Express 0.375 repeating as a common fraction.
To begin, let’s actually think about what this looks like if we were to write this out. So the line above the three, seven, and five means this is the part that’s going to repeat. So truly, this is 0.375375375 and on and on and on. So if we wanna write this as a common fraction, we need to use the part that doesn’t have the bar over top. So how could we somehow eliminate this repeating part?
Well, if we set our decimal equal to 𝑥 and we just say that 𝑥 is the fraction that we we’re wanting to find, we can eliminate this repeating part by moving the decimal place. And we can do that by moving it over to the right enough to where we’ve completely went by everything that repeats. So three, seven, and five all repeat. So if we wanna move the decimal place three places, we have to multiply by 1000 to each side because 1000 has three zeros which moves our decimal place three places to the right. So multiplying both sides by 1000, we have 1000 𝑥 equals 375.375375. And the three, seven, five keeps repeating, it keeps going from there.
So if we want to eliminate this repeating part, we already determined what the repeating part was equal to, .375375375 and so on, is just equal to 𝑥. So if we would take away 𝑥 from both sides, we would eliminate that repeating part. So on the left, we’ve already taken away 𝑥. However, instead of writing it as 𝑥 on the right-hand side, we can write it as its value, 0.375375 repeating. So all of that cancel, so we’re left with 375.
So to solve for 𝑥, we need to divide both sides by 999. So our last step would be to simplify. Both the numerator and denominator can be divided by three. So our final answer would be 125 divided by 333. This would be our common fraction.