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Dermot has €5000 and would like to invest it for two years - Junior Cycle Mathematics - Question 4 - 2014
Question 4
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 ye... show full transcript
Worked Solution & Example Answer:Dermot has €5000 and would like to invest it for two years - Junior Cycle Mathematics - Question 4 - 2014
Step 1
How much will the investment be worth at the end of one year, after tax is deducted?
Answer
To calculate the amount at the end of the first year, we first determine the gross interest earned from the investment. The gross interest is calculated as follows:
ext{Gross Interest} = ext{Principal} imes ext{Rate} = 5000 imes rac{3}{100} = 150Next, we need to deduct tax from this gross interest. The tax amount is calculated using:
ext{Tax} = ext{Gross Interest} imes rac{41}{100} = 150 imes rac{41}{100} = 61.50Thus the net interest earned after tax is:
Finally, the total amount after one year, including the principal and net interest, is:
Therefore, after tax is deducted, the investment will be worth €5088.50.
Step 2
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.
Answer
To find the rate of interest for the second year, we can use the information given and set up the equation based on the total amount at the end of the second year.
Let the gross interest for the second year be denoted as . The investment at the start of the second year is €5088.50. The equation can be written as follows:
5088.50 + igg(5088.50 imes rac{i}{100} imes rac{59}{100}igg) = 5223.60This equation accounts for the 41% tax on the gross interest, meaning only 59% of the gross interest is retained. Rearranging the equation gives:
5088.50 + rac{5088.50 i}{100} = 5223.60Now, isolating :
rac{5088.50 i}{100} = 5223.60 - 5088.50 rac{5088.50 i}{100} = 135.10Thus,
i = rac{135.10 imes 100}{5088.50} egin{aligned} \ ext{which simplifies to} \ i = 2.65 ext{ (approximately)} ext{ or } 4.5 ext{ (correct to one decimal place)} ext{ ext{(adding 41 ext{ tax) whether considered across two years as well).}}Therefore, the rate of interest for the second year is 4.5%.