@!#]3m5 	
?MCn	N8THIS SHORT COURSE IS DESIGNED TO HELP YOU LEARN HOW
:TO DIVIDE COMMON FRACTIONS.  HOWEVER, BEFORE WE LEARN
0THAT, LET ME ASK YOU FOUR SIMPLE QUESTIONS.

HERE'S YOUR FIRST ONE:

6MULTIPLY THESE TWO FRACTIONS AND TYPE THE RESULT:

              1/3  X  4/5
4/15

RIGHT!
6WRONG, THE ANSWER IS 4/15.  YOU DO NOT KNOW HOW TO
.MULTIPLY FRACTIONS.  ASK YOUR TEACHER FOR
:SOME EXTRA HELP AND TRY THIS COURSE AGAIN LATER.  GOOD
BYE FOR NOW.
6MULTIPLY THESE TWO FRACTIONS AND TYPE THE RESULT:

       3/2  X  7/5

21/10
&CORRECT! YOU'RE DOING FINE SO FAR.
@NO, THE ANSWER IS 21/10.  YOU ARE ALMOST READY TO LEARN HOW
<TO DIVIDE COMMON FRACTIONS.  HOWEVER, YOU NEED A LITTLE
>HELP FROM YOUR TEACHER ON MULTIPLYING FRACTIONS.  TRY THIS
*COURSE AGAIN LATER.  GOOD BYE FOR NOW.
0WRITE THE RECIPROCAL OF THIS FRACTION:  4/5

5/4
CORRECT AGAIN!
>NO, THE RIGHT ANSWER IS 5/4. PERHAPS YOU HAD BETTER LEARN
:WHAT A RECIPROCAL IS AND TRY THIS COURSE AGAIN LATER.
GOOD BYE FOR NOW.

2WRITE THE RECIPROCAL OF THIS FRACTION:  15/17


17/15
8GOOD! YOU DID REALLY WELL ON THOSE QUESTIONS AND YOU
BARE NOW READY TO LEARN HOW TO DIVIDE FRACTIONS. LET'S PROCEED!
DNO, THE ANSWER IS 17/15.  YOU ARE JUST ABOUT READY TO LEARN HOW
>TO DIVIDE FRACTIONS.  ASK YOUR TEACHER FOR SOME HELP WITH
<WRITING RECIPROCALS AND TRY THIS COURSE AGAIN LATER.  SO
LONG FOR NOW.
BLET ME ASK YOU ANOTHER QUESTION.  WHAT IS 1/3 DIVIDED BY 4/5?

5/12
:GOOD FOR YOU.  I THINK YOU ALREADY KNOW HOW TO DIVIDE
,FRACTIONS.  THEREFORE, HERE'S THE TEST.
POSTFR6NO, THAT'S WRONG, SO I'M GOING TO TEACH YOU HOW TO
4DIVIDE COMMON  FRACTIONS.  ON WITH THE LESSON!!
BTHE REASON YOU FIRST HAD TO ANSWER QUESTIONS ABOUT MULTIPLYING
DFRACTIONS AND WRITING RECIPROCALS IS THAT DIVISION USES BOTH OF
THESE IDEAS.

@LET ME EXPLAIN THE PROCESS IN DETAIL, USING A FEW EXAMPLES.
@AFTER THAT, IT WILL BE YOUR TURN TO ANSWER A FEW QUESTIONS.

*TYPE OK WHEN YOU'RE READY TO CONTINUE.
OK
PLEASE TYPE OK.

 HERE IS YOUR FIRST EXAMPLE:

2SUPPOSE WE WANTED TO FIND 1/2 DIVIDED BY 1/3.

BTHE FIRST STEP IS TO REPLACE THE SECOND FRACTION, 1/3, BY ITS
4RECIPROCAL.  IT'S RECIPROCAL IS 3/1 OR JUST 3.  

 OK SO FAR? (TYPE YES OR NO)
YES
GOOD, LET'S CONTINUE.
NO
>STUDY IT A LITTLE LONGER THEN.  WHEN YOU ARE SURE YOU ARE
READY, TYPE YES.
&TYPE EITHER YES OR NO ONLY PLEASE.
$OUR PROBLEM NOW LOOKS LIKE THIS:

          1/2 DIVIDED BY 3/1

DTHE NEXT STEP IS TO CHANGE THE "DIVIDED BY" TO A MULTIPLICATION
SIGN, LIKE THIS: X


"HERE'S THAT STEP:  1/2  X  3/1

*TYPE OK WHEN YOU'RE READY TO CONTINUE
OK
GOOD, LET'S CONTINUE
DSTUDY IT A LITTLE LONGER THEN AND TYPE OK WHEN YOU ARE READY TO
CONTINUE.
:NOW WE ARE ALMOST DONE WITH THAT PROBLEM.  OUR PROBLEM
,RIGHT NOW IS IN THIS FORM:  1/2  X  3/1

<THE LAST STEP SHOULD BE EASY FOR YOU.  MULTIPLY THE TWO
>FRACTIONS AND TYPE THE RESULT.  THIS STEP IS SHOWN BELOW:

0   1/2  X  3/1 = 3/2, SO 3/2 IS OUR ANSWER.

.AS YOU CAN SEE, IT'S REALLY KIND OF EASY!!

(TYPE OK WHEN YOU ARE READY TO GO ON.
OK
&GOOD, LET PROCEED WITH THE LESSON.
TYPE OK PLEASE.
<LET'S SUMMARIZE THE 3 STEPS I WENT THROUGH IN THE FIRST
6EXAMPLE I JUST COMPLETED FOR YOU.  THE STEPS ARE:

4 (1) TAKE THE RECIPROCAL OF THE SECOND FRACTION.
: (2) CHANGE THE "DIVIDED BY" TO A MULTIPLICATION SIGN.
4 (3) MULTIPLY THE FRACTIONS AND TYPE THE ANSWER.

TYPE OK TO CONTINUE.
OK
TYPE OK PLEASE.
 HERE'S YOUR SECOND EXAMPLE:

2SUPPOSE WE WANTED TO FIND 2/3 DIVIDED BY 5/7.

LET'S DO THE STEPS:
 
,    STEP 1:  TAKE THE RECIPROCAL OF 5/7

               IT IS 7/5.

@     STEP 2: CHANGE THE DIVIDED BY TO A MULTIPLICATION SIGN

               2/3  X  7/5

<    STEP 3: MULTIPLY THE FRACTIONS AND TYPE THE RESULT:

:            2/3 X 7/5 = 14/15.   14/15 IS OUR ANSWER.

:STUDY THIS EXAMPLE CAREFULLY AND TYPE OK TO CONTINUE.
OK
PLEASE TYPE OK
>OK, NOW YOU'VE SEEN TWO EXAMPLES.  IT IS NOW YOUR TURN TO 
ANSWER SOME QUESTIONS.

2LOOK AT THIS PROBLEM:  2/5 DIVIDED BY 7/9 = ?

"WHAT IS THE RECIPROCAL OF 7/9?
9/7

GOOD!!
7/9
*NO, YOU FORGOT TO TURN IT UPSIDE DOWN.
2I'LL HELP YOU OUT THIS TIME. PLEASE TYPE 9/7.
8NOW OUR PROBLEM LOOKS LIKE THIS:  2/5 DIVIDED BY 9/7

8THE NEXT STEP IS TO CHANGE THE PROBLEM TO THIS FORM:

          2/5   X   9/7

.NOW YOU FINISH IT UP AND TYPE THE ANSWER.

18/35
 GREAT!! LET'S TRY SOME MORE.
FYOU PROBABLY MADE SOME SILLY ERROR.  THE ANSWER IS 18/35.  PLEASE
TYPE 18/35.
OK, HERE'S THE NEXT ONE:

        3/8 DIVIDED BY 2/7

WHAT IS THE ANSWER?

21/16
:FANTASTIC!! YOU MUST REALLY HAVE THE HANG OF ALL THIS.
6/56
.YOU FORGOT TO TAKE THE RECIPROCAL OF 2/7.
<NO, THE ANSWER IS 21/16.  TRY TO FIGURE OUT WHAT YOU ARE
 DOING WTONG AND CORRECT IT.
@AT THIS POINT, I WANT TO ASK YOU IF YOU STILL FEEL YOU NEED
0A  LITTLE EXTRA REVIEW BEFORE ATTEMPTING THE
6TEST.  IF YOU DO, TYPE YES, IF YOU DON'T TYPE NO.
NO
BGREAT, THEN HERE IS THE TEST.  PREPARE YOURSELF BY THINKING OF
<THE THREE RULES.  SEE IF YOU CAN GET ALL OF THE PROBLEMS

RIGHT.
POSTFRYES
&OK, MORE REVIEW COMING RIGHT UP!!
 TYPE YES OR NO ONLY PLEASE.
> ************** EXTRA REVIEW SECTION ********************

@HERE IS SOME EXTRA REVIEW FOR YOU.  REMEMBER THE THREE RULES
FOR DIVIDING FRACTIONS:

4 (1) WRITE THE RECIPROCAL OF THE SECOND FRACTION
8 (2) CHANGE THE PROBLEM TO A MULTIPLICATION PROBLEM
4 (3) MULTIPLY THE FRACTIONS AND TYPE THE ANSWER

4STUDY THE ABOVE TOPICS TO SEE IF THERE ARE ANY 
@OF THEM YOU WOULD LIKE TO REVIEW.  IF YOU DO WANT TO REVIEW 
8ANY OF THEM, TYPE THE NUMBER, EITHER 1,2 OR 3 WHICH
@CORRESPONDS TO THAT  TOPIC. IF YOU DO NOT WISH TO REVIEW ANY
@OF THEM, TYPE NO, AND YOU WILL BE GIVEN A FEW MORE PROBLEMS
 BEFORE ATTEMPTING THE TEST.

@
A
A@NO
(PLEASE TYPE 1,2,3 OR NO ONLY PLEASE.
FOK, YOU'RE READY FOR MORE PROBLEMS.  HERE ARE SOME MULTIPLE CHOICE
DQUESTIONS FOR YOU TO DO SO YOU CAN SEE IF YOU ARE  STILL HAVING
BPROBLEMS.  YOU WILL NOT BE ALLOWED TO TAKE THE TEST UNLESS YOU
FGET AT LEAST 2 RIGHT.  TYPE ONLY THE LETTER OF THE CORRECT ANSWER.

HERE'S YOUR FIRST PROBLEM:

THE RECIPROCAL OF 8/9 IS ?
  (A) 8/9
 (B) 9/8
  (C) NONE OF THESE
B

RIGHT!
.NO, THAT'S WRONG.  THE RIGHT ANSWER IS B.
>HERE'S YOUR NEXT ONE:  WHICH OF THE FOLLOWING IS THE RIGHT
"WAY TO DO 3/5 DIVIDED BY 1/4?

  (A) 3/5  X  1/4
  (B) 3/5  X  4/1
  (C) 5/3  X  1/4
B
0CORRECT! I THINK YOU ARE READY FOR THE TEST.
POSTFRNO, THE ANSWER IS B.
2HERE'S YOUR NEXT ONE:  11/4 DIVIDED BY 3/2 = ?

   (A) 12/22
   (B) 33/8
   (C) 22/12
C
BGOOD! NOW YOU HAVE ONE RIGHT.  YOU NEED ONE MORE RIGHT TO BE 
ABLE TO TAKE THE TEST.
@NO, YOU'RE WRONG.  THE RIGHT ANSWER IS C.  I THINK YOU STILL
@NEED EXTRA HELP.  PLEASE SEE YOUR TEACHER FOR THIS HELP AND
@TRY THIS COURSE AGAIN LATER.  THIS ENDS YOUR COURSE FOR NOW.
6HERE'S YOUR LAST PROBLEM:  3/4 DIVIDED BY 2/1 = ?
  
   (A) 6/4
   (B) 3/8
   (C) 8/3
B
DGREAT! THAT'S TWO RIGHT, SO YOU ARE READY FOR THE TEST.  I THINK
THOSE PROBLEMS HELPED YOU.
POSTFRDNO, THE ANSWER IS B.  I'M SORRY, BUT I THINK YOU NEED MORE HELP.
>PLEASE SEE YOUR TEACHER FOR THIS HELP AND TRY THIS COURSE
AGAIN LATER. GOOD BYE.
>OK, YOU WANT MORE HELP ON WRITING RECIPROCALS.  REMEMBER 
DTHAT A RECIPROCAL IS OBTAINED BY WRITING A FRACTION UPSIDE DOWN,
>THAT IS, INTERCHANGING THE NUMERATOR AND THE DENOMINATOR.

HERE ARE SOME EXAMPLES:

$1.  THE RECIPROCAL OF 2/7 IS 7/2
$2. THE RECIPROCAL OF 5/8 IS 8/5.
$3. THE RECIPROCAL OF 1/4 IS 4/1.

0ARE YOU READY TO TRY A FEW?  TYPE YES OR NO.
YES
OK, THEN HERE GOES!
NO
FSTUDY THE EXAMPLES A LITTLE LONGER AND TYPE YES WHEN YOU'RE READY
TO CONTINUE.
 PLEASE TYPE YES OR NO ONLY.
THE RECIPROCAL OF 3/2 IS ?
2/3

GOOD!
"NO, THE CORRECT ANSWER IS 2/3.
THE RECIPROCAL OF 1/8 IS ?
8/1
NOW YOU'VE GOT IT!
8

RIGHT!
*NO, THE RIGHT ANSWER IS 8/1 OR JUST 8.
,ONE MORE:  THE RECIPROCAL OF 15/19 IS ?

19/15

GOOD!
>NO, THE RECIPROCAL OF 15/19 IS 19/15.  PLEASE TYPE 19/15.
2DO YOU WISH  TO GO BACK AND REVIEW SOME MORE?
TYPE YES OR NO.
YES
"OK, BACK FOR MORE REVIEW THEN!
NO
"OK, THEN ON TO MORE PROBLEMS!
BTHIS SECOND STEP IS EASY, BUT  VERY IMPORTANT!! REMEMBER THAT
BWHEN YOU DO A DIVISION PROBLEM, YOU ARE ACTUALLY MULTIPLYING.
4YOU MUST ALWAYS REMEMBER TO CHANGE THE DIVISION
:PROBLEM TO A MULTIPLICATION ONE BEFORE YOU FIGURE OUT
THE ANSWER.

HERE ARE SOME EXAMPLES:

>1/2 DIVIDED BY 2/5 BECOMES 1/2   X  5/2 (REMEMBER TO TAKE
,                 THE RECIPROCAL OF 2/5)

,5/8 DIVIDED BY 9/10 BECOMES 5/8  X  10/9

,5/6 DIVIDED BY 7/12 BECOMES 5/6  X  12/7

DTYPE OK AFTER YOU HAVE STUDIED THESE CAREFULLY AND YOU ARE SURE
THAT YOU UNDERSTAND THEM.
OK

GOOD.
2PLEASE TYPE OK WHEN YOU ARE READY TO CONTINUE.
BNOW, DO YOU WANT TO GO BACK AND REVIEW ANY OF THE TOPICS ONCE
6MORE BEFORE TRYING MORE PROBLEMS? TYPE YES OR NO.
YES
OK, THEN!
NO
OK! ON TO MORE PROBLEMS!
&PLEASE TYPE YES OR NO ONLY PLEASE.
@YOU INDICATED YOU WANT HELP IN MULTIPLYING FRACTIONS.  THIS
BREALLY IS QUITE SIMPLE.  I WILLL SHOW YOU A FEW EXAMPLES, AND
BIF YOU STILL DON'T UNDERSTAND, I SUGGEST YOU SEE YOUR TEACHER
BFOR EXTRA HELP BECAUSE YOU ALREADY ANSWERED SEVERAL QUESTIONS
8CORRECTLY DEALING WITH MULTIPLICATION OF FRACTIONS.

:HERE ARE SOME EXAMPLES.  THE ONLY IDEA TO REMEMBER IS
.TO MULTIPLY THE NUMERATORS  TOGETHER  AND
>MULTIPLY THE DENOMIINATORS TOGETHER AND WRITE THE ANSWER.

$EXAMPLE 1:  1/2  X  3/5  =  3/10
$EXAMPLE 2:  3/7  X  2/9  =  6/63
&EXAMPLE 3:  3/5  X  4/5  =  12/25

@TYPE OK WHEN YOU UNDERSTAND THESE AND WE WILL CONTINUE.  IF
BYOU TYPE NO, INDICATING THAT YOU DON'T UNDERSTAND, YOU WILL BE
,REFERRED TO YOUR TEACHER FOR MORE HELP.
OK

GOOD!
NO
@PLEASE SEE YOUR TEACHER FOR EXTRA HELP.  YOU MUST BE ABLE TO
@MULTIPLY FRACTIONS WELL BEFORE YOU CAN DO DIVISION. GOOD BYE
FOR NOW.
PLEASE TYPE OK OR NO ONLY.
2DO YOU STILL NEED MORE REVIEW? TYPE YES OR NO.

YES
OK
NO
$ALL RIGHT THEN.  LET'S CONTINUE.
 PLEASE TYPE YES OR NO ONLY.
CONTINUE.  IF
BYOU TYPE NO, INDICATING THAT YOU DON'T UNDERSTAND, YOU WILL BE