Question

John plans to compete in a regional bicycle race after he graduates, however, he needs to buy a racing bicycle. He finds a bicycle that costs 1100 euros (EUR). John has 950 EUR and invests this money in an account that pays 5% interest per year, compounded monthly. Determine the amount that he will have in his account after 3 years. Give your answer correct to two decimal places. The cost of the bicycle, C, can be modeled by, C = 20x + 1100, where x is the number of years since John invested his money. Find the difference between the cost of the bicycle and the amount of money in John’s account after 3 years. Give your answer correct to two decimal places. After m complete months Fatih will, for the first time, have enough money in his account to buy the bicycle. Find the value of m.

Answer 1

After 3 years, John will have approximately 1074.95 EUR in his account. The difference between the cost of the bicycle and the amount in John's account after 3 years is approximately 25.05 EUR. It will take John approximately 5.08 months to have enough money to buy the bicycle.

Step-by-step explanation:

To determine the amount John will have in his account after 3 years, we can use the formula for compound interest: A = P(1 + r/n)^(nt), where A is the final amount, P is the principal amount (initial investment), r is the annual interest rate (as a decimal), n is the number of times interest is compounded per year, and t is the number of years.

Given that John invested 950 EUR at an interest rate of 5% per year compounded monthly, we can plug in the values and solve for A: A = 950(1 + 0.05/12)^(12*3).

Calculating this, we find that the amount John will have in his account after 3 years is approximately 1074.95 EUR.

To find the difference between the cost of the bicycle and the amount of money in John's account after 3 years, we subtract the amount in John's account from the cost of the bicycle: C - A = 1100 - 1074.95.

Calculating this, we find that the difference is approximately 25.05 EUR.

To find the value of m, the number of complete months it will take for John to have enough money to buy the bicycle, we need to solve the equation C = P(1 + r/n)^(nt) for t. Given that C = 1100 EUR, P = 950 EUR, r = 5% (0.05), and n = 12, we can plug in the values and solve for t: 1100 = 950(1 + 0.05/12)^(12m).

Calculating this, we find that m is approximately 5.08 months.