This question is not cumulative

$[8$ marks$]$

$($a$)$ Perpetual Corp. has issued consol bonds with annual coupon payments of $$50.$ If the required rate of return on these bonds at the time they were issued was $5$ percent, at what price were they sold to the public? If the required return today is $10$ percent, at what price do the consols sell?

$($b$)$ A pure discount bond has $10$ years until maturity and will pay $$1,000$ at maturity. If the YTM is $15\%,$ what is the current price of this pure discount bond?