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

On November 14, 2019 by Raul Dinwiddie

– 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

MIGHT BE HELPFUL IS, SOMETIMES YOU MAY BE ASKED TO EXPRESS YOUR ANSWER

IN HOURS AND MINUTES. AND SINCE THERE ARE 60 MINUTES

IN ONE HOUR, 1/3 OF AN HOUR

WOULD BE 20 MINUTES SO WE COULD ALSO EXPRESS

THIS AS 5 HOURS 20 MINUTES. SO, THERE ARE SEVERAL WAYS

TO EXPRESS THIS ANSWER BASED UPON WHAT YOUR QUESTION

MIGHT STATE. ONE THING WE CAN MENTION THOUGH

IS THAT, NOTICE WHEN THERE EIGHT PEOPLE,

IT TOOK SIX HOURS SO BY ADDING ONE MORE PERSON, THE JOB WAS COMPLETED

40 MINUTES EARLIER. I HOPE YOU FOUND THIS HELPFUL.

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