Compound Interest problems?

I don't think you're substituting the right values in the right places in the formula. In the formula A = P(1 + r/n)^(nt), A is the accumulated amount, P is the principal amount (amount invested), r is the annual interest rate, n is the number of times it is compounded (number of compounding periods), and t is the time in years.

107.) A = P(1 + r/n)^(nt)

1400 = 1000(1 + r/360)^(360*2)

1400 = 1000(1 + r/360)^720 (divide both sides by 1000)

1.4 = (1 + r/360)^720 (place both sides to the 1/720 power)

1.00046743 = 1 + r/360 (subtract 1 from both sides)

0.00046743 = r/360 (multiply both sides by 360)

0.1682748 = r

ANSWER: The annual interest rate is 0.168 or 16.8%.

108.) A = P(1 + r/n)^(nt)

9000 = 5000(1 + r/360)^(360*4)

9000 = 5000(1 + r/360)^1440 (divide both sides by 5000)

1.8 = (1 + r/360)^1440 (place both sides to the 1/1440 power)

1.00040827 = 1 + r/360 (subtract 1 from both sides)

0.00040827 = r/360 (multiply both sides by 360)

0.1469772 = r

ANSWER: The annual interest rate is 0.147 or 14.7%.

Because of the wonders of compound interest, the solutions to your problems are actually quite straightforward. The annual interest rate j (as a fraction) is obtained by compounding the period rate r/n over the n periods in one year, that is,

1 + j = (1 + r/n)^n

When this is substituted into the formula for the accumulated amount A

A/P = (1 + r/n)^(n.t)

where t is the number of years the principal P has been invested, then

A/P = (1 + j)^t

as you might expect. So j is easily obtained from the data supplied from

1 + j = (A/P)^(1/t)

So the answers to problem 107 is

1 + j = (1400/1000)^(1/2) = 1.18321596 (for comparison purposes only)

or expressed as a percentage j = 18.32% pa

and for problem 108

1 + j = (9000/5000)^(1/4) = 1.15829219 equivalent to 15.83% pa.

These can be performed directly on any calculator which will do a^b, otherwise you may need to resort to taking logs, dividing by the exponent and antilogging (10^x).

Note added: Oh dear, hasn't my message got across?! There is no point in calculating r - which is not the annual interest rate or even the interest rate applied for each individual period (which is r/n). The method I gave above gives exactly the same values for the required annual rate as the correct ones produced by Peter but quickly and directly. If you really want to find r (and as the first answerer showed, it's a misleading and confusing quantity to understand), calculate the annual rate j first, and derive it from this with the first equation above.

107)

Hm, this sounds like the interest is being compounded almost daily. So, the equation we want to set up would be:

FV = P(1 + r / n)^(nt)

1400 = 1000(1 + r / 360)^(360 * 2)

1.4 = (1 + r / 360)^(720)

(1.4)^(1/720) = 1 + r / 360

(1.4)^(1/720) - 1 = r / 360

r = 360 * [(1.4)^(1/720) - 1]

r = 0.168275435

r is the nominal rate of interest compounded almost daily. So to find the effective annual rate of interest, we have that:

1 + i = (1 + r / 360)^360

1 + i = (1 + 0.168275435 / 360)^360

i = (1 + 0.168275435 / 360)^360 - 1

i = 0.183215957

So, the annual interest rate is about 18.32%.

108)

The steps are similar to the previous problem.

9000 = 5000(1 + r / 360)^(360 * 4)

9/5 = (1 + r / 360)^(1440)

(9/5)^(1/1440) = 1 + r / 360

(9/5)^(1/1440) - 1 = r / 360

r = 360[(9/5)^(1/1440) - 1]

r = 0.146976661

The effective annual rate of interest is:

1 + i = (1 + 0.146976661 / 360)^360

i = (1 + 0.146976661 / 360)^360 - 1

i = 0.158292185

Thus, the annual interest rate is about 15.83%.

Note: I am not sure whether the problems want the nominal rate of interest being compounded daily or the effective annual rate of interest.

You're on the right path, but now you have to break up the log.

1400 = 1000(1 + r/180)^360

1400/1000 = (1 + r/180)^360

7/5 = (1 + r/180)^360

ln (7/5) = 360 ln(1 + r/180)

ln (7/5) = 360 ln (1) + 360 ln (r/180)

ln (7/5) = 0 + 360 ln (r/180)

ln (7/5) = 360 ln (r/180)

ln (7/5) = [ln (r/180)]^360

Now drop the log from both sides:

7/5 = (r/180)^360

I don't have a calculator...can you figure out the rest from here? 108 is the same type.

Your first step is not to plug in the values. Your first step is to move the factors around in the equation until you get your unknown factor all by itself on one side of the equation.

Your unknown factor is 'r'. It is buried in the right side of the equation.

Just working with the equation, you can see that to move 'P' over to the left side, you need to divide both sides by 'P'. Thus, you have 'A' divided by 'P' equals (1 + r / n)^nt.

Next in order to get another factor over to the left, you need to raise both sides to the 1/2 power of nt. Given the font limitations that we have here, it is very difficult to write this out in numeric notation. To raise by a power of 1/2 times nt, you are taking the 'nt' root of the number.

Thus, the 'nt' root of (1 + r / n)^nt is (1 + r/n). The 'nt' is over on the left side of the equation now, taking the 'nt' root of A divided by P.

It's getting simpler now. Subtract one from both sides... carefully. The left side will now say: the 'nt' root of A divided by P and then subtract one from that result. You are not to take the root of A divided by P minus one. Imagine parentheses surrounding the expression: the 'nt' root of A divided by P... and then a minus one outside next to it.

All you have left is 'r/n' on the right. Multiply both sides by 'n' and you will finally have 'r' by itself on the right side.

Now you can plug in the values and the equation hands you the answer just like that.

In the end, the left side should say: 'n' times (the 'nt' root of (A divided by P) - 1).