Question Video: Converting Recurring Decimals to Fractions Mathematics • 7th Grade

Convert 0.354 recurring to a fraction.


Video Transcript

Convert 0.354 recurring to a fraction.

Here, we have a recurring decimal. The digits three, five, and four all repeat. So we could say that it’s equal to 0.354354354 and so on. Let’s recall the steps required to convert a recurring decimal to a fraction. Step one, we let 𝑥 be equal to our recurring decimal. We’re going to write out a few digits just to get an idea of the pattern. 𝑥 is 0.354354 and so on. Our second step is try and create another decimal number whose digits after the decimal point are identical to the original.

To achieve this, we’re going to multiply by some power of 10. That’s 10, 100, 1000, and so on so that the digits immediately after the decimal point match those of our original number. The pattern needs to sort of line up against the decimal point. We said that we had three digits that recur. So we’re going to need to figure out which power of 10 we multiply by to ensure that these digits move to the left three spaces. We end up with 354.354 and so on. Well, to achieve this, we multiply by 10 cubed or 1000. So let’s do the same to the 𝑥. In doing so, we get 1000𝑥 is equal to 354.354 and so on.

Then, our third step is to subtract. We’re hoping that in doing so, we completely eliminate the digits after the decimal point or at least the recurring bits. So we’re going to subtract the entire equation for 𝑥 from the entire equation for 1000𝑥. Note at this stage that we could do that the other way around. We just end up with two negatives. 1000𝑥 minus 𝑥 is 999𝑥. Then we notice that subtracting 0.354 recurring from 0.354 recurring gives zero. And so the sum becomes 354 minus zero, which is just 354. Remember, we were looking to eliminate the bit after the decimal. And we’ve done so.

Our fourth and final step is to solve this equation for 𝑥. We have 999𝑥 equals 354. So to solve, we’re going to divide both sides of this equation by 999. So 𝑥 is 354 divided by 999, which we can simply write as a fraction. Now, in fact, it’s not in its simplest form. We can divide both the numerator and denominator of this fraction by three. And when we do, we find 𝑥 is equal to 118 over 333. Remember, we originally defined 𝑥 to be equal to 0.354 recurring. But we’ve just shown that 𝑥 is equal to 118 over 333. So, in turn, we’ve shown that 0.354 recurring as a fraction in its simplest form is 118 over 333.

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