# A man has $80. Bank A is offering 2.5% compounded quarterly. Bank B offers simple interest. If the amount is to be deposited for 3 years what rate of interest should Bank B offer for the two to be...

A man has $80. Bank A is offering 2.5% compounded quarterly. Bank B offers simple interest. If the amount is to be deposited for 3 years what rate of interest should Bank B offer for the two to be equivalent.

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The formula for future value of compound interest is:

`FV_c= P(1+r_c/m)^(mt)`

where `FV_c ` = future value

P = present value (deposited)

`r_c` = interest rate in decimal form

m = number of compounding periods in a year

t = number of interest periods in "years"

The formula for simple interest is :

`FV_s = P(1+r_st)`

where `FV_s` = future value

P = present value (deposited)

`r_s` = interest rate in decimal form

t = number of interest periods in "years"

The given condition in the problem is that each bank accounts will have the **same accumulated value or future value** based from **same time frame (t) and deposited amount (P)**. The only thing that differs is the interest rate.

That is your clue that the working equation follows:` FV_c= FV_s.`

Equating the two formulas:

`P(1+r_c/m)^(mt) = P(1+r_st)`

Divide both side by "P".

`(P(1+r_c/m)^(mt))/P = (P(1+r_st))/P`

`(1+r_c/m)^(mt) = 1+r_st`

Subtract 1 from both sides:

`(1+r_c/m)^(mt) -1= 1+r_st -1`

Then` r_st =(1+r_c/m)^(mt) -1`

Divide both side by "t" to isolate `r_s` .

`(r_st)/t = ((1+r_c/m)^(mt) -1)/t`

Formula: `r_s = ((1+r_c/m)^(mt) -1)/t`

Note: The "t", "P" and "FV" are the same from the two bank accounts.

Recall the given values:

Compounded quarterly means m = 4

Compounding interest 2.5% :

`r_c = %/100`

`=(2.5%)/100`

=0.025

Time in years: t = 3

Plug-in the values in the formula: `r_s = ((1+r_c/m)^(mt) -1)/t`

`r_s = ((1+0.025/4)^(4*3) -1)/3`

Simplify.

`r_s = ((1+0.00625)^(12) -1)/3`

`= ((1.00625)^(12) -1)/3`

`=(1.077632599 -1)/3`

`= 0.077632599/3`

=0.02587533

Express in percentage.

simple interest rate `=r_s*100`

= 0.02587533*100

=2.587533

Final answer: 2.59% (approximated value)

Bank A is offering an annual interest rate of 2.5% compounded quarterly. In this case after every 3 months interest earned on the amount deposited initially is added to the initial principal and every term interest is earned on the interest earned in the earlier terms.

If an amount P is deposited initially at an interest rate r the total amount after n terms is equal to P*(1+r)^n. In the problem, P = 80, r = 0.025/4 = 0.00625 and n = 3*4 = 12. If $80 is deposited with Bank A, the amount after 3 years is equal to 80*1.00625^12 = 86.21

The final amount if P is deposited at an interest rate of r after n terms is equal to P*(1 + r*n). The rate that Bank B should offer is r where 86.21 = 80*(1+3r)

=> r = (86.21/80 - 1)/3

=> r = 2.58%

For the amount due after 3 years to be equivalent Bank B should offer an annual interest rate of 2.58%

First you should know the formulas for simple and compound interest. The formula for simple interest is I=PRT or principal/starting amount*rate*time. The formula for compound interest is PV(1+r)^n=FV or present value times (1 + rate)^number of periods equals future value. Now, substitute in your amounts.

**Bank A:** FV=($80)(1+.625%)^12

The rate is .00625 or .625% because it is compounded quarterly, or 4 times a year (.025/4), and the number of terms is 12 because you compound 4 times per year for 3 years (3*4).

FV=($80)(1.00625)^12

FV=($80)(about 1.078)

FV= about **86.21**, meaning you have earned $6.21 after 3 years in Bank A.

**Bank B:** I=($80)(.025)(3)

** I=$6**

For the two offers to be equivalent you would have to raise the rate for Bank B. Set your equation equal to $6.21 interest.

$6.21=($80)(R)(3) Solve for R

$6.21=240R

.025875=R

R= approximately **2.58%**

**Sources:**

Just a difference of 0.08%, I was expecting it to be larger.