We want to evaluate the expression.
Following the order of operations,
the first step is to simplify within the parentheses,
which means we begin by determining
the differences in the parentheses.
In order to add or subtract fractions,
we need to obtain a common denominator,
hopefully the least common denominator
which is the least common multiple of the denominators.
So for 2/25 minus 8/75,
the least common multiple of 25 and 75 is 75,
and therefore 75 is the least common denominator.
And therefore we need to write 2/25
as an equivalent fraction with a denominator of 75.
And since 25 times three is 75,
we multiply the numerator and denominator by three.
And now for 8/15 minus 2/45,
the least common multiple of 15 and 45 is 45,
and therefore 45 is the least common denominator,
which means we need to write 8/15
as an equivalent fraction with a denominator of 45.
And since 15 times three is 45,
we multiply the numerator and denominator
of 8/15 by three.
Notice for the first difference
we have a common denominator of 75,
and for the second difference
we have a common denominator of 45.
Notice 2/25 is equivalent to 6/75,
giving us 6/75 minus 8/75.
And then we have divided by, for the second difference,
notice 8/15 is equivalent to 24/45,
giving us 24/45 minus 2/45.
And now we can go ahead and subtract.
For 6/75 minus 8/75 the denominator remains 75,
and the numerator is six minus eight
which is negative two,
giving us a difference of negative 2/75.
And then we have divided by 24/45 minus 2/45.
The denominator remains 45,
and the numerator is 24 minus two which is 22.
Notice neither fraction simplifies
because the only common factor
between the numerator and denominator is one.
And now we divide, but dividing by a fraction
is equivalent to multiplying by the reciprocal,
and therefore dividing by 22/45
is equivalent to multiplying by 45/22.
So let's write the quotient as a product,
which is negative 2/75
times the reciprocal of 22/45,
which, again, is 45/22.
To multiply fractions
we multiply the numerators and denominators.
Multiplying in the numerator,
we have negative two times 45.
In the denominator we have 75 times 22.
Before we determine the products though,
we will now simplify out the common factors
other than one between the numerator and denominator.
Notice two and 22 share a common factor of two.
And since there's one two in two
or because two divided by two is one,
the two simplifies to one.
And because there are 11 twos in 22
or because 22 divided by two is 11,
the 22 simplifies to 11.
But also notice how 45 and 75
share a common factor of 15.
And since 45 divided by 15 is three
or because there are three 15s in 45,
the 45 simplifies to three.
And because 75 divided by 15 is five
or because there are five 15s in 75,
the 75 simplifies to five.
And now we go ahead and multiply.
And we can't forget about the negative here.
Negative one times three is negative three.
In the denominator we have five times 11 which is 55.
The given expression simplifies to negative 3/55.
I hope you found this helpful.