# Math Help - Need help with interest (Nominal)

1. ## Need help with interest (Nominal)

I need help with how to calculate nominal annual rate.

1) A sum of $850 is invested for 10 years and the interest is compounded quarterly. There is$1000 in the account at the end of 10 years. What is the nominal annual rate?

So to set this up is the equation:
1000 = 850(1+r/4)^(4*10) and we solve for r? I don't know if that equation is correct. But when I solve for it the answer comes out to be 0.0163 or 1.63%. Please confirm with me if that answer is correct.

2) An investment grows by 3% per year for 10 years. By what percent does it increase over the 10-year period?

Is the formula: P(1 + 0.03/1)^(1*10) and we are trying to find the effective rate. So should the answer be 34.39%?

3) If the balance, M, at time t in years, of a bank account that compounds its interest payments monthly is given by
M = M0(1.07763)^t
Note: In the M0, the 0 is a subscript.

What is the nominal annual rate?
Again I need help with nominal annual rate. In the back of the book it says the nominal annual rate is 7.5%. But I do not get how to find that answer. Please shed some light on this issue and help me with finding out how to solve this problem.

Thank you so much.

2. Originally Posted by krzyrice
I need help with how to calculate nominal annual rate.

1) A sum of $850 is invested for 10 years and the interest is compounded quarterly. There is$1000 in the account at the end of 10 years. What is the nominal annual rate?

So to set this up is the equation:
1000 = 850(1+r/4)^(4*10) and we solve for r? I don't know if that equation is correct. But when I solve for it the answer comes out to be 0.0163 or 1.63%. Please confirm with me if that answer is correct.

2) An investment grows by 3% per year for 10 years. By what percent does it increase over the 10-year period?

Is the formula: P(1 + 0.03/1)^(1*10) and we are trying to find the effective rate. So should the answer be 34.39%?

3) If the balance, M, at time t in years, of a bank account that compounds its interest payments monthly is given by
M = M0(1.07763)^t
Note: In the M0, the 0 is a subscript.

What is the nominal annual rate?
Again I need help with nominal annual rate. In the back of the book it says the nominal annual rate is 7.5%. But I do not get how to find that answer. Please shed some light on this issue and help me with finding out how to solve this problem.

Thank you so much.
1. $A = P(1 + r)^t$.

Since this is being compounded quarterly for 10 years, there will be 40 time periods and the rate will be the rate every quarter.

So $1000 = 850(1 + r)^{40}$

$\frac{20}{17} = (1 + r)^{40}$

$\left(\frac{20}{17}\right)^{\frac{1}{40}} = 1 + r$

$r = \left(\frac{20}{17}\right)^{\frac{1}{40}} - 1$.

This is the quarterly interest rate. So multiply by 4 to get the yearly interest rate.

3. Originally Posted by krzyrice
2) An investment grows by 3% per year for 10 years. By what percent does it increase over the 10-year period?

Is the formula: P(1 + 0.03/1)^(1*10) and we are trying to find the effective rate. So should the answer be 34.39%?
Year 0: $A = P$

Year 1: $A = 1.03 \cdot P$

Year 2: $A = 1.03\cdot 1.03 \cdot P$

Year 3: $A = 1.03 \cdot 1.03 \cdot 1.03 \cdot P$.

Can you see that for year $n$ the amount will be

$A = 1.03^n\cdot P$.

So in 10 years: $A = 1.03^{10}\cdot P$

$\approx 1.3439 \cdot P$.

So it has increased by just over $34.39\%$.

4. Thank you so much. I got the same answers but wasn't sure if I did them right.