The sum of the prices of a pen, an eraser and a notebook is 100 rupees. The price of a notebook is g

Ezekiel Yoder

Ezekiel Yoder

Answered question

2022-06-16

The sum of the prices of a pen, an eraser and a notebook is 100 rupees. The price of a notebook is greater than the price of two pens. The price of three pens is greater than the price of four erasers and the price of three erasers is greater than the price of a notebook. If all of the prices are in integers, find each of their prices.
Let the prices of a pen, an eraser and a notebook be p, e and n respectively. Then we have the following:
p + e + n = 100 n > 2 p 3 p > 4 e 3 e > n
We have p > 4 3 e and n > 2 p > 8 3 e.
So,
4 3 e + e + 8 3 e < 100 e < 20
Similarly, we get
p 27         and n 56
Setting the values of e and p and with a bit brute forcing, I got e = 19, p = 26 and n = 55 which I think is the only solution.
Is the solution correct?

Answer & Explanation

iceniessyoy

iceniessyoy

Beginner2022-06-17Added 27 answers

p + e + n = 100 n > 2 p 3 p > 4 e 3 e > n
As you found n 56 , p 27
Now we write upper bound of p and e in terms of n.
p < n 2 , e < 3 n 8
So, n 2 + 3 n 8 + n > 100 n > 160 3
So, 54 n 56. Also, 3 e > n e 19 and we already have e < 20.
So
e = 19 is the only solution.
You can also use p > 4 e 3 26 p 27.
p = 27 , n = 54 does not satisfy n > 2 p. So there is only one solution as you wrote.
Leland Morrow

Leland Morrow

Beginner2022-06-18Added 11 answers

The AM-GM inequality and the Cauchy-Schwarz inequality deal with products of quantities, and everything in this problem is linear; so I highly doubt that they will find any application here.

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