Question Video: Converting Recurring Decimals to Fractions

Convert 0.347 recurring to a fraction.

03:07

Video Transcript

Convert 0.347 recurring to a fraction.

We have a recurring decimal with a bar just above the seven. So that tells us the seven is the only bit that recurs. It’s 0.34777 and so on. So let’s recall the steps that we take to convert a recurring decimal to a fraction. Our first step is to define 𝑥. We let 𝑥 be equal to our recurring decimal. And at this stage, it can be helpful to write out a few digits of the recurring part just to get an idea of the pattern. Then, our second step is to multiply this by some power of 10 so that the digits after the decimal point match.

We do have a little bit of a problem here. The only bits that recur are the seven. So we’re actually going to need to do this twice. We want to create two numbers whose digits after the decimal point match. Well here, that’s going to be seven recurring. So let’s begin by multiplying by some power of 10 so that the digits move twice to the left, in other words, so that we get 34.7 recurring. Well, the only way to achieve this is to multiply by 100. So let’s multiply 𝑥 by 100. 𝑥 times 100 is 100𝑥, so we get 100𝑥 equals 34.7 recurring.

Let’s do this again. Now we could multiply our original number by something. However, if we look carefully, we notice that if we multiply 34.7 recurring by 10, the digits move to the left one space. And we’ll still end up with 0.7 recurring. 34.7 recurring times 10 is 347.7 recurring. And 100𝑥 times 10 is 1000𝑥. Notice that this is the same as multiplying our original value for 𝑥 by 1000. That would’ve moved the digits three spaces to the left.

Now that we’ve created two numbers whose digits after the decimal point perfectly match, we subtract these two numbers. In other words, we’re going to subtract the entire equation for 100𝑥 from the equation for 1000𝑥. In doing so, we notice that the recurring part of the decimal disappears. 0.7 recurring minus 0.7 recurring is zero. And so we simply need to work out 347 minus 34. That’s 313. Similarly, 1000𝑥 minus 100𝑥 is 900𝑥. So we have an equation for 𝑥. It’s 900𝑥 equals 313.

Our fourth and final step will always be to solve this equation for 𝑥. We perform inverse operations to do so. We’re going to divide both sides of this equation by 900. 𝑥 is therefore equal to 313 divided by 900, which we can write as a fraction as shown. Note that we originally defined 𝑥 to be equal to 0.347 recurring. But we’ve now shown that 𝑥 is equal to 313 over 900. This must mean that in its fractional form, 0.347 recurring is 313 over 900.

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