Calculating annuities and monthly payments

A family applies for a loan of $5.000 \euro$ to buy a second-hand motorbike. The bank grants them the loan at an annual interest rate of $5 \%$, which they must repay in 4 equal annual instalments payable at the end of each year. Determine the amount of this annual loan repayment.

Lucía is planning to attend a music festival when she finishes Baccalaureate in 3 years' time. To cover the expenses, she decides to open a youth savings account offering $3 \%$ annual compound interest. If she deposits $250 \euro$ at the beginning of each year, what total capital will she have accumulated for her trip at the end of the three years?

A family applies for a loan of $15.000$ € to renovate their kitchen and bathroom. The bank grants them the loan at an annual interest rate of $5\%$ to be repaid over $4$ years in fixed monthly instalments due at the end of each month.

1. a) Determine the monthly instalment amount they will have to pay.
2. b) Calculate the total amount they will end up paying back to the bank after $4$ years.
3. c) What percentage of the total amount paid corresponds to interest?

Calculate the monthly instalment required to repay a debt of $2.400 \euro$ incurred from buying a professional camera, if the loan is repaid over 2 years at a nominal annual interest rate of $4,8 \%$ with monthly payments.

Lucas wants to buy an electric motorbike that costs $4.200$ €. In order not to deplete his savings, he decides to finance it at a nominal annual interest rate of $6\%$, committing to pay monthly instalments of $120$ €.

1. a) Use the loan repayment formula to calculate how many months it will take him to pay off the motorbike (you will need to use logarithms).
2. b) If he decides to pay off the motorbike in exactly $3$ years, what should the fixed monthly instalment be?

A school e-sports club decides to save up to replace their chairs. They deposit $100 \euro$ at the beginning of each quarter into an account yielding $4 \%$ annual compound interest. If compounding is quarterly ($n=4$ periods per year), calculate the capital they will have saved after one full year (4 deposits).

A 22-year-old entrepreneur decides to plan her financial independence. Her goal is to have saved $100.000$ € by the time she turns 35. To do this, she sets up the following savings plan in an account with annual compound interest of $4,5 \%$:

1. During the first 5 years, she will make monthly contributions of $a$ euros at the beginning of each month.

2. During the next 8 years, she will increase her savings capacity and contribute $2a$ euros at the beginning of each month.

Calculate the value of the initial monthly payment $a$ needed to achieve her goal.