There are 9.46 times 10 to the power of 15 metres in one light-year. What is 1.14 times 10 to the power of 17 metres in light-years? Give your answer to the nearest light-year.
Okay, so, the first sentence of this problem statement is telling us that 9.46 times 10 to the power of 15 metres is equivalent to one light-year. And based on that information, we need to try and work out how much 1.14 times 10 to the power of 17 metres is in light-years.
So, let’s say that the quantity that we’re trying to work out is 𝑥, the number of light-years in this many metres. Well, we can go about doing this by taking a ratio between this quantity 𝑥 and this quantity one light-year. In other words, then, we’re dividing 𝑥 by one light-year. But then, we can say that this fraction is equal to 1.14 times 10 to the power of 17 metres in the numerator because, remember, 𝑥 is the same thing as that. And then, we divide that by the number of metres in one light-year, which is 9.46 times 10 to the power of 15 metres.
And so, essentially, what we’ve done in going from the left-hand side to our right-hand side is to say that 𝑥 divided by one light-year is the same thing as this quantity divided by this quantity. Because 𝑥 is equal to 1.14 times 10 to the power of 17 metres and one light-year is equal to 9.46 times 10 to the power of 15 metres. So, now that we’ve set up this equation, we can rearrange to solve for 𝑥.
We can do this by multiplying both sides of the equation by one light-year. That way, on the left-hand side, the one light-year cancels with the one light-year in the denominator. So, all we’re left with is 𝑥 on the left-hand side and this fraction multiplied by one light-year on the right-hand side.
Now at this point, we can notice that the unit of metres appears both in the numerator and the denominator of the fraction on the right-hand side of the equation. In this situation, those units cancel. And so, this fraction just becomes a unitless quantity. It’s just a number. But then, when we multiply it by one light-year, that then gives us the units of light-years. And that works perfectly because, remember, 𝑥 is the number of light-years corresponding to this many metres.
So, all we have to do is to evaluate the fraction on the right-hand side of this equation. When we do this, we find out that it ends up being 12.0507 dot dot dot so on and so forth light-years. But, remember, this is not our final answer. We need to give our answer to the nearest light-year.
In another words, then, we need to round this number. And in order to do that, we need to look at the number in the first decimal place. That number is zero. Zero is less than five. And so, the value before the decimal place will stay the same. And so, we’ve found our final answer. 1.14 times 10 to the power of 17 metres is equivalent to 12 light-years to the nearest light-year.