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This problem consists of four independent parts, each related to the business activities of Eleanor Corporation. Compute the answer $($to the nearest dollar$)$ for each part by using the tables in the Appendix at the end of this chapter.

$(1)$ Eleanor plans to accumulate $$500,000$ on December $31.$ Year $10,$ to retire preferred stock. Eleanor deposited $$125,000$ in a fund on January $1,$ Year $1,$ that will earn interest at $10\%$ compounded quarterly, and wants to know what additional amount it has to deposit in the fund at the end of each quarter for $10$ years $($starting on March $31,$ Year $1)$ to have $$500,000$ available at the ead of Year $10.$ The periodic deposits also will earn interest at $10\%$ compounded quarterly.

$(2)$ Eleanor plans to make five equal annual deposits in a fund beginning June $1,$ Year $1,$ in order to be able to withdraw $$50,000$ at six annual intervals beginning June $1.$ Year $6.$ The amount on deposit will earn interest at $12\%$ annually until the fund is exhausted. Compute the equal deposits that should be made to the fund.

$(3)$ On June $30,$ Year $1,$ Eleanor acquired a machine for $$80,000.$ The down payment was $$10,000,$ and the balance is to be paid in $48$ equal monthly payments, including interest at $18\%$ compounded monthly. What is the amount of the monthly payment if the first payment is due one month from the date of acquisition?

On April $1,$ Year $2,$ Eleanor made a deposit of $$100,000$ in a fund and left the fund undisturbed for four years to earn compound interest at a rate that did not change during the four$-$year period. At the end of four years, the fund had a balance of $$132,088\mathrm{.}60.$ If interest was compounded quarterly, what was the annual rate of interest earned on the deposit?