### Video Transcript

Venus takes 225 days to orbit the Sun. How long is this period in Earth years? Use a value of 365 for the number of days in an Earth year. Give your answer to two decimal places. How long does Earth take to orbit the Sun in Venus years? Give your answer to two decimal places.

This question tells us that it takes 225 days for the planet Venus to orbit the Sun. And the first part of this question asks how long this period is in units of Earth years. If we know that it takes 225 days for Venus to orbit the Sun, then this means that one year on Venus, which we’ve labeled 𝑦 subscript V, is equal to 225 days. We should clarify that when we’re talking about days, we are referring to days as measured on Earth. The question tells us to use 365 as the number of days in one Earth year. So we can say that one year on Earth, which we’ve labeled 𝑦 subscript E, is equal to 365 days.

Now we want to convert this value here into units of Earth years. Currently, we have units of days, but we also have this conversion factor between days and Earth years. What this statement is telling us is that there are 365 days per Earth year or equivalently that there is one over 365 Earth years per day. What this statement here means is that each one of the 365 days in the Earth year makes up one three hundred and sixty-fifth of that year. So if we want to take this value of 225 days and convert it to units of Earth years, then we take the value in units of days and multiply by one over 365 Earth years per day.

If we look at the units, we see that we have days that cancel with per day and we are left with units of Earth years. Evaluating this expression gives a result of 0.6164 Earth years, where the ellipses indicate that there are further decimal places. Looking back at the question, we see that we were asked to give our answer to two decimal places. Rounding our result to two decimal places gives us 0.62 Earth years. And so our answer to the first part of the question is that Venus’s orbital period of 225 days, as measured in units of Earth years, is equal to 0.62 years.

If we now look at the second part of the question, we see that we’re being asked to work out how long Earth takes to orbit the Sun in Venus years. We know that Earth orbits the Sun in a time equal to one Earth year. And in answering the first part of this question, we found a conversion between Earth years and Venus years. If one Venus year is equal to 0.6164 et cetera Earth years, then if we divide through by the factor 0.6164 on both sides of the equation, then on the right-hand side, 0.6164 divided by 0.6164 gives one. Going from this step to this one here, all we’ve done is swapped the left- and right-hand sides of the equation over so that we have one Earth year is equal to one over 0.6164 Venus years.

It’s worth mentioning that we were careful to use a precise value of the conversion factor between Venus years and Earth years rather than our rounded value of 0.62. The reason for this is to avoid carrying any errors through arising from the rounding process into the next part of the calculation. When we evaluate the fraction on the right-hand side of this expression, we get that one Earth year is equal to 1.62 recurring Venus years. The question asks for our answer to two decimal places. So to two decimal places, we have that one Earth year is equal to 1.62 Venus years. And so our answer to the second part of the question is that, in units of Venus years, the time that it takes for Earth to orbit the Sun is equal to 1.62 years.