Answer question below?

Question

It takes the earth 1.0 years to travel around the sun. It takes Venus 0.615 years to orbit the sun. If the planets are in a straight line with the sun now, how long will it be before this is the case again? The answer must be accurate to the nearest hundredth of a year. (Note: Assume that the orbital planes of all the planets are the same.)

Answer 1

Venus takes less time to orbit the sun than does earth, so the question is the same as asking "if Venus passes earth today, when does Venus next pass earth?" By the time Venus gets to where Earth is today, Earth will have moved partway along its orbit, and Venus has to catch up. At that point, Earth will have moved x degrees around its orbit, and Venus will have moved 360 (full circle) plus x degrees around its orbit.

Figure Venus's orbital speed in degrees per year (which is 360/0.615, call it V) and Earth's orbital speed in degrees per year (which is 360). You want to find Y, the number of years for which YV + 360 = 360Y. This is the same as
360Y - YV = 360
And
Y(360-V) = 360
And
Y = 360/(360-V). Carry the division to as many decimal places as you want.

Answer 2

You want the synodic period of Venus. The source tells you how to calculate it. Basically, if the year of the Earth is 1 and the year of the other planet is P years, then the synodic period is 1/|1-1/P|. The answer is 1/|1-1/.615| = 1.5974 years.