Norma has recently purchased a science book, a history book and an atlas from her neighborhood bookstore for a total of $100.   It is known that:

(I) The science book costs more than twice the history book.

(II) Three copies of the history book cost more than four copies of the atlas.

(III) Three copies of the atlas cost more than the science book.

All prices are a whole numbers of dollars.   How much does each item cost?



Solution to the Problem:


The price of the science book is $55, the price of the history book is $26 and the price of the atlas is $19.

[EXPLANATION]:
Let the respective costs of the science book, history book and the atlas be $S, $H, and, $A.
Then, we must have:
S>2H,
3H>4A,
3A>S.

Now we get
S>2H>(8/3)A
H>(4/3)A>(4/9)S.
A>S/3>2H/3.
Substitute these in S+H+A=100, we get

S+S/2+(3/8)S>100 => S>160/3=53.33.
S+(4/9)S+S/3<100 => S<56.25 => S={54,55,56}

H+(3/4)H+(9/4)H>100 => H>25
H+2H+2H/3 <100 => 11H/3<100 =>H<300/11 => H<27.27.
H={26,27}

A+3A+(3/2)A>100 => A>200/11 => A>18.18..
A+(8/3)A+(4/3)A<100 =>A<20 => A=19

But S+H+A=100 => S+H=81. Now Consider the values of S,H
S=54, H=27 is not possible as S>2H
Hence S=55,H=26 is the only solution for S+H=81.
Hence {S,H,A}={55,26,19}

Click here for Carlos de Armas' excellent graphical solution


Correctly solved by:

1. Davit Banana Istanbul, Turkey
2. Elizabeth Pleacher Bastrop, Texas
3. Colin (Yowie) Bowey Beechworth, Victoria, Australia
4. Sudhir Bavdekar Mumbai, India
5. Carlos de Armas Barcelona City, Spain
6. Ivy Joseph Pune, Maharashtra, India
7. Seth Cohen Concord, New Hampshire
8. Rob Miles Northbrook, Illinois