Video Transcript
Answer the following questions for
the recurring decimal 0.4 recurring, that is, 0.44444 and so on. Let 𝑥 be equal to 0.4
recurring. Find an expression for 10𝑥. Subtract 𝑥 from 10𝑥 to find an
expression for nine 𝑥. Find 𝑥.
By breaking the question down into
three individual steps, it’s showing us the process for converting a simple
recurring decimal into a fraction. We have a simple recurring
decimal. It’s 𝑥 equals 0.4 recurring. That is, 𝑥 equals 0.444 and so
on. It can be easier to write out a few
digits of the recurring part so we can get an idea of the pattern. The question asks us to find an
expression for 10𝑥. Well, to get to 10𝑥 from 𝑥, we’re
clearly going to need to multiply it by 10. And since 𝑥 is equal to 0.4
recurring, it follows that we’ll find an expression for 10𝑥 by multiplying 0.4
recurring by 10.
𝑥 multiplied by 10 is 10𝑥 as
required. And to multiply a decimal number by
10, we move the digits to the left exactly one space. So 10𝑥 is equal to 4.444 and so
on. The expression for 10𝑥 is
therefore 4.4 recurring. The second part of this question
tells us to subtract 𝑥 from 10𝑥. In doing so, we’re going to
subtract the entire equation for 𝑥 from the entire equation from 10𝑥. 10𝑥 minus 𝑥 is nine 𝑥 as
required.
Now let’s look at our recurring
decimals. Notice how the digits after the
decimal point are identical in both numbers. So that means when we subtract each
four, we end up getting zero. So we see that 4.4 recurring minus
0.4 recurring is simply four. And so our expression for nine 𝑥
is simply four. The third part of this question
asks us to find 𝑥. What it’s really saying is solve
this equation for 𝑥. Solve the equation nine 𝑥 equals
four.
Well, to solve the equation, we’ll
perform a series of inverse operations. Currently, nine 𝑥 means nine times
𝑥. So we’re going to divide both sides
of our equation by nine. And so, when we do, we find 𝑥 is
four divided by nine or four-ninths. 𝑥 is equal to four-ninths. Note that we originally defined 𝑥
to be equal to 0.4 recurring, but we’ve just written that 𝑥 is equal to
four-ninths. That must mean that 0.4 recurring
is four-ninths.
