Bank of America borrowed $$80,000,000$ at $5\%$ simple interest for $180$ days from a Chinese investment house. At the same time, the bank made the following loans, each for the exact same $180-$day period.

A $7\%$ simple interest note for $$38,000,000$ to a Canadian firm that extracts oil from Canadian tar sands;

An $8\mathrm{.}2\%$ simple discount note for $$27,500,000$ to a British contractor building a factory in South Africa; and

An $8\%$ simple discount note for $$14,500,000$ to a Louisiana company building minesweepers in New Orleans for the Mexican government.

$($a$)$ Find the difference between interest received and interest paid by the bank on these funds.

$($b$)$ The bank did not loan out all $$80,000,000.$ Find the amount it actually loaned out.

$($c$)$ Find the effective rate of interest earned by the bank.