Evaluate 0.65 minus one-fifth,
giving the answer as a fraction in its simplest form.
In this question, we are asked to
find the difference between two rational numbers, one given as a decimal and the
other given as a fraction. We need to give our answer as a
fraction in its simplest form. To find the difference between
rational numbers, it is a good idea to write them in the same form. This means that one way that we can
answer this question is by rewriting both numbers as decimals. We know that one-fifth is 0.2. So, this difference is equal to
0.65 minus 0.2. We can then evaluate this
difference to get 0.45. We need to write this as a fraction
in its simplest form.
To do this, we can start by writing
the decimal as a fraction by noting that it is equal to 45 over 100. We can then cancel the shared
factor of five in the numerator and denominator to obtain nine over 20. This cannot be simplified further,
so it is our final answer.
While this method works for this
particular question, in general, fractions may have difficult decimal expansions to
add or subtract. So, in general, it is a good idea
to convert both numbers into fractions rather than decimals.
To do this, let’s clear some space
and then rewrite 0.65 as a fraction. We can note that it is equal to 65
over 100. We can then cancel the shared
factor of five in the numerator and denominator to get 13 over 20. Therefore, the difference is equal
to 13 over 20 minus one-fifth. We can then recall that we can find
the difference between two fractions if they have the same denominator by finding
the difference in their numerators. In general, we have 𝑎 over 𝑐
minus 𝑏 over 𝑐 is equal to 𝑎 minus 𝑏 all over 𝑐.
We want to rewrite both of our
fractions to have the same denominator. To do this, we need to find the
lowest common multiple of the denominators. In this case, 20 is a multiple of
five, so the lowest common multiple is 20. The first fraction already has a
denominator of 20. We can find an equivalent fraction
to one-fifth with a denominator of 20 by multiplying the numerator and denominator
by four. Evaluating the products then gives
us 13 over 20 minus four over 20.
Now that the denominators of the
fractions are equal, we can find their difference by finding the difference in their
numerators. We get 13 minus four all over
20. We can then evaluate to obtain nine
over 20. There are no nontrivial shared
factors between the numerator and denominator, so we cannot simplify the fraction