 # Ex: Inverse Variation Application – Number of Workers and Job Time

– THE TIME, T,
REQUIRED TO DO A JOB VARIES INVERSELY WITH THE NUMBER
OF PEOPLE, P, WORKING ON THE JOB. IF IT TAKES SIX HOURS FOR EIGHT
WORKERS TO COMPLETE A JOB, HOW LONG WOULD IT TAKE
IF THERE WERE NINE WORKERS? SO, SOMETIMES
WHEN WE HAVE AN INVERSE VARIATION APPLICATION LIKE THIS,
IT’S A LITTLE MORE CHALLENGING TO FIND
THE INVERSE VARIATION EQUATION BECAUSE THE IMPORTANT
INFORMATION IS HIDDEN
WITHIN THIS FIRST SENTENCE. WE NEED TO BE ABLE TO READ
THIS FIRST SENTENCE AND REALIZE THE MOST IMPORTANT
INFORMATION IS THAT T VARIES INVERSELY WITH P, WHERE T IS THE TIME
AND P IS THE NUMBER OF PEOPLE. THE GENERAL INVERSE VARIATION
EQUATION IS Y EQUALS K DIVIDED BY X WHERE
K IS THE VARIATION CONSTANT, BUT FOR THIS EQUATION, WE WOULD
Y VARIES INVERSELY WITH X. HERE, WE HAVE T
VARIES INVERSELY WITH P, SO THE INVERSE VARIATION
EQUATION THAT WILL USE HERE IS
T=K DIVIDED BY P. FROM HERE, WE’RE TOLD
IT TAKES 6 HOURS FOR 8 WORKERS TO COMPLETE A JOB. SO, IF T=6, THEN P=8,
AND AGAIN, WHERE T IS TIME, WHICH WE’LL ASSUME IS IN HOURS,
AND P IS THE NUMBER OF PEOPLE. SO, WE CAN USE THIS INFORMATION TO DETERMINE
THE VARIATION CONSTANT AND THEN ANSWER THE QUESTION ABOUT HOW LONG IT WILL TAKE
IF THERE ARE 9 WORKERS. SO, WE’LL SUBSTITUTE 6 FOR T
AND 8 FOR P. SO, TO SOLVE THIS EQUATION
FOR K, WE’D MULTIPLY BOTH SIDES
OF THE EQUATION BY 8, REMEMBER THIS IS 8/1. SO, THIS SIMPLIFIES TO 1 HERE. SO, WE HAVE 8 x 6 OR 48=K
OR K=48. NOTICE WHAT WE DID HERE
IS MULTIPLIED BOTH SIDES OF THIS EQUATION BY P, SO WE COULD HAVE FOUND K BY
FINDING THE PRODUCT OF P AND T, WHICH WE DID HERE
BY SOLVING THIS EQUATION. BUT THERE’S ALSO THE SECOND
INVERSE VARIATION EQUATION DOWN HERE BELOW THAT SAYS
X x Y=K, WHICH WE COULD HAVE USED
USING THE VARIABLES P AND T. EITHER WAY, FOR THIS SITUATION, THE EQUATION RELATES TO TIME AND THE NUMBER OF WORKERS
IS T EQUALS 48 DIVIDED BY P. AND NOW TO ANSWER THE QUESTION, HOW LONG IT WOULD TAKE
IF THERE WERE NINE WORKERS, WE’RE GOING TO SUBSTITUTE 9
FOR P AND THEN DETERMINE T. SO, WE’D HAVE T=48
DIVIDED BY 9. NOTICE HOW THIS QUESTION DOES NOT STATE
HOW TO EXPRESS THE ANSWER, SO WE’LL EXPRESS
THIS A COUPLE OF WAYS. IF WE’RE ALLOWED TO EXPRESS
THIS IN DECIMAL FORM, WE CAN GO AHEAD AND JUST PERFORM
THIS DIVISION. LET’S START WITH THAT. WE’D HAVE 48 DIVIDED BY 9, WHICH WOULD BE 5.3
REPEATING IN DECIMAL FORM. REMEMBER 0.3 REPEATING=1/3, SO IN DECIMAL FORM,
WE COULD EXPRESS THIS AS 5.3 REPEATING
AND THIS WOULD BE HOURS. IF WE’RE ASKED TO ROUND THIS
TO THE NEAREST 10th, WE COULD SAY IT’S APPROXIMATELY
5.3 HOURS, BUT AGAIN, BECAUSE 0.3
IS REPEATING, WE COULD ALSO EXPRESS THIS
AS 5 AND 1/3 HOURS AND THE REASON THIS 