Rachita enters a shop to buy ice-creams, cookies and pastries. She has to buy atleast 9 units of each. She buys more cookies than ice-creams and more pastries than cookies. She picks up a total of 32 items. How many cookies does she buy? (a) Either 12 or 13 (b) Either 11 or 12 (c) Either 10 or 11 (d) Either 9 or 11 (e) Either 9 or 10

Solution

To solve this, let’s denote:

• \(I\) for ice-creams,
• \(C\) for cookies, and
• \(P\) for pastries.

Given conditions are:

• \(I \geq 9\),
• \(C \geq 9\),
• \(P \geq 9\),
• \(C > I\),
• \(P > C\),
• \(I + C + P = 32\).

Since Rachita buys a minimum of 9 units of each item, the smallest numbers she can buy, respecting the given conditions (\(C > I\) and \(P > C\)), are:

• \(I = 9\),
• \(C > 9\),
• \(P > C\).

Given \(I + C + P = 32\) and \(I = 9\), we have:

• \(9 + C + P = 32\),
• \(C + P = 23\).

To satisfy \(C > I\) and \(P > C\) with the least numbers:

• If \(C = 9\), then \(P\) cannot be equal to \(9\) since \(P > C\), contradicting \(C > I\).
• \(C\) must be more than 9, so the next possible value is \(C = 10\) or higher, and \(P\) must be higher than \(C\).

Let’s examine the possible combinations under these constraints:

• If \(C = 10\), then \(P = 13\) (since \(P > C\) and they must sum to 23), which satisfies all conditions.
• If \(C = 11\), then \(P = 12\) (still satisfies \(P > C\)), which also meets all requirements.

Therefore, based on the constraints and the need for \(C\) to be more than \(I\) and \(P\) to be more than \(C\), Rachita can buy either 10 or 11 cookies to keep the total count to 32 while adhering to all given conditions.

The correct answer is (c) Either 10 or 11.