Haley likes to look at the stars with her telescope at night. she can see the stars best when rhe moon is new. at this time, the moon is at its dimmest. there is a first quarter moon in the sky tonight. how long will haley have to wait until the next new moon?

Well, that's easy to answer when we realize that "First Quarter" isn't talking about   a 'quarter' of the Moon being illuminated.  It's talking about a 'quarter' of the time that the complete cycle of phases takes.

"First quarter" means that 1/4 of the time from one New Moon to the next one has passed, and there are 3/4 of the cycle left until the next New Moon.

-- roughly 3/4 of a month to go

-- roughly 3 weeks to go

-- actually 3/4 of (29.531 days)

-- actually  22 days 3.6 hours until the instant of the next New Moon.

But Haley doesn't actually need to wait that long.  You can't see the moon in the sky for 1-2 days before the New Moon and 1-2 days after it.  So Haley really only has to wait like 20 days until there's no moon in the sky to interfere with her observations.

Did you notice what I noticed ?

What if Haley stays interested in Astronomy for many years, gets really serious about doing observations regularly, and one day she discovers a comet !  These days, comets are  named for their discoverers, so if that happened, we would have TWO of um !