Buying Mousse Cake Slices
One day, I had a number of dollars in my wallet. I went into a cake shop and saw this special limited edition mousse cake. I went to an employee in the shop and had a conversation with him. The following conversation ensued:
Me : Hello there! May I ask how much that mousse cake is?
Employee : Oh, it's ...... dollars per slice.
I checked my wallet to see how much money I had.
Me : Oops, I can only buy 4 of them.
Employee : That's a pity! The mousse cake is really good. Fine, I lower the price by 2 dollars just for you!
Me : Wow thanks! Now I will be able to buy at most 5 of them.
Employee : I think you should buy more. I lower the price again by 3 dollars just for you, okay?
Me : You're so nice! I will be able to buy at most 6 of them!
Employee : Just nice! Our mousse cake is always cut into 6 slices. I will prepare the mousse cake for you!
Note: The "......" part is blanked out for you to solve this problem.
If the amount of money I had in my wallet and the amount per slice of the mousse cake are both positive integers, find the smallest amount that I had in my wallet.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
Let X be the amount I have in my wallet. Let Y be the price per slice.
So 4Y "< or =" X < 5Y,
5(Y - 2) "< or = X" < 6(Y - 2), and
6(Y - 5) "< or =" X < 7(Y - 5).
We want 5Y to be more than 5(Y - 2) and 6(Y - 5), 6(Y - 2) to be more than 6(Y - 5) and 4Y, and 7(Y - 5) to be more than 4Y and 5(Y -2).
5Y > 5Y - 10 is definitely correct. 5Y > 6Y - 30, 30 > Y.
6Y - 12 > 6Y - 30 is definitely correct too. 6Y - 12 > 4Y, 2Y > 12, Y > 6.
7Y - 35 > 4Y, 3Y > 35, Y > "11.6666666.....". 7Y - 35 > 5Y - 10, 2Y > 25, Y > 12.5
So let's join up all the inequalities related to Y:
12.5 < Y < 30
The least possible positive integer for Y is 13, so let's go back to the linear equations:
52 "< or =" X < 65,
55 "< or =" X < 66, and
48 "< or =" X < 56.
The only possible positive integer for X is 55.