Problem of the Month for February 2023

An International Affairs book and a Crime Fiction novel are available from a renowned wholesaler.
Each costs an integer number of dollars.
The cost of 620 copies of the Crime Fiction novel is greater than that of 340 copies of the International Affairs book but less than 341 copies of the book.

Will $700 be enough for a retailer to purchase 11 copies of the International Affairs book and 5 copies of the Crime Fiction novel?
Provide valid reasoning for your answer.

Solution to the Problem:
No. $700 will not be enough for the retailer to purchase 11 copies of the International Affairs book and 5 copies of the Crime Fiction novel.

Explanation:
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Let the respective cost of one copy of the International Affairs book and one copy of the Crime Fiction novel be n dollars and c dollars.

Then by the given conditions we must have:
340n < 620c < 341n ............(i)

Now, from (i), we have:
17n < 31c, and 20c < 11n

Therefore, we must have :
17n+1 ≤ 31c .............(ii)
20c ≤ 11n -1 .............(iii)

Multiplying (ii) by 20 and (iii) by 31, we have:
340n+20 ≤ 620c ≤ 341n - 31
or, n≥ 51

Then, in terms of (ii), we must have:

31c ≥ 17n+1 ≥ 1751 +1 = 868
or, c ≥ 28

Therefore, we have:
11n+5c ≥ 1151 + 5*28 = 701

Accordingly, the retailer would require at least $701 to buy the requisite number of copies of the two items.

Consequently, $700 will not be enough for the wholesaler to purchase 11 copies of the International Affairs book and 5 copies of the Crime Fiction novel.

Here is Ritwik Chaudhuri's solution:

Correctly solved by:

1. Colin (Yowie) Bowey	Beechworth, Victoria, Australia
2. Davit Banana	Istanbul, Turkey
3. Dr. Hari Kishan	D.N. College, Meerut, Uttar Pradesh, India
4. Ritwik Chaudhuri	Santiniketan, West Bengal, India
5. Eloise A. Michael	Greenfield, Massachussetts (Hinsdale High School)
6. Ivy Joseph	Pune, Maharashtra, India