   Chapter 4.3, Problem 14E Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203

Solutions

Chapter
Section Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203
Textbook Problem

Tenure Professor Walt is up for tenure and wishes to submit a portfolio of written student evaluations as evidence of his good teaching. He begins by grouping all the evaluations into four categories: good reviews, bad reviews (a typical one being "GET RID OF WALT! THE MAN CAN'T TEACH!"), mediocre reviews (such as “I suppose he’s OK, given the general quality of teaching at this college”), and reviews left blank. When he tallies up the piles, Walt gets a little worried: There are 280 more bad reviews than good ones and only half as many blank reviews as bad ones. The good reviews and blank reviews together total 170. On an impulse, he decides to even up the piles a little by removing 280 of the bad reviews, and this leaves him with a total of 400 reviews of all types. How many of each category of reviews were there originally?

To determine

To calculate: The number of reviews of each category. The categories are good reviews bad reviews, mediocre reviews and blank reviews. There are 280 more bad reviews than good ones and half as many blank reviews as bad ones: The good reviews and blank reviews together total 170, removing 280 of the bad reviews leaves him with a total of 400 reviews of all types.

Explanation

Given Information:

The categories are good reviews bad reviews and mediocre reviews and blank reviews. There are 280 more bad reviews than good ones and half as many blank reviews as bad ones: The good reviews and blank reviews together total 170, removing 280 of the bad reviews leaves him with a total of 400 reviews of all types.

Formula Used:

Elimination method:

In this method, we first combine the equations in a way such that one of the variables gets eliminated. Then we substitute this obtained value in either of the equation to get the value of the other variable.

Calculation:

Let the number of reviews of each category good, bad, mediocre and blank be x,y,z,w respectively.

As from the statement, there are 280 more bad reviews than good ones.

Thus, we can say that:

yx=280 …… (1)

Also, there are half as many blank reviews as bad ones.

Thus,

y=2w …… (2)

Now the sum of good reviews and blank reviews is 170. Therefore,

x+w=170 …… (3)

Also, he removed 280 of bad reviews which leaves him with 400 reviews of all type.

Thus,

x+y+z+w=400+280x+y+z+w=680 …… (4)

Apply the elimination method to find the solution of the given system of the equations:

Subtract third equation from forth to eliminate variables x and w.

x+y+z+wxw=680170y+z=510z=510y …… (5)

Put equation (2) in (5) to eliminate y.

z=5102w

Now add equations first and third then subtract the second equation from it to eliminate x and y

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