A moon is orbiting a planet. I am given the sidereal month of the moon and sidereal year of the planet. How do I go about calculating the delay...

I think the simplest way to think about this is to convert them from periods to angular velocities. Then you can just add and subtract angular velocities. The conversion each way is `(2pi)/x` . (If all you wanted was hints, you can stop there. Full solution follows.)

The sidereal month is 27.321661 days. This is an angular velocity of `(2pi)/27.321661 = 0.2299708` radians per day.

The sidereal year is 365.25636 days. This is an angular velocity of 0.017202 radians per day.

Since the Earth and the Moon are spinning the same direction, the synodic month is longer than the sidereal month, and thus we should subtract these velocities.

The synodic angular velocity is `0.2299708 - 0.017202 = 0.2127688` , which corresponds to a period of `(2pi)/0.2127688 = 29.53` days.

Look it up, and you will see that yes, indeed, the synodic month is 29.53 days long.