both sides by 4:
r = 4 * [(1.52941)^(1/48) - 1]
Using a calculator, we can find that (1.52941)^(1/48) is approximately 1.0127.
Substituting the value:
r = 4 * (1.0127 - 1)
Simplifying further:
r = 4 * 0.0127
Calculating the final answer:
r ≈ 0.0508
Therefore, the interest rate required is approximately 0.0508 or 5.08% compounded quarterly.