Download Solving problems using the Least Common Denominator2

Name ______________________
Susan E. Wagner High School
Mr. Gary Giordano, Principal
Ms. L. Silver, AP
Department of Mathematics
Mr. R. Gargano, Teacher
MR21 – Review Notes for Exam 1-2MP
Solving problems using the Least Common Denominator:
To find the LCD … factor denominators to obtain prime factors; then take all the different prime
factors and raise them to the power which is equal to the highest number of times the factor
appears in any ONE denominator.
Uses…
I: Addition / Subtraction (keep the LCD as the new denominator)
Arithmetic Example
1. Prime factor the 2
denominators (10 and
Add
12).
Prime factors of 10
are 5 • 2.
Prime factors of 12
are 3•2•2
Find the LCD
2. There are 3
different prime
factors: 5, 3, and 2.
The most 5 and 3
appear in any one
denominator is once;
but the most 2
appears is twice.
Therefore the LCD
will be 5•3•2•2,
which equals 60.
Raise the numerator
by the fraction’s new
factors.
Thus
=
and
=
+
Algebraic Example
1. Prime factor the 2
denominators.
Prime factor a2 – 1 and get
(a – 1)*(a + 1).
The prime factor for a + 1, is
itself.
Add
Find the LCD
=
=
2. There are two different
factors, (a – 1) and (a + 1).
They appear at most only one
time in any denominator.
Therefore, the LCD is
(a – 1)*(a + 1) which equals
a2 – 1
Raise the numerator
by the fraction’s
new factors
Thus,
and
=
=
So,
+
=
=
OR
II. To simplify a complex fraction (a fraction which has a fraction as the numerator and/or
denominator.
1. Simplify
2. Multiply the
numerator and the
denominator of the
complex fraction by
the LCD. This is to
eliminate the
denominators in the
complex fraction.
3 and 2 are the prime
factors so the LCD =
2(3) = 6
(I put the 6 over
1 just to see this
example more clearly:
of course,
)
Simplify
Multiply the numerator
and the denominator of
the complex fraction
by the LCD. This is to
eliminate the
denominators in the
complex fraction.
The prime factors of
the “small”
denominators are “2”
and “x”; so the LCD =
2x
3. Multiply by the LCD
to remove the small
denominators and then
cancel.
2
3
=
(answer… and
note that the original
denominators, the “2”
and “3” were removed),
so the numerator and
the denominator are
whole numbers.)
The reduced fraction
is the answer.
III. To solve equations which contain fractions. The easiest way to solve fractional equations is to
remove the denominator. Do this by multiplying each term in the equation by the LCD.
1. Determine the LCD for the fractions…
Solve for
the LCD is
2. Multiply each term by the LCD and the
denominators will be canceled.
3. After canceling multiply across each
numerator with the factor(s) that were
not canceled. This will give an equivalent
equation without denominators.
Then solve this equation.
4. Because of the fact that fractional
operations are not valid if denominators
are 0, we must check the result.
So the solution is