Welcome back for two more examples
on how to convert repeating decimals to fractions.
For the first example
we have 0.86 with just the six repeating.
Step one is to set the repeating decimal equal to X
which gives us X equals 0.86 with the six repeating.
Let's label this equation one from step one.
Step two, if needed, multiply both sides of the equation
by multiples of 10 until just the repeating
digits are to the right of the decimal.
Notice right now we have this eight that does not repeat,
and it's to the right of the decimal.
We want our decimal where we just have
the six repeating to the right.
Notice for this to happen we have to move
the decimal one place to the right, which is equivalent
to multiplying by 10, which means for equation two
from step two,
we multiply both sides of this equation by 10.
10 times X is 10X, and 10 times 0.86 with the six repeating
is equal to 8.6 repeating.
Notice now we just have the repeating six
to the right of the decimal,
which is required for step two.
And now for step three perform another equation
by multiplying both sides of the equation
from step one or step two by multiples of ten
so that one set of the repeating digit or digits
are now to the left of the decimal.
Because the six is repeating, we need our decimal
where we have one six to the left of the decimal,
looking at equation two for that to happen
you have to move the decimal one place to the right
which, again, is equivalent to multiplying by 10.
So for step three or equation three,
we multiply both sides of equation two by 10.
10 times 10X is 100X, and 10 times 8.6 repeating
is equal to 86.6 repeating.
Again, now notice how we have one six to the left
of the decimal, which is required for step three.
Step four, we now subtract the equation from step two
from the equation from step three.
So beginning with equation three, we have 100X equals
86.6 repeating, and now we subtract equation two,
which means we subtract 10X from the left
and we subtract 8.6 repeating on the right,
and now we subtract 100X minus 10X is 90X.
On the right, 86.6 repeating minus 8.6 repeating
is equal to 78.
And now we solve for X by dividing both sides by 90.
Simplifying on the left,
this quotient is one, one times X is X.
We have X equals 78 90ths, which does simplify.
78 and 90 share a common factor of two,
so we divide the numerator and denominator by two.
78 divided by two is equal to 39.
90 divided by two is equal to 45.
Now we have X equals 39 45ths,
and X is equal to the repeating decimal
so now we know that 0.86
with the 6 repeating is equal to 39 45ths.
And of course we can easily check this on the calculator.
39 divided by 45 does give us a repeating decimal
of 0.86 with the six repeating.
Next we have 3.152 with just the five and two repeating.
Step one we set X equal to the repeating decimal,
so we'll leave off the three for right now,
and the equation is X
equals 0.152 with the five and two repeating.
This is equation one.
For equation two, we multiply both sides
of the equation by multiples of 10
until just the repeating digits
are to the right of the decimal.
Notice right now we have this one to the right
which doesn't repeat.
We only the five and two to the right,
which means we have to move the decimal
one place to the right which is equivalent
to multiplying by 10.
For step two we multiply both sides of the equation by 10.
10 times X is 10X.
And 10 times 0.152 with the five and two repeating
is equal to 1.52 with the five and two repeating.
Notice now just the repeating digits
are now to the right of the decimal.
Step three, we now form another equation
by multiplying both sides of the equation
from step one or two by multiples of 10
so that one set of the repeating digit or digits
are now to the left of the decimal.
Which means now we want our decimal
where we have a five and two
to the left of the decimal.
In order to do this we have to move the decimal
two places to the right, which is equivalent
to multiplying by 100.
So for step three or equation three
we multiply both sides of equation two by 100.
100 times 10X is equal to 1,000X,
and 100 times 1.52 with the five and two repeating
is equal to 152.53 repeating.
For step four we subtract equation two
from equation three.
So beginning with equation three
we have 1,000X equals 152.52 repeating,
and now we subtract equation two,
so we subtract 10X on the left, and we subtract 1.52
with the five and two repeating on the right.
Subtracting 1,000X minus 10X is 990X equals
subtracting on the right, 152.52 repeating
minus 1.52 repeating is equal to 151.
Now we solve for X by dividing both sides by 990.
Simplifying, this quotient is one, one times X is X.
We have X equals 151 990ths.
And this actually doesn't simplify
so now we know the repeating decimal
of 0.152 with the 5 and 2 repeating is equal to 151 990ths.
So now we know 3.152 with the five and two repeating
is equal to three the whole number
plus the repeating decimal
as a fraction which is 151 990ths,
or as a mixed number, 3 and 151 990ths.
If we did want the fraction
rather than the mixed number,
the denominator would be 990 and the numerator
would be 990 times three plus 151
which is equal to 3,121 990ths.
And again we can check this on the calculator.
3,121 divided by 990 does give us
the original repeating decimal.
Just be aware the calculator
is rounding in the rightmost digit.
I hope you found this helpful.