Introduction: Information Gathering
This paper provides a solution to the problem, which involves the student quitting smoking and investing the saved money at a certain percentage. Available data includes an age of 20 years and an investment period limited to 65 years. The price of a pack of cigarettes remains unchanged and is not subject to inflation, respectively, the 16 packs per month that a student left would be stable at $6.50. Finally, an annual contribution rate of 4% is known, which ensures the growth of savings received as a result of quitting smoking. Accordingly, preliminary calculations will include the following formulas:
From the data obtained, it follows that a student who quit smoking would invest $104 per month in a 4 percent deposit for 45 years. It should be borne in mind that the contribution grows annually, which means that the accrued interest will also change from year to year, which forces the use of spreadsheets.
Reasonable Answer
It is quite difficult to make a forecast in this regard, since any accurate approximation requires serious calculations due to the presence of compound interest and fractional exponents. However, one can try to predict the order of the final number by finding the student’s annual contribution and its approximate increase. The student contributes $1,248 per year, and with the addition of 4%, the amount can grow to almost $1,300. The progression of interest is approximately $50 per year, and if we assume n equal to 45 using the formula of the first n-terms of the arithmetic progression, then it comes out to about 105 thousand dollars after 45 years. Accordingly, the exact model developed below will be compared with preliminary calculations and a forecast using the formula for the sum of members of an arithmetic progression.
Solving
Using the formula for the sum of members of an arithmetic progression can only give an approximate estimate, since it does not take into account the complexity of calculating interest on a deposit. Therefore, in this work it is more reasonable to use either a spreadsheet or a mathematical model in the form of an equation, although the calculations may take more time. It is taken into account that interest is accrued annually on the balance of the deposit, and, accordingly, the deposit is replenished annually by 1248 dollars. Spreadsheets were used for the calculation, since building a mathematical model that takes into account both compound interest and monthly or even annual replenishment is quite difficult. In this regard, a table was built in Excel, where in the first year it is entered separately by the formula “=104*12*($B$3+1)”: cell B3 contains a deposit rate equal to 4%. Then, the formula “=B2*(1+$B$3)+1248” is entered into the next cell, which is then stretched to the end of the period of 45 years. Cell B2 in this case reflects what was left on the deposit at the time of the previous year and is multiplied by the rate, then a new replenishment of $1248 is added. After a period of 45 years, another cell is added that no longer adds $1248, but adds interest at the time of the balance after 45 years of investment.
Conclusion
As a result of solving this problem using Excel spreadsheets, it turned out that after 45 years of monthly investment in a 4 percent contribution, a student would earn $ 157,378.10.