Maximus invests a sum of money in a bank account which earns 2%, and increases by 1% every 6 months. The interest is compounded monthly. He pulls out $4300 out

Question

Maximus invests a sum of money in a bank account which earns 2%, and increases by 1% every 6 months. The interest is compounded monthly. He pulls out $4300 out of his account after 21 months. How much did he initially invest?

Answer 1

Take a systematic approach. If you start with a certain amount of money, x_initial. The money receives a 2% interest rate every year which is compounded monthly (12 times a year). So after the first month he has x_{1 month} dollars in his account given by

 

x_{1 month} = x_initial*(1+0.02/12months) = xinitial*(1.001666)

 

Note that 0.02 comes from 2%, meaning 2/100 = 0.02; and 0.001666 comes from 0.02/12 months = 0.001666 per month which is what the interest is when compounded for the month. 

 

For the second month,

 

x_{2 month} = x_{1 month*}(1.001666) = x_initial*(1.001666)²

 

For the third month,

 

x_{3 month} = x_{2 month}*(1.001666) = xinitial*(1.001666)³

 

And so on until month 6 which is

 

x_{6 month} = x_initial*(1.001666)⁶

 

Now the interest rate changes for month 7 to 3% so instead of 0.001666, the interest is 0.03/12 = 0.0025 every month. So for month 7

 

x_{7 month} = x_{6 month}*(1.0025) = x_initial*(1.001666)⁶*(1.0025)¹

 

This leads to the 12th month when another increase happens to the interest from 3% to 4%. So month 12 has

 

x_12month = x_initial*(1.001666)⁶*(1.0025)⁶

 

The 6 in the exponent next to the 1.001666 and 1.0025 denotes how many months the interest rate was applicable. Now instead of 0.0025 per month it is 0.04/12 = 0.003333 so at 18 months you have

 

x_{18 month} = x_initial*(1.001666)⁶*(1.0025)⁶*(1.003333)⁶

 

and at 21st month when the interest is raised to 5% for the last 3 months, the equation is

 

x_{21 month} = x_initial*(1.001666)⁶*(1.0025)⁶*(1.003333)⁶*(1.004166)³

 

Now solve for x_initial knowing that x_{21 month} is $4,300.

 

 

Answer 2

Principal amount invested is $4050

 

Interest month 1-5 is 2%, month 6 -11 is 3%, Month 12-17 is 4%, month 18-21 is 5%

 

Create and excel spreadsheet

 

Month 1 = Principal * 2%/12+ principal

Month 2= Month 1*2%/12 + month 1

and so on

 

month 6 = Month 5*3%/12 + month 5

and so on

 

Month 12 =Month 11*4%/12 + month 11

and so on

 

Month 18=Month 17*5%/12 + month 17

and so on till month 21.

 

Have a cell reference for Principal

 

If the amount in Cell 21 should be $4300 (it includes principal plus compounded interest).

I played with the Principal amount to see if the cell under month 21 was close to $4300. I typed in $4000 and it was close, then 4100 and it went over $4300, then tried $4050 and it was exact at $4300 at end of month 21

 

Hope this helps