A company must replace machinery every two years at a cost of $100000.$ It is decided to set aside equal amounts at the beginning of each quarter. If interest is $5\mathrm{.}75\%$ per annum, calculate the size of the quarterly deposits.

Starting on her $35$ th birthday, a woman saves $200$ at the end of each month with a view to retiring on her $50$ th birthday.

$($a$)$ What is the value of her savings if the interest rate is $8\%$ per annum?

$($b$)$ From the age of $50,$ how much should she withdraw each month if the fund isto last for the next $15$ years, assuming the interest rate is $9\%$ per annum?

An education fund of $40000$ is set up for a child on his $4$ th birthday. The fund is to mature on his $18$ th birthday. Assuming an interest rate of $10\%$ per annum, calculate $($a$)$ the present value of the fund assuming annul compounding. $($b$)$ the size of monthly deposits $($in arrears$)$ required to achieve the fund of $40000,($c$)$ the size of quarterly deposits required to achieve the fund of $40000.$