Answer the following questions for
the recurring decimal 0.265 recurring; that’s 0.2656565 and so on. Let 𝑥 be equal to 0.265
recurring. Find an expression for 10𝑥. Find an expression for 1000𝑥. Subtract 10𝑥 from 1000𝑥 to find
an expression for 990𝑥. And find 𝑥.
The question has defined our
recurring decimal to be equal to 𝑥. And it wants us to begin by finding
an expression for 10𝑥. It can be useful to write out a few
digits of the recurring number to get an idea of the pattern. To get from 𝑥 to 10𝑥, we’re going
to need to multiply by 10. And so let’s do the exact same
thing to our recurring number. That’s 2.656565 and so on. Remember when we multiply by 10, we
move the digits to the left one space. And so the expression for 10𝑥 here
is 2.65 recurring.
Next, we need to find an expression
for 1000𝑥. Well, this time to get from 𝑥 to
1000𝑥, we’re going to need to multiply by 1000. So we’ll do the same to our
recurring decimal. This time, when we do so, the
digits move to the left three spaces. So 1000𝑥 is 265.6565 and so
on. 1000𝑥 is therefore 265.65
recurring. Notice that we now have two numbers
whose digits after the decimal point are identical. And so we’re ready for the next
part of this question. We’re going to subtract 10𝑥 from
1000𝑥. And in turn, we’re going to
subtract their decimals.
So we’re going to work out 265.65
recurring minus 2.65 recurring. When we subtract these numbers, we
notice that the recurring part gives us zero. 0.65 recurring minus 0.65 recurring
is zero. 265 minus two is 263. And 1000𝑥 minus 10𝑥 is 990𝑥. And so our expression of 990𝑥 is
263. The very final part of this
question says to find 𝑥. In other words, we’re going to
solve our equation 990𝑥 equals 263. To do so, we need to divide through
by 990. 𝑥 is therefore equal to 263
divided by 990, which we can write as a fraction as shown.
Now remember, we originally defined
𝑥 to be equal to 0.265 recurring. But we’ve just shown that it’s
equal to 263 over 990. And that means the fraction
equivalent of the recurring decimal 0.265 recurring must be 263 over 990.