Last June $16,$ Mr$.$ Smith bought a vacation package for his family for $$10,000.$ He paid for the package with his credit card. His card billing period runs from the $16$th of each month to the $15$th of the following month. The package is the only purchase he made with his card during the period June $16$ July $15.$ On July $16$ an account statement was issued by the credit card company requesting that he pay $$10,000$ by August $6.$Mr$.$ Smith decided to pay $$9,999\mathrm{.}99$ on August $6,$ opting to repay the remaining $$0\mathrm{.}01$ with the next statement. He did not make any purchase during the billing cycle July $16$ August $15.$ On August $16$ a new statement was issued. The statement showed a balance of $$336\mathrm{.}88.$ Mr$.$ Smith immediately called his credit card company to declare that a mistake had been made. The company checked his file and replied that there was no mistake. Prove to Mr$.$ Smith that the company$$s assertion is correct.