A sum of $850 is invested for 10 years and the interest is compounded quarterly. There is $1100 in the account at the end of 10 years. What is the nominal annual rate?
P = Po(1+r)^n.
Po = $850, r = Quarterly % rate expressed as a decimal.
n = 4Comp./yr. * 10yrs. = 40 Compounding periods.
P = 850(1+r)^40 = 1100,
(1+r)^40 = 1100/850 = 1.294,
Raise both sides to the 1/40 power: 1+r = 1.294^(1/40) = 1.00647, r = 1.00647 - 1 = 0.00647 = Quarterly % rate expressed as a decimal.
APR = 4 * 0.00647 = 0.0259 = 2.59 %.
Well, let's crunch some numbers together! We have $850 that magically grows to $1100 in 10 years. That means it must be a really talented dollar. Bravo, dollar!
Now, since the interest is compounded quarterly, it means that the interest is added four times a year. So, over the course of 10 years, there would have been a total of 40 times that the interest is added.
To find the nominal annual rate, we can use the formula:
A = P(1 + r/n)^(n*t)
Where:
A = the final amount ($1100)
P = the principal amount ($850)
r = the nominal annual interest rate (what we're trying to find)
n = the number of times compounded per year (4 times, since it's quarterly)
t = the number of years (10 years)
Now, let's put our clown noses on and do some math fun!
1100 = 850(1 + r/4)^(4*10)
Now comes the tricky part of solving for r. But don't worry, I'm here to make it less scary!
We'll divide both sides of the equation by 850 first:
1100/850 = (1 + r/4)^(4*10)
1.294 = (1 + r/4)^(40)
To get rid of that pesky exponent, we can take the 40th root of both sides:
(1.294)^(1/40) = 1 + r/4
1.018 = 1 + r/4
Subtracting 1 from both sides:
0.018 = r/4
Now, let's multiply both sides by 4 to isolate r:
0.072 = r
So, the nominal annual rate, my friend, is approximately 7.2%. Sure, it may not be as impressive as a clown juggling chainsaws while riding a unicycle, but it's still pretty good for a dollar!
To find the nominal annual rate, we can use the formula for compound interest:
A = P(1 + r/n)^(n*t)
Where:
A = the future value of the investment ($1100)
P = the initial investment ($850)
r = the nominal annual interest rate (unknown)
n = the number of compounding periods per year (4, since interest is compounded quarterly)
t = the number of years (10)
Plugging in the given values, we have:
1100 = 850(1 + r/4)^(4*10)
To solve for r, we need to isolate it on one side of the equation. Let's divide both sides by 850:
1100/850 = (1 + r/4)^(4*10)
1.2941 ≈ (1 + r/4)^40
Next, let's take the 40th root of both sides:
(1.2941)^(1/40) ≈ 1 + r/4
Now, subtract 1 from both sides:
(1.2941)^(1/40) - 1 ≈ r/4
Multiply both sides by 4 to isolate r:
4 * [(1.2941)^(1/40) - 1] ≈ r
Using a calculator, evaluate the left side of the equation:
4 * [(1.2941)^(1/40) - 1] ≈ 0.0461
Therefore, the nominal annual rate is approximately 0.0461, or 4.61%.
To find the nominal annual rate, we can use the formula for compound interest:
A = P*(1 + r/n)^(nt)
Where:
A is the final amount
P is the principal amount (initial investment)
r is the annual interest rate (in decimal form)
n is the number of times that interest is compounded per year
t is the number of years
We are given:
P = $850 (principal amount)
A = $1100 (final amount)
t = 10 years
n = 4 (quarterly compounding)
Substituting these values into the formula, we have:
$1100 = $850*(1 + r/4)^(4*10)
Now we need to solve for r, the nominal annual rate. To do this, we need to rearrange the equation and isolate r:
(1 + r/4)^(40) = $1100/$850
Taking the 40th root of both sides, we get:
1 + r/4 = (1100/850)^(1/40)
Subtracting 1 from both sides, we have:
r/4 = (1100/850)^(1/40) - 1
Finally, multiplying both sides by 4 gives us:
r = 4*((1100/850)^(1/40) - 1)
Calculating this expression will give us the nominal annual rate.
Please note that this calculation assumes a constant interest rate over the entire 10-year period.