Write the following number as ratios of integers

Question

asked Apr 10, 2019 24.5k views

1 Answer

Damathryx · Answer 1 · 2019-04-15T11:05:24+0000

We're going to "cut" the repeating part here in a few steps. First, we're going to put the number in a variable:

$x=-2.0\overline{42}$

Next, to get rid of the negative, we can multiply either side by -1 to get

$-x=2.0\overline{42}$

Now, we won't actually use this -x directly; instead, we want to create two new values, one by multiplying either side by 10:

$-10x=20.\overline{42}$

and the other by multiplying either side by 1000:

$-1000x=2042.\overline{42}$

Next, we can get rid of the repeated part of the number by subtracting -10x from -1000x:

And finally, we can divide either side of the equation by -990 to find that

x= (2022)/(-990)=-(2022)/(990)/ (6)/(6)=-(337)/(165)