Transcript section 1.3 B

Math 010 online work that was due
today at the start of class:
• Section 1.3A Online Homework
•
•
•
•
Please log in to MyLabandMastering
Click on “Gradebook”
Click “Review” by HW 1.3A
Notice that you can look at each problem and
answer, and even work similar problems again
for practice, even after the deadline, without
affecting your grade.
Any questions on any of the problems from
Section 1.3A?
Some comments about entering
answers for fractions:
1. Enter an answer like 12/1 just as 12. (If the
denominator is 1, just write the answer as a
whole number without the denominator.)
2. Check to make sure all fraction answers are
completely simplified.
Example:
How would you write the answer if it comes out to
42/66? (Answer: 7/11)
• After you enter the answer to a problem, click “Check Answer”
to see if it’s correct. For most problems, you’ll get three tries
to get it right.
• Once you’ve clicked “check answer” on a problem, that
problem’s result is stored on the system and will be retained
even if you don’t click “Save”.
IMPORTANT: Even if you get a problem wrong on each of your
three tries, you can still go back and do it again by clicking
“similar exercise” at the bottom of the exercise box. You can
do this nine times, for a total of 30 tries (3 tries at each of 10
different problems. You should always work to get 100% on
each assignment!
25%
Remember: Homework counts for about
of your entire
course grade, so scoring 100% on each assignment really
makes a difference.
Things to remember:
Take notes as you do each homework problem.
Write down all steps (show your work!). Again,
this helps tremendously when you’re studying
for tests.
• If you are having trouble with a problem,
check the on-line help available for each
problem:
• Help Me Solve This
• Textbook Pages
• Video Clip (for some problems)
THIS IS SO IMPORTANT I’M
SAYING IT ONE MORE TIME:
You can go back in and work on an
assignment even after saving it,
provided the deadline has not expired, so
again, you should always work to get 100%
on each assignment.
NOW
CLOSE
YOUR LAPTOPS
(You may reopen them when I finish the
lecture, at which time we will help you
get started on the homework
assignment.)
Section 1.3, Part B
Last time we covered:
• Factoring numbers into prime factors
• Simplifying fractions using prime factors
• Multiplying fractions
• Dividing fractions
Problems in Section 1.3A homework used these
concepts, as do problems #3 & #5 in the Gateway
Homework and on all regular Gateway Quizzes.
Today we will explore how to work with mixed
fractions and how to add and subtract fractions.
• In this course, we will usually need to change mixed numbers
into improper fractions in order to do our calculations.
• Also, fraction answers with a numerator larger than the
denominator should be entered into the computer in that form
(improper fraction) rather than as a mixed number.
NOTE: Gateway problems 4 & 6 and
several of today’s homework problems
using mixed numbers all start
with the same step.
A mixed fraction (mixed number) consists of an
integer part and a fraction part. We want to
covert the mixed number into an improper
fraction. This is done by multiplying the integer
part by the denominator of the fraction part,
then adding that product to the numerator of
the fraction and putting that sum over the
original denominator.
Converting a Mixed Number
Into an Improper Fraction:
Example: Convert the mixed number 5 14 into
an improper fraction:
Solution: First, note that
Then: 5
1
 
1
4
54
14
5 14  5  14  15  14
 
1
4
20
4
 
1
4
21
4
Converting a Mixed Number
Into an Improper Fraction
Another way to look at it:
To convert 5 ¼:
1. Multiply the denominator of the fraction part (4)
by the whole number part (5) 5 ∙ 4 = 20
2. Add the numerator of the fraction part (1)
to this result:
1 + 20 = 21
3. Write this number over the denominator of the
original fraction : ANSWER: 21/4
Sample Gateway Problem #4: Multiplying mixed numbers
Step 1: Convert the mixed number 5 23 into
an improper fraction: (Note that 5 23  5  23  15  32 )
.
5
1
 
2
3
53
13
 
2
3
15
3
 
2
3
17
3
So 5 23  76 becomes 173  76 , which we can then solve the same
way we did our fraction multiplication problems in the last
lecture and in HW 1.3A that was due today.
Sample Problem #4 (continued)
17
3

6
7
Step 2: Factor both the numerators and denominators into
prime factors, then write each fraction in factored form:
First fraction:
Second fraction:
17 and 3 are both prime
6 = 2∙3 and 7 is prime
So you can write 17 ∙ 6 as 17 ∙ 2∙3 .
3 7
3 7
Step 3: Now just cancel any common factors that appear in both
numerator and denominator. Once you multiply out any remaining
factors, the result is your simplified answer.
17 ∙ 2∙3/ = 17∙2 =
3/
7
7
34
7
.
14
Sample Gateway Problem #6: Dividing with mixed numbers
Step 1: Convert the mixed numbers into improper fractions:
7 7   
1
7
1
7
7
1
1
7
77
17
 
1
7
49
7
12 12  12  12  121  12  12122  12 
Now we can rewrite the problem as:
Then convert from division
to multiplication by using the
reciprocal of the second fraction:
 
50
7
 12 
25
2
24
2
1
7
7 17 12 12  507  252
50
7

25
2

50
7

2
25
Sample Problem #6 (continued) 7 17 12 12 
50
7
 252  507  252
Step 2: Factor both the numerators and denominators into
prime factors, then write each fraction in factored form:
First fraction:
50 = 2∙5∙5 and 7 is prime
Second fraction:
2 is prime and 25 = 5∙5
So you can write 50 • 2 as 2∙5∙5 • 2
7 25
7
5∙5
.
Step 3: Now just cancel any common factors that appear in
Both numerator and denominator. Once you multiply out any
remaining factors, the result is your simplified answer.
2∙5∙5
/ / • 2 = 2∙2 = 4
7
5∙5
7
7
/ /
16
Now we’ll move on to adding and subtracting
fractions, which is usually a little more work
than multiplying or dividing fractions, because
before you add or subtract, both fractions have
to be converted so they have the same
denominator.
If your two fractions already have the same
denominator, just add (or subtract) the
numerators and put the result over that
denominator:
a c ac
 
b b
b
a c ac
 
b b
b
Adding or subtracting factions with the SAME denominator:
1.
2.
Add (or subtract) the numerators together and write the sum (or
difference) over the common denominator.
Simplify the fraction.
Example
3
9
12



20 20
20
Add the following fractions.
2 23 3

2 25 5
But what if you need to add fractions in which the two
denominators are different?
• Then you have to find a COMMON (same) DENOMINATOR
before you can add the numerators together. Simplifying your
answer will be MUCH easier if you use the smallest possible
(“least”) denominator that works for both fractions.
Steps to follow for finding the
least common denominator (LCD) of two fractions:
1. Factor both denominators into primes.
2. List all the primes in the first denominator (with
multiplication signs between each number)
3. After these numbers, list any NEW primes that appear
in the second denominator but not the first.
4. Multiply this whole list of primes together. This is your
LCD.
Finding the least common denominator (LCD)
of two fractions:
Example: Find the LCD of 3/4 and 7/18:
1.
Factor both denominators into primes.
4 = 2*2
2.
18 = 2*9 = 2*3*3
Start with all the primes in the first denominator (with multiplication
signs between each number). If any prime number appears more than
once in the first denominator, include each one in the LCD.
2*2
3.
After these numbers, list any NEW primes that appear in the second
denominator but not the first.
2*2*3*3
4.
Multiply this whole list of primes together. This is your LCD.
2*2*3*3= 4*9 = 36
NOTE: Gateway problems 1 & 2 on
adding and subtracting fractions as well as many of the
problems on today’s homework assignment can all be
done using the same set of steps.
Adding fractions and subtracting fractions both
require finding a least common denominator
(LCD), which as we just saw is most easily done
by factoring the denominator (bottom number)
of each fraction into a product of prime
numbers (a number that can be divided only by
itself and 1).
21
Sample Gateway Problem #1: Adding Fractions
Step 1: Factor the two denominators into prime factors, then
write each fraction with its denominator in factored form:
10 = 2∙5
and
35 = 5∙7,
so
.
3 + 2 = 3 + 2
10 35 2∙5 5∙7
Step 2: Find the least common denominator (LCD):
LCD = 2∙5∙7
22
.
Sample Problem #1 (continued)
Step 3: Multiply the numerator (top)and denominator of each fraction
by the factor(s) needed to turn each denominator into the LCD.
LCD = 2∙5∙7
3∙7 + 2 ∙2
2∙5∙7
5∙7∙2
.
Step 4: Multiply each numerator out, leaving the denominators in
factored form, then add the two numerators and put them over the
common denominator.
21 + 4 = 21 + 4 = 25 (note that 5∙7∙2 = 2∙5∙7 by
2∙5∙7 5∙7∙2
2∙5∙7
2∙5∙7
the commutative property)
Step 5: Now factor the numerator, then cancel any common factors
that appear in both numerator and denominator. Once you multiply
out any remaining factors, the result is your simplified answer.
= 25 = 5∙5 = /5∙5 = 5 = 5
/
2∙5∙7 2∙5∙7 2∙5∙7
2∙7
14
.
23
Sample Gateway Problem #2: Subtracting Fractions
Step 1: Factor the two denominators into prime factors, then
write each fraction with its denominator in factored form:
14 = 2∙7
and
35 = 5∙7,
so
5 - 2
2∙7
5∙7
Step 2: Find the least common denominator (LCD):
LCD = 2∙7∙5
24
Sample Problem #2 (continued)
Step 3: Multiply the numerator and denominator of each fraction by
the factor(s) needed to turn each denominator into the LCD: form:
LCD = 2∙7∙5
5∙5
2∙7∙5
- 2 ∙2
5∙7∙2
Step 4: Multiply out the numerators, leaving the denominators in
factored form, then add the two numerators and put them over the
common denominator.
25 - 4
= 25 - 4 = 21
2∙5∙7 5∙7∙2
2∙5∙7 2∙5∙7
.
Step 5: Now factor the numerator, then cancel any common factors
that appear in both numerator and denominator. Once you multiply
out any remaining factors, the result is your simplified answer.
21 = 3∙7 = 3∙7/ = 3 = 3
2∙5∙7 2∙5∙7 2∙5∙7
10
/ 2∙5
.
25
You may now OPEN your LAPTOPS.
If there’s still time left in the class
session after lecture, you should stay in
the classroom to work on your
homework till the end of the session.
If you have finished the homework
already, or if you get it finished before
the end of the class period, show your
on-screen 100% score to the teacher or
TA and you may then work on Practice
Gateway Quiz problems.
26
Reminder:
The homework assignment
on section 1.3B is due
at the start of our next
class session.
Visit the Math TLC Open Lab
for homework help!