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SECTION 1.2 IS A REVIEW OF FRACTIONS.
NOW IF YOU RECALL,
ALMOST THE ENTIRE COURSE OF MATH 112 IS ON FRACTIONS,
SO ALL OF YOU ALREADY HAVE CREDIT FOR 112
OR YOU ARE TESTING TO 115,
SO YOU SHOULD HAVE SOME UNDERSTANDING ABOUT FRACTIONS.
SO I'M NOT GONNA GO THROUGH EVERY DETAIL ABOUT FRACTIONS.
I'LL JUST TOUCH AND GO ON THE DIFFERENT OPERATIONS.
SO FIRST AND FOREMOST,
YOU NEED TO REMEMBER HOW TO REDUCE FRACTIONS
TO THEIR LOWEST TERMS. OKAY, SO LOWEST TERMS.
SO WHAT DO WE MEAN BY LOWEST TERMS?
SO TAKE, FOR EXAMPLE,
IF YOU HAVE A FRACTION, SAY, 120 OVER 480, OKAY?
THE FIRST THING YOU WANT TO UNDERSTAND IS ZEROES.
THEY CAN BE CANCELED.
SO ONE FROM THE TOP AND ONE FROM THE BOTTOM.
SO NOW I'M LOOKING AT 12 AND 48.
SO IS THERE ANY NUMBER THAT COULD DIVIDE 12 AS WELL AS 48?
SO THAT'S HOW YOU HAVE TO THINK.
SO THE ANSWER IS, YES, I CAN DIVIDE BY 4,
FOR EXAMPLE.
NOW IF I DIVIDE BY 4, THIS 12 BECOMES...
SO I'M DIVIDING THE TOP BY 4 AND THE BOTTOM BY 4.
SO 12 DIVIDED BY 4 IS 3 AND 48 DIVIDED BY 4 IS 12.
SO NOW I'M LOOKING AT 3 AND 12.
YOU NOTICE THAT I CAN DIVIDE BY 3 AGAIN,
SO I CAN DIVIDE BY 3.
SO 3 DIVIDED BY 3, REMEMBER, IS NOT 0.
IT'S 1, AND 12 DIVIDED BY 3 IS 4.
SO THE FINAL ANSWER IS 1 OVER 4.
SO THIS IS WHAT WE MEAN BY REDUCE A FRACTION
TO ITS LOWEST TERM.
SO WHAT ELSE DO YOU NEED TO KNOW IN FRACTIONS?
YOU NEED TO KNOW HOW TO MULTIPLY.
SO MULTIPLYING FRACTIONS, THE RULE IS "A" x B,
"A" OVER B x C OVER D IS THE SAME AS AC OVER BD.
SO ALTHOUGH IT'S TWO SEPARATE FRACTIONS,
BUT YOU CAN LOOK UPON IT LIKE ONE.
THEN YOU STILL REDUCE IF POSSIBLE.
SO REMEMBER, REDUCE IF POSSIBLE.
SO TAKE, FOR EXAMPLE,
IF YOU GIVE ME A FRACTION,
SAY, 3 OVER 4 MULTIPLIED BY,
SAY, 12, OKAY?
SO MAKE SURE YOU UNDERSTAND
THAT 12 REALLY IS THE SAME AS 12 OVER 1.
SO IS THERE SOMETHING I CAN REDUCE?
THE RULE OF THE GAME
AS ALWAYS IS ONE FROM THE TOP, ONE FROM THE BOTTOM,
SO IT DOESN'T HAVE TO BE TOP AND BOTTOM.
IT COULD BE CRISS-CROSSED LIKE THIS.
SO YOU NOTICE THAT I COULD USE 4 TO DIVIDE THEM.
SO 12 DIVIDED BY 4 IS 3 AND 4 DIVIDED BY 4 IS 1.
SO WHAT YOU END UP WITH IS 3 x 3,
SO REMEMBER, IT'S MULTIPLICATION.
SO REALLY IF YOU LIKE, 3 x 3 WHICH IS 9
AND 1 x 1 IS 1,
BUT WE NEVER WRITE IT AS 9 OVER 1.
WE SIMPLIFY FURTHER.
SO ALWAYS SIMPLIFY AS MUCH AS YOU CAN,
SO SIMPLIFY AS MUCH AS YOU CAN.
SO THAT'S THE FINAL ANSWER YOU SHOULD STOP AT.
SO THE NEXT RULE IS WHAT ABOUT DIVIDING FRACTIONS?
NOW DIVIDING FRACTIONS IS WHERE YOU HAVE TO REMEMBER
YOU HAVE TO FIRST CONVERT DIVISION INTO MULTIPLICATION.
BUT IN DOING SO,
YOU'RE GOING TO TAKE THE RECIPROCAL OF THE DIVISOR
WHICH IS THE SECOND FRACTION.
SO CONVERT TO MULTIPLICATION OF RECIPROCAL OF DIVISOR.
SO THE DIVISOR IS THE ONE THAT COMES AFTER THE DIVISION SIGN.
SO WHAT HAPPENS IS THE DIVISION BECOMES MULTIPLICATION
AND THE FRACTION IS BEING FLIPPED OVER,
SO INSTEAD OF C OVER D, IT IS NOW D OVER C.
SO NOW WE'RE BACK INTO MULTIPLICATION
AND, AGAIN, SIMPLIFY IF POSSIBLE.
LET'S TAKE AS AN EXAMPLE,
SO WHAT IF I HAVE THE FRACTION 15 OVER 14
AND I WISH TO DIVIDE THIS BY 5?
SO REMEMBER SOMETHING WE MENTIONED JUST NOW,
5 IS THE SAME AS 5 OVER 1. SO BY DIVIDING 5 AS 5 OVER 1,
I DIDN'T REALLY DO ANYTHING WITH THE DIVISION SIGN,
SO IT STILL STAYS AS DIVISION.
SO REMEMBER, YOU NEED TO CONVERT TO MULTIPLICATION,
KEEP THE FIRST FRACTION,
BUT TURN THE DIVISION INTO MULTIPLICATION
AND FLIP THE SECOND FRACTION, WHICH IS THE DIVISOR.
SO YOU NEED TO TAKE THE RECIPROCAL
SO IT BECOMES 1 OVER 5.
SO YOU'RE GONNA REDUCE IF POSSIBLE,
SO I'M GONNA KEEP REMINDING YOU TO REDUCE IF POSSIBLE.
SO 15 AND 5, DIVIDE 5 BY 5, 5 DIVIDED BY 5 IS 1,
15 DIVIDED BY 5 IS 3, SO THIS IS MULTIPLICATION.
IN MULTIPLICATION, YOU DO NOT NEED COMMON DENOMINATOR,
SO YOU JUST MULTIPLY ACROSS.
3 x 1 IS 3 AND 14 x 1 IS 14. AND THAT'S IT.
SO THIS ESPECIALLY THE CASE IF THESE WERE MIXED NUMBERS.
SO, FOR EXAMPLE, IF I HAVE TWO AND ONE-THIRD
AND I WISH TO MULTIPLY TO THREE AND ONE-QUARTER.
SO YOU CANNOT MULTIPLY MIXED NUMBERS.
YOU NEED TO CONVERT TO FRACTIONS FIRST.
SO HOW DO WE CONVERT TWO AND ONE-THIRD?
THE RULE IS YOU'RE GONNA MULTIPLY THE 2 AND THE 3,
SO 2 x 3.
SO THIS IS HOW YOU HAVE TO THINK.
2 x 3, AND THEN YOU HAVE TO ADD THE 1.
SO 2 x 3 AND ADD 1,
AND YOU DIVIDE BY THE COMMON DENOMINATOR
WHICH IS THE OLD DENOMINATOR, WHICH IS 3.
AND YOU DO THE SAME WITH THE OTHER ONE, SO LET'S DO IT AGAIN.
3 x 4 AND THEN PLUS 1,
BUT YOU KEEP THE DENOMINATOR,
SO IT'S IMPORTANT TO KEEP THE DENOMINATOR.
SO 2 x 3 IS 6, 6 + 1 IS 7 OVER 3 MULTIPLIED BY,
LET'S SEE, 3 x 4 IS 12, 12 + 1 IS 13.
SO 13 OVER 4.
SO THIS IS WHERE YOU'RE LOOKING FOR PAIRS TO MULTIPLY,
SO YOU NOTICE THERE'S NO PAIRS TO LOOK FOR.
7 AND 4, 13 AND 3, 7 AND 3,
SO THERE'S NO PAIR TO MULTIPLY.
SO IF THERE IS NOTHING, YOU CAN MULTIPLY ACROSS.
SO 13 x 7, WHAT IS THIS?
1 AND 20, SO IT'S 91.
SO 91 FOR THE TOP AND 4 x 3 IS 12, AND THAT'S IT.
SO IT'S ALL RIGHT TO LEAVE YOUR ANSWER AS IMPROPER FRACTION.
THERE IS NO NEED TO CONVERT TO MIXED NUMBERS
UNLESS TOLD TO DO SO.
SO UNLESS THE QUESTION SAYS CONVERT TO MIXED NUMBERS,
IF POSSIBLE, OTHERWISE YOU CAN STOP.
OKAY, LET'S TRY ONE WITH DIVISION.
SO WHAT IF THIS WAS, LET'S SAY, 1 AND 5/12 DIVIDED BY,
LET'S SAY, 1 AND 3/7.
SO LET'S TRY THIS AGAIN.
YOU'RE SUPPOSED TO 1 x 12 AND YOU HAVE TO ADD 5,
BUT REMEMBER TO KEEP THE DENOMINATOR,
SO IT'S 12 DIVIDED BY 1 x 7 + 3.
YOU CAN DO ALL THAT IN YOUR HEAD,
BUT I'M JUST WRITING IT OUT.
SO 12 x 1 IS 12, 12 + 5 IS 17,
SO 17 OVER 12 DIVIDED BY 1 x 77, 7 + 3 IS 10,
SO 10 OVER 7.
SO NOW YOU NOTICE THAT ALL THIS WE ARE JUST CONVERTING
FROM THE MIXED NUMBER TO AN IMPROPER FRACTION,
SO WE'RE JUST CONVERTING FROM MIXED NUMBERS TO FRACTIONS.
WE'VE STILL YET TO DO THE DIVISION.
SO REMEMBER THE RULE.
YOU HAVE TO KEEP THE FIRST FRACTION
AND YOU HAVE TO FLIP OVER THE SECOND FRACTION, OKAY.
WHICH IS WHAT I JUST DID.
NOW AGAIN, WE'RE LOOKING FOR PAIRS
AND IT DOESN'T LOOK LIKE THERE IS ANYTHING WE CAN SIMPLIFY,
SO 17 x 7, WHAT IS THIS, 49?
SO IT'S A 4 AND IT'S 11,
SO THIS COMES OUT TO BE 119
AND 12 x 10 IS 120 AND THAT'S IT.
OKAY, NOW THAT YOU'VE SEEN MULTIPLICATION AND DIVISION,
LET'S TALK ABOUT ADDING AND SUBTRACTING FRACTIONS.
SO WHAT YOU NEED TO REMEMBER
IS YOU HAVE TO HAVE SOMETHING IN COMMON.
MUST HAVE COMMON DENOMINATOR.
THEN WHAT YOU WOULD DO IS YOU'LL ADD OR SUBTRACT THE NUMERATOR,
BUT YOU HAVE TO KEEP THE DENOMINATOR.
VERY IMPORTANT.
AND AS ALWAYS, YOU HAVE TO REDUCE IF POSSIBLE.
SO LET'S START WITH A FRACTION ADDITION
THAT IS OF THE SAME DENOMINATOR.
SO LET'S SAY WE HAVE 7 OVER 15 PLUS 6 OVER 15
AND MINUS 3 OVER 15.
SO YOU NOTICE THAT ALL THE DENOMINATORS ARE THE SAME,
SO YOU'RE NOT SUPPOSED TO DO ANYTHING WITH THE DENOMINATOR.
VERY IMPORTANT.
YOU NEED TO KEEP THE DENOMINATOR SO IT STAYS AS 15,
BUT YOU'RE GONNA ADD AND SUBTRACT THE NUMERATORS.
SO YOU'RE LOOKING AT 7, 6 AND 3.
REMEMBER TO TAKE THE SIGNS WITH YOU,
SO YOU'RE LOOKING AT, WHAT'S 7 + 6 - 3?
SO 7 + 6 IS 13 AND 13 - 3 IS 10,
SO THE ANSWER IS 10 OVER 15,
BUT IF YOU STOP THERE, YOU'RE STILL PARTIALLY WRONG
BECAUSE YOU CAN REDUCE.
SO YOU CAN DIVIDE TOP AND BOTTOM BY 5 IN THIS CASE.
SO IF YOU DIVIDE BY 5, IT'S A 2 AND THAT'S A 3,
SO THE ANSWER IS 2 OUT OF 3.
SO THE BIG QUESTION IS WHAT HAPPENS
IF THE DENOMINATORS ARE NOT THE SAME?
SO IF THE DENOMINATORS ARE DIFFERENT,
WE MUST FIND THE LCD FIRST.
SO IF YOU RECALL WHAT'S LCD,
THAT'S REFERRING TO THE LEAST COMMON DENOMINATOR.
SO TAKE, FOR EXAMPLE, WE'RE ADDING TWO-THIRDS WITH,
LET'S SAY, THREE-QUARTERS
AND LET'S SAY MINUS ONE-SIXTH, FOR EXAMPLE.
SO WHAT YOU NEED TO LOOK FOR IS THE LCD
AND THERE IS A FEW WAYS TO DO IT.
ONE WAY IS TO PRIME FACTORIZE YOUR NUMBERS
OR ANOTHER WAY IS TO DO DIVISION.
SO DEPENDING ON WHAT METHOD YOU PREFER,
ONE WAY IS TO THINK OF MULTIPLES OF THREE.
OKAY, SO MULTIPLES OF THREE. WHAT AM I THINKING OF?
THINGS LIKE 3, 6, 9, 12, 15 AND SO ON.
WHAT ARE MULTIPLES OF 4?
MULTIPLES OF 4 ARE LIKE 4, 8, 12 AND SO ON.
WHAT ARE MULTIPLES OF 6?
6, 12, 18, AND SO ON.
SO YOU NOTICE ON THE LIST,
12 IS THE VERY FIRST NUMBER WHERE THEY ALL AGREE ON.
SO THAT TELL YOU THAT THE LCD IS 12.
SO THIS IS A METHOD THAT'S PRETTY GOOD
IF THE NUMBERS ARE NOT TOO BIG.
BUT IF THE NUMBERS ARE BIGGER,
YOU MIGHT WANT TO CONSIDER PRIME FACTORIZATION.
SO NOW THAT WE'VE DECIDED WE WANT THE LCD TO BE 12,
SO THE WAY TO DO IT IS WE'RE GONNA MAKE EVERYBODY 12.
SO YOU HAVE TO GO BACK INTO THE ORIGINAL QUESTION,
THE ORIGINAL FRACTIONS,
AND ASK YOURSELF WHAT MUST YOU MULTIPLY TO 3 TO MAKE IT A 12?
SO 3 x 4 WILL MAKE IT 12.
THAT MEANS I HAVE TO MULTIPLY 4 TO THE TOP AS WELL,
SO 4 x 2 BECOMES 8.
THE NEXT FRACTION DENOMINATOR IS 4.
SO YOU'RE SUPPOSED TO LOOK AT THE DENOMINATOR
AND USE THAT TO DECIDE WHAT TO MULTIPLY.
SO 4 x 3 WILL GIVE ME 12,
SO I HAVE TO DO THAT TO THE TOP AND IT MAKES IT A 9.
OKAY, 6.
6 x WHAT WILL GIVE ME 12?
SO IT'S TIMES 2, SO THAT WILL MAKE IT 1 x 2 WHICH IS 2.
SO WHAT I HAVE IS NOW COMMON DENOMINATOR 12
THAT I KEEP JUST LIKE THE EARLIER EXAMPLE
AND NUMERATORS I'M GONNA ADD, SO 8 + 9 - 2.
SO 8 + 9 IS 17 AND 17 - 2 IS 15.
SO MY ANSWER IS 15 OVER 12,
BUT DON'T FORGET YOU HAVE TO REDUCE.
SO MAKE SURE TO REDUCE.
WHAT CAN DIVIDE 15 AND 12?
SO YOU'RE GONNA DIVIDE BY 3, FOR EXAMPLE,
SO IN YOUR MIND YOU THINK OF 3,
BUT WHAT YOU'RE WRITING IS REALLY THE RESULT,
SO THE ANSWER IS 5 OVER 4, AND YOU'RE DONE.
SO LET ME SHOW YOU ADDITION AND SUBTRACTION IN MIXED NUMBERS.
TAKE, FOR EXAMPLE,
IF YOU HAVE 1 AND 2/3
AND YOU WISH TO ADD 2/4 AND 2/5.
SO THERE IS A FEW METHODS TO DO THIS,
BUT ONE OF THE METHODS IS TO CONVERT TO FRACTIONS FIRST,
WHICH IS THE METHOD WE USE FOR MULTIPLICATION AND DIVISION.
SO YOU CAN USE IT HERE AS WELL.
SO THE METHOD WAS WE'RE SUPPOSED TO MULTIPLY 1 x 3,
SO REMEMBER THE RULE.
1 x 3 + 2, BUT KEEP THE DENOMINATOR.
SO 4 x 5 + 2, BUT KEEP THE DENOMINATOR.
SO LET'S SEE.
3 + 2, SO THERE'S 5 OVER 3 AND, WHAT IS THIS?
20 + 2, SO IT'S 22 OVER 5.
SO REMEMBER WHERE TO LOOK.
YOU'RE LOOKING FOR LCD.
REMEMBER WHAT D STANDS FOR IS DENOMINATOR.
SO WE ARE FOCUSING ON THE DENOMINATOR
AND WE'RE THINKING OF MULTIPLES.
SO IN MY MIND, I'M THINKING IN MULTIPLES OF 3
AND ALSO MULTIPLES OF 5
AND THE VERY FIRST ONE THEY BOTH AGREE ON IS 15.
SO 15 WORKS FOR BOTH NUMBERS,
SO THAT TELLS ME I HAVE TO MAKE YOU A 15
AND YOU A 15 AS WELL.
SO THIS 3 I HAVE TO MULTIPLY BY 5 TO MAKE IT 15,
SO THE TOP, 5 x 5 MAKES IT 25.
THE BOTTOM IS A 5 ON THIS ONE,
SO I'M GONNA x 3 TO THE BOTTOM TO MAKE IT 15,
BUT NOW 22 I HAVE TO MULTIPLY BY 3, WHICH MAKES IT 66.
SO ONCE THE DENOMINATORS ARE THE SAME,
YOU KEEP IT AND THEN YOU'RE GONNA ADD THE NUMERATORS.
SO 25 AND 66, IF YOU ADD THOSE,
COMES TO 91, SO IT'S 91 OVER 15.
NOW BE CAREFUL. WHEN THERE'S BIG NUMBERS LIKE THAT,
IT'S VERY EASY TO SAY, OKAY, IT CANNOT BE DIVIDED.
THERE'S NOTHING TO REDUCE IT.
NOT ENTIRELY TRUE.
MAKE SURE YOU CHECK IT CAREFULLY
BECAUSE SOMETIMES IT'S STILL DIVISIBLE.
BUT IN THIS CASE, 91.
SO THIS IS ONE WAY TO CHECK.
HOW DO I KNOW IT'S DIVISIBLE?
LOOK AT THE SMALLER NUMBER.
DO YOU SEE THAT 15 IS 3 x 5?
SO REALLY THE ONLY TWO NUMBERS
THAT CAN BE DIVIDE BY 15 IS 3 AND 5.
SO I'M ASKING MYSELF IS 91 DIVISIBLE BY 5?
SO THE ANSWER IS NO BECAUSE DIVISIBILITY
TELLS ME THAT THE LAST DIGIT HAS TO BE A 5 OR A 0
FOR YOU TO BE DIVISIBLE BY 5.
IS 91 DIVISIBLE BY 3?
SO DO YOU REMEMBER WHAT'S THE DIVISIBILITY FOR 3?
DIVISIBILITY TEST FOR 3
IS THE SUM OF DIGITS MUST BE DIVISIBLE BY 3.
SO IF YOU ADD 9 AND 1, IT'S 10 AND 10 IS NOT DIVISIBLE,
SO I'M DONE.
OKAY, SO THAT'S ONE WAY TO CHECK.
ALL RIGHT, LET'S TRY FOR SUBTRACTION.
LET'S TRY THIS.
NEGATIVE 3 AND 2/7
AND POSITIVE, LET'S SAY, 4 AND 3/4.
SO IF IT'S NEGATIVE, YOU SHOULD JUST LEAVE THE SIGN ALONE
AND, SAME THING, MULTIPLY 3 AND 7 AND PLUS THE 2
AND DIVIDE BY 7.
SO WHERE I'M PUTTING THE SIGN IS RIGHT IN THE MIDDLE
OF THE FRACTION.
PLUS, SO IT'S 4 x 4 + 3 AND OUT OF 4.
SO 21, I'M NOT LOOKING AT IT AS NEGATIVE 21.
I'M LOOKING AT IT LIKE THIS.
SO IT'S 21 + 2, SO IT'S NEGATIVE 23 OVER 7.
16 + 3, SO THAT'S 19 OVER 4.
SO THE LCD IN THIS CASE IS MULTIPLES OF 7 AND 4.
WHAT WORKS WILL BE 28, SO I'M GONNA MULTIPLY.
MAKE THIS A 28 AND MAKE THIS A 28
AND, IN THIS WHOLE PROCESS, I'M LEAVING THE SIGNS ALONE.
SO I'M GONNA MULTIPLY BY 4 TO THIS,
SO 23 x 4 IS 92,
AND THEN I'M GONNA MULTIPLY 7 TO THIS,
SO IT'S 19 x 7, SO IT'S 63, SO WHAT IS THIS?
133?
OKAY, ONCE THE DENOMINATORS ARE THE SAME,
YOU WRITE IT, KEEP IT AS IT IS.
THE BIGGER NUMBER IS 133, THE SMALLER NUMBER IS 92,
BUT IT'S NEGATIVE.
SINCE THE SIGNS ARE DIFFERENT,
I WILL KEEP THE TOP SIGN, BUT I WILL SUBTRACT.
SO THIS IS WHAT WE TALKED ABOUT IN 1.1.
SO LET'S GO AHEAD AND SUBTRACT.
3 - 2 IS 1 AND 13 - 9 IS 40,
SO THIS IS 41,
AND THE ONLY NUMBERS THAT CAN DIVIDE 28
ARE LIKE 2s AND 7s WHICH BOTH DON'T WORK FOR 41, SO I'M DONE. �