Answer to Problem of the Month for July 2023

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:

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

July 2023 Problem of the Month

Correctly solved by:

July 2023
Problem of the Month