SOLUTION: Dermot has �5000 and would like to invest it for two years. A special savings account is offering a rate of 3% for the first year and a higher rate for the second year, if the mone

Question 919005: Dermot has �5000 and would like to invest it for two years. A special savings account is offering a rate of 3% for the first year and a higher rate for the second year, if the money is retained in the account. Tax of 41% will be deducted each year from the interest earned.
(a) How much will the investment be worth at the end of one year, after tax is deducted?
(b) Dermot calculates that, after tax has been deducted, his investment will be worth �5223�60 at the end of the second year. Calculate the rate of interest for the second year.
Give your answer as a percentage, correct to one decimal place.
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
in the first year.
starting balance is 5000.
interest rate is .03 so interest is .03 * 5000 = 150
tax on that is .41 * 150 = 61.5
net interest after taxes is therefore equal to 150 - 61.5 = 88.5
add that to 5000 and you get ending balance of 5088.5 after taxes.
this is your starting balance at the beginning of the second year.

in the second year.
starting balance is 5088.5
interest rate is x so interest is x * 5088.5
tax on that is .41 * x * 5088.5 which is equal to x * 2086.285
net interest after taxes is therefore equal to x * 5088.5 - x * 2086.285.
factor this to get net interest after taxes is equal to x * (5088.5 - 2086.285) which is equal to x * 3002.215.

at the end of the second year you are given that you have an ending balance of 5223.6 after taxes.

subtract 5088.5 from that and your net interest after taxes is equal to 135.1.

since your net interest after taxes is also equal to x * 3002.215, you get the following equation:

135.1 = x * 3002.215.

divide both sides of this equation by 3002.215 to get:

x = .0450001083

that should be your interest rate.

confirm by going back to the original problem and solving it using the second year interest rate of .0450001083.

for the first year, start with 5000.
multiply that by .03 to get net interest of 150 before taxes.
subtract .41 * 150 from that to get net interest of 88.5 after taxes
add that to starting balance of 5000 to get ending balance of 5088.5 after taxes.
your net value at the end of the first year is 1088.5.

for the second year, start with 5088.5
multiply that by .0450001083 to get net interest of 228.9830508 before taxes.
subtract .41 * 228.9830508 from that to get 135.1 net interest after taxes.
add that to starting balance of 5088.5 to get ending balance of 5223.6 after taxes.

this agrees with the ending balance at the end of the second year that you were given so it looks like the interest rate in the second year has been calculated correctly.

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