SOLUTION: I know the answer, which is 97, because I found it in the back of the book, but I can't figure out how to solve the problem and I need to show work. Please help me with the steps.

Question 113462This question is from textbook
: I know the answer, which is 97, because I found it in the back of the book, but I can't figure out how to solve the problem and I need to show work. Please help me with the steps. The question is:
Verna decided to sell her collection of paperback books. To Fred, she sold 2 books, and one fifth of what was left. Later to Joan she sold 6 books, and one fifth of what then remained. If she sold more books to Fred than to Joan what was the least possible number of books in her original collection? This question is from textbook

Found 2 solutions by edjones, rajagopalan:
Answer by edjones(8007) (Show Source): You can put this solution on YOUR website!
I know the answer, which is 97, because I found it in the back of the book, but I can't figure out how to solve the problem and I need to show work. Please help me with the steps. The question is:
Verna decided to sell her collection of paperback books. To Fred, she sold 2 books, and one fifth of what was left. Later to Joan she sold 6 books, and one fifth of what then remained. If she sold more books to Fred than to Joan what was the least possible number of books in her original collection?
.
Let b be the amt. she started with.
Sold to Fred: 2+(b-2)/5
Sold to Joan: 6+(b-6-2+((b-2)/5))/5
2+((b-2)/5))>6+(b-6-2-((b-2)/5))/5
10+b-2>30+b-6-2-((b-2)/5)) multiply both sides by 5
50+5b-10>150+5b-30-10-b+2 multiply both sides by 5 again
5b+40>112+4b
b>72
At b=72 Fred and Joan have an equal number of books (16 each)
At b=73 Fred will have more books but he can't have a round number because of the division by 5. Both Joan and Fred can't both have a round number except for every 25 because of the second division by 5.
72+25=97
.
That was super tough!!!
Ed

Answer by rajagopalan(174) (Show Source): You can put this solution on YOUR website!
This a beautifully coined problem.
Solution:
First we find the solution for both Fred and Joan getting equal books.
Then we add one more to Fred logically.
Let the total number of books with Verna = x
Books got by Fred = 2+((x-2)/5) ...first two plus one fifth of (x-2)
Books got by Joan=6+((1/5)x(4/5(x-2)-6)...
equaiting both
1/5(4/5(x-2)-6)+6=((x-2)/5)+2
multiply thro out by 5
(4/5)(x-2)-6+20=x-2
4(x-2)-30+100=5x-10 ...(again x5)
4x-8-30+100=5x-10
100-38=x
72=x
so with 72 books both get equal number of books (14 each) (2 + 70/5)=2+14=16
so for Fred to get one more than Joan you have to add 5 more to Fred ( as denomoiantor is 5 in both cases.
which leads to 2+(F/5)=21
F=95
and total books =95+2=97
Also you can use reduxio ad absurdem mehtod ie from 72 u keep on trying 77, 82 etc till u get a full number for Joan.
Answer 97 books

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