Welcome to a lesson on an alternative method for adding mixed numbers that
require an exchange. Let's first review the two most common methods taught for
adding mixed numbers. Let's consider 6 and 2/3 + 2 and 4/5. The first step is to
obtain a common denominator, hopefully the least common denominator,
which is a least common multiple of 3 and 5, which is 15. We begin by writing 2/3
and 4/5 as equivalent fractions with a denominator of 15. Notice for 2/3 we
multiply the numerator denominator by 5 and for 4/5 we multiply the numerator
and denominator by 3 and then we add the fractions and add the whole numbers,
which gives us 8 and 22/15. But now we need to recognize that 22/15 is an improper
fraction, which must be converted to a mixed number and then added to 8, which
gives us a final sum of 9 and 7/15. The second method would be to convert to
improper fractions once we have a common denominator. Once
we convert to improper fractions, we add and then convert back to a mixed number.
Both of these methods can be somewhat involved and therefore let's take a look
at a new method called the compensation method. The idea behind the compensation
method is that it's much easier to add a mixed number and a whole number or than
two mixed numbers. The first step in the compensation method is to obtain a
common denominator, hopefully the least common denominator,
which we already know in this case is 15. So looking at the equivalent of sum in
this form, the next step is to add a proper fraction or a fraction to one of
the addends to make a whole number. In this case let's say we want the second
addend of 2 and 12/15 to be a whole number. Because 12/15 + 3/15 is equal to 1, if we
add 3/15 to the 2nd addend we would have 2 and 15/15 which is equal to 2 + 1
or 3. But by doing this the sum is going to be too large and therefore the next
step is to subtract the same fraction from step two from the other addend. So if
we want an equivalent sum if we add 3/15 to the second then we must
subtract 3/15 from the first addend. So again going back to the second addend, if
we add 3/15 so we have 2 and 12/15 + 3/15, which is 2 and 15/15, which is equal to 2
plus 1 or 3. then if we subtract 3/15 from the other addend we have 6 and 10/15
minus 3/15 which gives us 6 and 7/15. And now we determine the sum because 6 plus
3 is 9, we know the sum is nine and seven fifteenth's. Let's take a look at a
second example. Let's consider 7 and 3/4 + 4 and 2/3. The least common denominator is
12 and therefore the given some is equivalent to 7 and 9 twelfths
+ 4 and 8 twelfths. Notice how if we add the fractions we would have an improper
fraction. To avoid this let's use the compensation method. In this case let's
make the first addend or 7 and 9/12 the whole number by adding three twelfths. 7
and 9 twelfths + three twelfths is 7 and 12/12 which equals 7 plus 1 or 8. But if
we add three twelfths to the first ad end to create an equivalent sum, we
must have subtract three twelfths from the second addend. And 4 and 8 twelfths minus
3/12 is 4 and 5 twelfths. And now we can more easily determine the sum because 8
plus 4 is 12. The sum is 12 and 5 twelfths. Just keep in mind there's no reason to
apply the compensation method unless when adding the fractions,
the result is an improper fraction. we know this is not always the case when
adding mixed numbers. For example if we have 6 and 1/3 + 2 and 2/5 after
obtaining a common denominator we have 6 and 5/15 + 2 and 6/15. Notice how if we add
the fractions we have 11/15 which is a proper fraction, and therefore there's no
need for the compensation method. If we did apply the compensation method we can
still find the sum, but it would take a little extra work.
Notice here the sum is simply 8 and 11/15. Thank you for watching.