a box of munchkins contains chocolate and glazed donut holes. if Gloria ate 2 chocolate , then 1/11 of the remianing munchkins would be chocolate. if she added 4 glazed to the box, 1/7 of the box would chocolate. how many munchkins are in the original box?
g = glazed holes
c = chocolate holes
c+g-2 = 11(c-2)
c+g+4 = 7c
Solve these to get
c=4
g=20
Eat 2c, leaving 2c/22 total = 1/11
Add 4g, gives 4c+24g = 4/28 choc.
Let's assume the original number of munchkins in the box is represented by N.
Given that Gloria ate 2 chocolate munchkins, the remaining chocolate munchkins would be N - 2.
According to the given information, 1/11 of the remaining munchkins would be chocolate. So, we can write the equation:
(N - 2) / N = 1/11
By cross-multiplying, we get:
11(N - 2) = N
11N - 22 = N
Simplifying the equation, we find:
10N = 22
N = 22/10
N = 2.2 (This means that the original number of munchkins cannot be a decimal.)
Since we cannot have a decimal number of munchkins, we need to check the second condition.
Given that Gloria adds 4 glazed munchkins to the box, the total number of munchkins would be N + 4.
According to the given information, 1/7 of the box would be chocolate. So, we can write the equation:
(N - 2 + 4) / (N + 4) = 1/7
By cross-multiplying, we get:
7(N - 2 + 4) = N + 4
7N - 14 + 28 = N + 4
7N + 14 = N + 4
6N = -10
N = -10/6
N = -1.6667
Again, we have a decimal number of munchkins, which is not possible. So, there is no whole number solution to this problem.
To solve this problem, let's assume the number of munchkins in the original box is "x".
According to the given information, Gloria ate 2 chocolate munchkins. So, the remaining number of munchkins becomes (x - 2).
Now, it is stated that 1/11 of the remaining munchkins are chocolate. This means that (x - 2) * (1/11) munchkins are chocolate. Therefore, the number of chocolate munchkins in this case is (x - 2) * (1/11).
Next, Gloria adds 4 glazed munchkins to the box, making the total number of munchkins become (x - 2 + 4) = (x + 2).
According to the second information given, 1/7 of the box would be chocolate. So, (x + 2) * (1/7) munchkins are chocolate. Therefore, the number of chocolate munchkins in this case is (x + 2) * (1/7).
Now, we have two expressions for the number of chocolate munchkins, which we can equate:
(x - 2) * (1/11) = (x + 2) * (1/7)
To solve this equation for x, let's multiply both sides by 77 (L.C.M of 11 and 7) to get rid of the fractions:
77 * (x - 2) * (1/11) = 77 * (x + 2) * (1/7)
7(x - 2) = 11(x + 2)
Simplifying further:
7x - 14 = 11x + 22
14 + 22 = 11x - 7x
36 = 4x
x = 36 / 4
x = 9
Therefore, the original box contained 9 munchkins.