Math 20 – Lesson 58
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A PORTLAND COMMUNITY COLLEGE MATHEMATICS TELECOURSE.
A COURSE IN ARITHMETIC REVIEW. PRODUCED AT PORTLAND COMMUNITY COLLEGE. BEING ABLE TO CONVERT UNITS OF COURSE HAS
ALWAYS BEEN THE VERY BREAD AND BUTTER OF SCIENCE AND ENGINEERING. WHAT MOST PEOPLE DON’T REALIZE
IS THAT IT’S ALSO CURRENTLY THE BREAD AND BUTTER OF BUSINESS. BUT AT THE PORT OF THE
CITY WHERE I LIVE, THE LONGSHOREMAN THEMSELVES ARE ALWAYS HAVING TO CONVERT METRIC TONS TO
DRY TONS, KILOLITERS TO GALLONS, ETC. AS WE IMPORT AND EXPORT OUR GOODS, THE VERY LIFE
BLOOD OF MUCH OF OUR COMMUNITY. AND AS LONG AS WE’RE COMMUNICATING WITH COMPANIES AND
COUNTRIES USING DIFFERENT SYSTEMS, IF WE WISH TO TRADE WE WILL HAVE TO TRANSLATE NOT ONLY
LANGUAGE BUT IN UNITS OF MEASURE. AS IT TURNS OUT, TO BE ABLE TO CONVERT UNITS
IS A VERY SIMPLE TASK. AND WE’LL CONCLUDE THIS PORTION OF THE TAPE BY SHOWING YOU HOW
TO CONVERT 60 MILES AN HOUR, SAY, INTO FEET PER MINUTE, WHICH IS THE USUAL WAY THAT AN
AMERICAN PHYSICIST WOULD MEASURE A SPEEDING PARTICLE RATHER THAN THIS. ACTUALLY IN PURE
PHYSICS WE WOULD MEASURE IT IN CENTIMETERS PER SECOND OR METERS PER SECOND. BUT LET’S
SEE WHAT KIND OF TECHNIQUE WE WILL USE OR BE INVOLVED WITH TO DO THIS CONVERSION. NOW, FIRST YOU HAVE TO BE GIVEN A CONVERSION
CHART OR MEMORIZE THE EQUIVALENCY OF VARIOUS UNITS. THEN IN ADDITION TO THIS, YOU MUST
REMEMBER A VERY FEW SIMPLE FACTS. TRY TO MEMORIZE THESE IMMEDIATELY; THEY’RE VERY SIMPLE. FACT
#1, LET’S REVIEW OUR CANCELLATION LAW ON THE MULTIPLICATION OF FRACTIONS. IT SAYS IF YOU’RE
MULTIPLYING FRACTIONS AND TOP AND BOTTOMS ARE MULTIPLICATION OR SINGLE NUMBERS, YOU
MAY DIVIDE ANYTHING YOU WISH INTO A TOP MULTIPLIER. SO I’LL DIVIDE THIS 25 BY 5. PROVIDING I CAN
DIVIDE SOME BOTTOM MULTIPLIER, A SINGLE BOTTOM MULTIPLIER, BY THE SAME AMOUNT. SO I CAN DIVIDE
15 BY 5, OR 35 BY 5. IT’S MY CHOICE. SO I’LL DIVIDE 15 BY 5. THAT PROCESS IS WHERE THE
WORD CANCELLATION COMES FROM. SO REMEMBER CANCELLATION IS REALLY A DIVISION. NOW, NOTICE I COULD DIVIDE THIS 5 BY 5 ONCE
AGAIN. IT GOES IN 1. BUT THEN I HAVE TO DIVIDE A BOTTOM MULTIPLIER BY 5, AND I CAN SEE THAT
I CAN DO IT WITH 35. SO THERE’S ONE MORE CANCELLATION. AND NOW I SEE I CAN DIVIDE THIS 7 AND THIS
14 BY 7, GOES IN ONCE HERE, TWICE HERE. THIS ONE YOU MIGHT NOT SEE SO READILY. 13 GOES
INTO BOTH 26 AND 39. GOES INTO 26 TWICE, 39 THREE TIMES. NOW, THERE’S NOTHING THAT WILL DIVIDE INTO
A TOP MULTIPLIER THAT WILL ALSO DIVIDE INTO A BOTTOM. SO THAT MEANS I CAN MULTIPLY THE
TOPS TO GET THE NEW TOP, THE BOTTOMS TO GET THE NEW BOTTOM, AND MY RESULT IS REDUCED.
SO THE CANCELLATION LAW SAYS, IN EFFECT, IF I AM MULTIPLYING I MAY REDUCE BEFORE I MULTIPLY,
AND I CAN REDUCE OR CANCEL ANYWHERE AS LONG AS IT’S TOP FOR BOTTOM. SO KNOW THE CANCELLATION
LAW. THAT’S GOING TO BE THE KEY TOOL IN THIS PROCESS. THE NEXT IDEA TO KEEP IN MIND IS TO TREAT
A MEASUREMENT AS THOUGH IT WERE A MULTIPLICATION. INSTEAD OF THINKING 3 CENTIMETERS, BEGIN TO
THINK 3 TIMES CENTIMETERS. CERTAINLY, IF I HAD 3 TIMES A VARIABLE, WHICH IS SOME NUMBER,
WE WOULD KNOW THAT THIS IMPLIED PROXIMITY HERE MEANS TIMES. SO THAT’S 3 TIMES SOME NUMBER
CALLED C. WE WISH US TO TREAT THE UNITS OF MEASUREMENT THE SAME WAY, AS THOUGH THEY WERE
TIMES. NOW IF WE TREAT THIS AS TIMES, AND THIS OF COURSE IS TIMES, THE CONDITIONS ARE
LEGAL FOR ME TO CANCEL. SO NOW I CAN DIVIDE 2 INTO THIS 6, 2 INTO THIS 8. BUT NOW I CAN
DIVIDE CENTIMETER INTO CENTIMETER ONCE AND CENTIMETER INTO CENTIMETER ONCE. BECAUSE ANYTHING
DIVIDES INTO ITSELF ONCE. WHICH BRINGS US TO OUR SECOND PRINCIPAL, OR
THIRD, WHICH IS SIMPLY THE READILY KNOWN FACT THAT ANYTHING DIVIDED BY ITSELF IS EQUAL TO
ONE. AN OBVIOUS FACT, BUT IT’S GOING TO BE IMPORTANT. AND IT’S GOING TO ALLOW US TO SAY
ONE IN SOME RATHER STRANGE WAY. FOR INSTANCE, CONSIDER THIS: WE KNOW FROM OUR TABLES THAT
5,280 FEET IS THE SAME MEASUREMENT AS 1 MILE. NOW WATCH THIS: I’M GOING TO CLAIM THAT THIS
FRACTION INDEED IS EQUAL TO THE NUMBER ONE. IT CERTAINLY DOESN’T LOOK LIKE IT, DOES IT?
BUT NOTICE WE KNOW FROM THIS CONVERSION EQUIVALENCY THAT THIS MEASUREMENT AND THIS MEASUREMENT
ARE THE SAME. AND WE KNOW FROM THIS STATEMENT UP HERE ANYTHING DIVIDED BY ITSELF IS 1. SO
I HAVE A MEASUREMENT DIVIDED BY ITS EQUIVALENT, SO IT MUST BE ONE. AND INDEED IT IS. I COULD HAVE ALSO WRITTEN THAT 1 THIS WAY.
THAT’S WHY A COUPLE OF UNITS AGO WE MADE SUCH A BIG ISSUE OUT OF THE FACT OF NOT CONSIDERING
THIS AS 5,280 WITH THE WORD STUCK BEHIND IT. IT IS A NUMBER AND THAT NUMBER IS EQUIVALENT
TO THAT MEASUREMENT NUMBER EVEN THOUGH THEY’RE EQUAL AND THE SAME THING AS HERE. SO AS LONG
AS THE TOP IS EQUIVALENT TO THE BOTTOM, YOU INDEED HAVE A FANCY FORM OF 1. AND OUR FOURTH FACT TO BEAR IN MIND IS THE
OBVIOUS STATEMENT THAT ANY NUMBER TIMES 1 IS STILL THE SAME NUMBER, OR MULTIPLICATION
BY 1 DOES NOT REALLY CHANGE WHAT THE NUMBER REALLY IS. NOW LET’S BEAR THOSE FOUR FACTS
STRONGLY IN MIND. FIRST, THE CANCELLATION LAW. THEN THAT WE WILL CONSIDER A UNIT OF
MEASURE AS A SORT OF MULTIPLICATION, WHICH WILL ALLOW US TO CANCEL UNITS AS WELL AS INTO
NUMBER MULTIPLIERS. THIRD, ANY QUANTITY DIVIDED BY ITS EQUIVALENT IS EQUAL TO ONE. AND FINALLY,
1 TIMES ANY NUMBER WILL NOT CHANGE WHAT IT IS EVEN THOUGH IT MIGHT LOOK DIFFERENT. THOSE
FOUR FACTS, AND ONLY THOSE FOUR, ARE THE TOOLS FOR CONVERSION OF UNITS. LET’S SAY FOR SOME REASON YOU NEEDED TO KNOW
HOW MANY CENTIMETERS IS EQUIVALENT TO 48 INCHES. NOW THE FIRST PART OF OUR TASK IS TO FIND
IN A CONVERSION TABLE SOME RELATIONSHIP, SOME EQUIVALENT RELATIONSHIP, BETWEEN INCHES AND
CENTIMETERS. LOOKING THAT UP WE MIGHT FIND THIS. THERE ARE SEVERAL OTHERS THAT WOULD
ALSO WORK. THEN WE SIMPLY START WITH WHAT WE HAVE, AND I’LL WRITE THAT AS A FRACTION.
48 TIMES INCHES OVER 1, THAT’S A FRACTION. NOW I THINK THIS WAY, IF I WERE MULTIPLYING
BY ANOTHER FRACTION AND BECAUSE OF THE CANCELLATION LAW IF I HAD AN INCH ON THE BOTTOM, I COULD
CANCEL THAT INCH OFF. SO I CONCENTRATE FIRST NOT ON WHAT I WANT BUT WHAT I WANT TO GET
RID OF. SO IN MY CONVERSION EQUIVALENCY, I NOTICE THAT ONE INCH IS EQUIVALENT TO 2.54
CENTIMETERS. SO IF I WRITE 1 ON TOP OF THE OTHER, I REALLY HAVE A FORM OF 1. AND IF I’M
MULTIPLYING BY 1, I DON’T CHANGE ANYTHING. BUT NOW BECAUSE I’M MULTIPLYING, I CAN CANCEL
THE INCHES OFF, AND WHAT I HAVE LEFT IS CENTIMETERS. AND IT SAYS THAT I HAVE 48 TIMES 2.54 CENTIMETERS.
SO IT’S NOT ONLY TELLING ME THAT I HAVE CENTIMETERS, IT’S TELLING ME WHAT TO DO WITH THOSE NUMERICAL
COEFFICIENTS OR MULTIPLIERS. AND MULTIPLYING, AND I HAVE MY ANSWER. THE 48 INCHES IS 121.9
CENTIMETERS. REALLY THAT’S THE WHOLE SHOW RIGHT THERE. SO LET’S BACK UP AND SEE WHAT WE DID. FIRST,
WE CONCENTRATE WHAT WE WANTED TO GET RID OF. WE TRIED TO FIND AN EQUIVALENCY IN A CHART
THAT GAVE ME WHAT I WANTED TO GET RID OF AND A RELATIONSHIP WITH WHAT I WANTED. OKAY, I
PUT INSIDE OF THE EQUATION WITH THE THING THAT I WANT TO GET RID OF ON THE OPPOSITE
SIDE OF THE EQUATION SIGN SO THAT IT’LL CANCEL. I PUT THE OTHER SIDE OF THE EQUIVALENCY ON
THE OTHER SIDE, TOP IN THIS CASE, SO THAT I WOULD KNOW THAT THIS IS EQUAL TO THIS. AND
THAT’S A VERY IMPORTANT POINT. I CANNOT DO ANY OF THIS IF EVER THE TOP IS NOT EQUAL TO
THE BOTTOM. THEN I CANCEL WHAT I WANT TO GET RID OF AND THAT BRINGS IN WHAT I WANT. SO
AGAIN YOU CONCENTRATE ON WHAT YOU WANT TO GET RID OF AND EQUIVALENCIES. THEN, IN EFFECT,
YOU’RE REALLY MULTIPLYING BY ONE. LET’S TRY THAT SAME APPROACH NOW WITH A LONGER
PROBLEM. THEN AFTER THAT WE’RE GOING TO COME BACK TO THIS SAME PROBLEM AND DO IT FROM A
SLIGHTLY DIFFERENT APPROACH. NOW LET’S DO THIS ONE. REALLY THIS IS NOT A LONGER PROBLEM,
IT’S TWO PROBLEMS. WE’RE GOING TO WANT TO CONVERT MILES, WHICH IS LENGTH, TO FEET, WHICH
IS LENGTH, AND THEN HOURS, WHICH IS TIME, TO MINUTES, WHICH IS TIME. REMEMBER OUR STRATEGY.
WE’RE GOING TO CONCENTRATE ONE BY ONE ON THE UNIT OF MEASURE WHICH YOU WISH TO GET RID
OF. AND THINK, I WANT THIS SITUATION SET UP TO BE SUCH THAT THAT THING WILL CANCEL OUT.
BEAR THAT IN MIND AND EVERYTHING STAYS CLEAR AND SMOOTH. LET’S CONCENTRATE ON MILES FIRST AND LEAVE
THE HOURS ALONE. THIS STRATEGY SAYS I WANT SOMETHING WITH MILES ON THE BOTTOM. HOPEFULLY
THAT ALSO HAS FEET SO THAT I GO FROM MILES TO FEET IN ONE MOVE, BUT IT’S MILES THAT I
CONCENTRATE ON. SO CHECKING MY EQUIVALENCY CHART I SEE THAT 5,280 FEET IS ONE MILE, SO
THE SIDE THAT HAS WHAT I WANT TO GET RID OF I WILL PUT ON THE BOTTOM, SO THAT BEING ON
THE BOTTOM IT WILL CANCEL THE THING I WANT. SO THERE IT IS. THE ONE MILE IS ON THE BOTTOM,
AND I CAN SEE THAT WHAT I WANTED TO GET RID OF IS GOING TO GO. BUT NOW I NEED AN EQUAL
AMOUNT, ALWAYS AN EQUAL AMOUNT, ON TOP. SO I’LL PUT THE 5,280 FEET ON TOP. SO I DID CANCEL
OUT MY MILES AND I DID END UP WITH FEET AND THAT’S PART OF WHAT I WANTED. I WANTED FEET;
I HAVE FEET. SO NOW WE’LL CONCENTRATE ON TRYING TO CONVERT
THE HOURS TO MINUTES. BUT REMEMBER YOU CONCENTRATE ON WHAT YOU KNOW ABOUT WHAT YOU WANT TO GET
RID OF. SO WE ASK WHAT DO WE KNOW ABOUT HOURS? WELL, WITHOUT EVEN LOOKING IT UP WE KNOW THAT
1 HOUR IS 60 MINUTES. BUT NOTICE THE THING I WANTED TO GET RID OF, HOURS, IS ON THE BOTTOM.
SO I MUST PUT SOMETHING WITH HOURS ON TOP AND THEN SOMETHING THAT’S EQUAL TO IT ON THE
BOTTOM. NOW THE TOP IS EQUAL TO THE BOTTOM. IF THE TOP IS NOT EQUAL TO THE BOTTOM, I’M
ON THE WRONG TRACK. BUT NOW NOTICE THE HOUR ON TOP CANCELS THE HOUR ON THE BOTTOM, AND
I’M LEFT WITH MINUTES. SO I HAVE A UNIT OF MEASURE WHICH IS MEASURED IN FEET PER MINUTE.
FEET PER MINUTE. AND THAT’S EXACTLY WHAT I WANT. NOW AS FAR AS ARITHMETIC IS CONCERNED,
I SIMPLY TEND TO ANY REMAINING CANCELLATION. SO SIMPLY READING WHAT I HAVE LEFT. I HAVE
5,280 FEET, TIMES FEET, OVER 1 MINUTE. SO I HAVE THAT 60 MILES PER HOUR IS THE SAME
THING AS 5,280 FEET PER MINUTE. NOW LET’S SAY OUR PROBLEM SUDDENLY CHANGED
ON US. WHAT IF SOMEONE CHANGES THEIR MIND AND SAYS, I DON’T WANT TO KNOW HOW MANY FEET
PER MINUTE IT IS, I WANT TO KNOW HOW MANY FEET PER SECOND THAT IS. WELL, WE SIMPLY MOVE
ONE MORE STEP. NOW WE’RE SIMPLY SAYING, I WANT TO GET RID OF MINUTES, AND HOPEFULLY
PULL IN SECONDS. AND THEN WE RECALL THAT 60 SECONDS IS 1 MINUTE. SO WE THINK IT’S A MINUTE
I WANT TO GET RID OF. IT’S ON THE BOTTOM; I NEED A MINUTE ON TOP. THEREFORE I’LL PUT
THIS MINUTE SIDE OF THE EQUIVALENCY ON TOP, AND AN EQUAL EQUIVALENCY ON THE BOTTOM. NOW
MINUTE CANCELS WITH MINUTE, AND I’M LEFT WITH FEET PER SECOND. SIXTY GOES INTO HERE. ONE
GOES INTO HERE 88. SO I HAVE 88 TIMES FEET ON TOP. AND 1 TIMES SECOND ON THE BOTTOM.
SO WE HAVE THAT 60 MILES PER HOUR IS THE SAME RATE AS 88 FEET PER SECOND. REMEMBER OUR KEY IDEA: CONCENTRATE ONE BY
ONE UPON THE UNIT OF MEASURE WHICH YOU WISH TO GET RID OF AND THINK, I WANT THE SITUATION
TO BE SUCH THAT IT WILL CANCEL OUT. SO IN THIS CASE I WANTED MINUTE ON TOP TO CANCEL
MINUTE ON THE BOTTOM, BUT ALWAYS REMEMBER THE SITUATION THAT MAKES THAT WORK IS THE
TOP MUST BE EQUIVALENT TO THE BOTTOM, WHICH WE GET FROM TABLES. NOW LET’S ATTACK A SIMPLER PROBLEM BY YET
ANOTHER METHOD. LET’S GO BACK TO OUR FIRST PROBLEM AND RECALL OUR WORK WITH PROPORTIONS.
RECALL THAT WE HAVE A SITUATION HERE, WE CALL THAT SITUATION ONE. WE HAVE 48 INCHES IS EQUIVALENT
TO AN UNKNOWN AMOUNT OF CENTIMETERS. THE TABLE, IN EFFECT, GIVES US A SECOND SITUATION AGAINST
WHICH WE CAN COMPARE IT. SO REMEMBER WE CAN WRITE THIS IN ANY ORDER WE WANT, INCH TO CENTIMETER
OR CENTIMETER INCHES, AS LONG AS THE OTHER SITUATION IS WRITTEN IN THE SAME ORDER. SO
I WANT INCHES HERE. SO FROM THE TABLE I HAVE 1 INCH. I WANT CENTIMETERS HERE AND FROM OUR
TABLE THAT 1 INCH IS 2.54 CENTIMETERS. NOW WE HAVE INCH, INCH, CENTIMETER, CENTIMETER.
ONE OF THEM IS UNKNOWN. HERE’S OUR PROPORTION. TOP TIMES BOTTOM EQUALS BOTTOM TIMES TOP.
AND WE GET THIS. AND DOING OUR MULTIPLICATION WE GET THE SAME ANSWER WE HAD BEFORE: 121.9
CENTIMETERS. NOW WHEN YOU HAVE A PROBLEM WHICH YOU CAN
GO FROM ONE TO THE OTHER DIRECTLY, THE PROPORTIONS SOMETIMES IS A BIT FASTER. HOWEVER, WHEN YOU
HAVE MANY STEPS TO TAKE, THIS MULTIPLICATION PROPERTY OF ONE IS THE TECHNIQUE WE WISH TO
USE. AND WE, ONE BY ONE, KEEP GETTING RID OF WHAT WE DON’T WANT UNTIL WE END UP WITH
WHAT WE DO WANT. LET’S NOW SHOW THAT WE CAN COMBINE THIS LESSON
WITH THE LAST LESSON. IN THIS CASE, HOPEFULLY YOU’LL RECOGNIZE THAT I’M GOING FROM A METRIC
UNIT, HECTOGRAM, INTO ANOTHER METRIC UNIT, DECIGRAMS. WE RECOGNIZE THAT THE LAST LETTER
TELLS ME WHAT KIND OF UNIT I’M IN, GRAM, THAT’S THE WEIGHT UNIT. THIS IS TELLING ME HOW MANY
I WANT INTO ANOTHER WAY OF MEASURING THAT SAME WEIGHT UNIT. SO RATHER THAN USING THE
TECHNIQUE WE DID ON THE LAST LESSON, WHICH IS A VERY GOOD ONE, WE COULD USE THE TECHNIQUE
OF THIS LESSON. WE RECOGNIZE THAT HECTO, ONE HECTOGRAM. BUT HECTO MEANS A HUNDRED, SO HECTOGRAM
MEANS 100 GRAMS. SO LAST TIME WE SIMPLY REPLACED THE H BY 100,
AND WE GET THE 100 GRAMS. BUT WE ALSO WANT TO GET INTO DECIGRAMS. AND THIS SAYS THAT
ONE DECIGRAM, BUT DECI MEANS A 10TH, SO THAT MEANS 1/10TH OF A GRAM. NOW USING THE TECHNIQUE
OF THIS PARTICULAR LESSON, WE START WITH THE UNITS WE WANT TO MOVE OUT OF, WRITE IT AS
A FRACTION IF IT ISN’T ALREADY. WE’RE GOING TO MULTIPLY. WHAT WE MULTIPLY BY MUST BE LOGICALLY
EQUIVALENT TO 1 OR I CAN’T DO IT. WE CONCENTRATE ON WHAT I WANT TO GET RID OF, WHICH IS HG.
SO WE FIND ANYTHING WITH AN HG AND PUT IT ON THE BOTTOM, BECAUSE WE CAN ONLY CANCEL
TOP WITH BOTTOM OR BOTTOM WITH TOP. SO HERE IS AN HG, ONE OF THEM COULD HAVE BEEN ANYTHING.
BUT NOW WE MUST PUT AN EQUAL AMOUNT ON TOP. AND WE ASK ONLY ONE QUESTION AT THE START:
DID I GET RID OF WHAT I WANTED TO GET RID OF? IN THIS CASE, HECTOGRAM. THE ANSWER’S
YES. IS THE TOP ACTUALLY EQUAL TO THE BOTTOM? THE ANSWER’S YES. THEN THAT TELLS ME I’M ON
THE RIGHT TRACK. THEN I ASKED, DID I GET WHAT I WANTED? I GOT GRAMS BUT I WANTED DECIGRAMS.
IF I DON’T HAVE WHAT I WANT, THEN THAT DOESN’T MEAN THAT I’M WORKING WRONGLY, IT SIMPLY MEANS
I’M NOT DONE. I HAVE ADDITIONAL STEPS. SO NOW I CONCENTRATE ON GETTING RID OF GRAMS.
SO I FIND ANYTHING WITH GRAM IN IT, G, TO PUT IT ON THE BOTTOM TO CANCEL THAT G ON TOP.
HOPEFULLY I FIND SOMETHING THAT ALSO HAS WHAT I WANT. SO IN THIS CASE HERE’S A GRAM AND
HERE’S A DG OR DECIGRAM. SO I PUT THIS AMOUNT ON THE BOTTOM. WHY? BECAUSE IT GETS RID OF
THIS ONE I WANT TO GET RID OF. AND THEN I PUT AN EQUIVALENT AMOUNT ON TOP TO GUARANTEE
THAT THIS WHOLE MULTIPLIER IS STILL LOGICALLY EQUAL TO ONE. NOW AGAIN, THE TOP WAS EQUAL
TO THE BOTTOM SO I AM MULTIPLYING BY 1. I DID GET RID OF WHAT I WANTED TO GET RID OF,
THAT TELLS ME I’M ON THE RIGHT TRACK. AND THE FACT THAT I ENDED WITH WHAT I WANTED TO
END WITH TELLS ME THAT IN FACT I’M DONE. IMPORTANT FACT: DO YOU RECOGNIZE THAT IT’S
ATTENTION TO THE UNITS THAT’S TELLING ME WHAT TO DO AND HOW TO DO IT AND NOT THE NUMBERS?
THE NUMBER PART IS THAT WHICH WE TAKE CARE OF SORT OF AFTER THE FACT, AFTER OUR UNITS
HAVE ALREADY DIRECTED MY TRAFFIC FLOW FOR WORKING THE PROBLEMS. SO THIS MATTER OF UNITS,
AS IT TURNS OUT, IS THE MOST IMPORTANT PART OF THE MEASUREMENT NUMBER. MANY OF US IN OUR
EARLIER SCHOOLING WILL SORT OF WORK WITH THE NUMBERS AND JUST STICK THIS ON WHEN WE’RE
ALL DONE TO SORT OF PLEASE THE TEXTBOOK ANSWER OR THE TEACHER. AS IT TURNS OUT, THESE UNITS
ARE THE MOST IMPORTANT PART OF THE WORK. NOW AFTER THE FACT THAT WE KNOW WE HAVE WORKED
PROPERLY, WE KNOW THAT WE’VE GOTTEN RID OF WHAT WE WANTED TO GET RID OF. WE KNOW THAT
WE END WITH WHAT WE WANT. THEN AND ONLY THEN DO WE BACKTRACK AND TAKE CARE OF THE ARITHMETIC.
IN THIS CASE, SINCE WE’RE MULTIPLYING BY POWERS OF 10S AND DIVIDING BY POWERS OF 10S, WE UTILIZE
WHAT WE LEARNED IN OUR CHAPTER ON DECIMALS AND SIMPLY MOVE THE DECIMAL POINT. MULTIPLYING
BY A POWER OF 10 SAYS TO MAKE BIGGER BY TWO PLACES. AND HERE WE CAN EITHER DIVIDE BY .01
OR I CAN THINK THIS WAY, IF I WERE TO MULTIPLY THIS BY 10, IT WOULD BECOME 1, BUT THEN I
WOULD HAVE TO MULTIPLY THE TOP BY 10 WHICH NOW SAYS TO MAKE BIGGER BY ONE MORE PLACE,
WHICH IS HERE. SO WE END WITH 5 DECIGRAMS. SO .0052 OF A HECTOGRAM IS 5 DECIGRAM. LET’S DO ONE OF THESE, BECAUSE ONCE YOU SEE
THE PATTERN IT’S A REMARKABLY SIMPLE PROCESS. DO YOU RECOGNIZE THIS AS BEING METRIC AND
THIS AS BEING ENGLISH? FREQUENTLY THE ENGLISH ABBREVIATIONS WILL END WITH A PERIOD. A METRIC
SYSTEM NEVER WILL. IN THE METRIC SYSTEM, IT’S THE LAST SINGLE LETTER THAT TELLS ME WHAT
KIND A UNIT I HAVE, IN THIS CASE, A METER. AND THE PREFIX DA TELLS ME HOW LARGE OR HOW
SMALL OF UNITS OF METER I WANT. WE RECALL, HOPEFULLY, FROM MEMORIZATION NOW THAT DA IS
10 METERS. SO LET’S MAKE A NOTE OF THAT 1 DA IS 10 MS. AND IF WE WISH TO GO INTO INCHES,
WE NOTICE THAT ANOTHER TABLE SAYS A 2.54 CM, CENTIMETER, IS 1 INCH. SO THIS WILL LET ME
GO FROM DA TO M, BUT I DON’T WANT M, I WANT INCHES. WELL, THIS WILL LET ME GO FROM CM TO INCH,
BUT I NEED SOMEHOW TO GET FROM METER TO CENTIMETER. WELL, HERE WE AGAIN HAD MEMORIZED THAT C MEANS
A HUNDRED, SO ONE CENTIMETER IS ALSO A HUNDREDTH OF A METER. SO WE GET A PLAN OF OPERATIONS
HERE. WE CAN GO FROM DA TO M. WE CAN GO FROM M TO CM. AND NOW WE CAN GO FROM CM TO INCH
AND WE’RE DONE. BUT WE DON’T REALLY HAVE TO FRET THAT MUCH
ABOUT IT, BECAUSE BY USING THE STRATEGY WE HAD JUST SET UP, WE HAD NOTED THAT WE’RE ALWAYS
MULTIPLYING BY 1, AND WE ALWAYS CONCENTRATE ON WHAT I WANT TO GET RID OF. SO I WANT TO
GET RID OF THIS SO I NEED SOME OF THIS ON THE BOTTOM. THIS WILL DO IT FOR ME. I HAVE
A, SHOULD BE DAM. SHOULDN’T IT? I HAVE ONE DAM, DECAMETER, IS EQUAL TO 10 METERS. I DID
GET RID OF WHAT I WANTED TO GET RID OF, BUT I DIDN’T END WITH WHAT I WANTED. SO NOW I
CONCENTRATE ON GETTING RID OF M. SO HERE IS AN M. I PUT IT NOW ON THE BOTTOM SO THAT ITS
M WILL CANCEL THE M ON TOP. BUT THEN I MUST PUT AN EQUAL AMOUNT ON TOP TO GUARANTEE THAT
I AM MULTIPLYING BY 1. SOMETHING OVER ITS EQUIVALENT. BUT I STILL DIDN’T END WITH WHAT I WANTED
TO END. SO NOW I CONCENTRATE ON GETTING RID OF THIS. AND I ALSO NOTE THAT I WANT IT ON
THE BOTTOM, SO IT’LL CANCEL THAT. SO HERE IS SOMETHING WITH CENTIMETERS. I’LL PUT IT
ON THE BOTTOM AND AN EQUAL AMOUNT ON TOP. NOW I DID GET RID OF WHAT I WANTED TO GET
RID OF. AND NOW, IN THIS CASE, I ENDED WITH A UNIT I WANTED TO. SO SINCE ALL OF THESE
ARE LOGICALLY EQUAL TO 1, I’M MULTIPLYING BY 1 WHICH TELLS ME NOTHING’S CHANGED. SO
ALL THAT REMAINS FOR ME IS TO DO THE ARITHMETIC AFTER THE FACT. SO I’M GOING TO MULTIPLY THIS
BY 10, DIVIDE BY A 100, THEN DIVIDE AGAIN BY 2.54, AND THAT WILL CONVERT THIS TO INCHES.
AND WE CAN DO THE MULTIPLICATIONS AND DIVISIONS IF WE WISH ON A CALCULATOR. AND DOING SO,
I GOT AN UNENDING DECIMAL, WHICH TOLD ME I NEEDED TO ROUND OFF AT SOME DESIGNATED POINT. ONCE YOU SEE THE STRATEGY, YOU CAN REALIZE
I CAN TRANSLATE FROM ANYTHING I WISH TO ANYTHING ELSE I WISH BY REPEATED MULTIPLICATIONS OF
1 AND BY FACTS WHICH I GET OUT OF CONVERSION TABLES. IT’S REALLY JUST THAT SIMPLE. THIS
IS YOUR MATH HOST BOB FINNELL. GOOD LUCK.

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