There are purple, pink and white cups in Nancy's storeroom. The number of purple cups is 7 times the number of white cups, and there are 20 pink cups. Knowing that the purple cups make up 2/3 of all cups, find the number of the white cups in the storeroom
Answers (3)
p = Number of purple cups
k = Number of pink cups
w = Number of white cups
Given are the following conditions:
k = 20
p = 7 * w
p = (2/3) * (p + k + w)
We are looking for w, (and just for fun calculate p and the total, too)
Solution:
p:= (2/3) * (p + k + w) = 7 w | times 3; Insert known conditions
2 (7 w + 20 + w) = 21w | expand
14w + 20 + 2w = 21w
16w + 20 = 21w
20 = 5w
w = 4
Now we know k = 20 and w = 4. So, what's the number of p?
p = (2/3) * (p + k + w) | insert values of w and k
p = (2/3) * (p + 24) | divide by (2/3) or, which is the same, multiply by (3/2)
(3/2)p = p + 24 | minus p [that would be here (2/2)p]
(1/2)p = 24 | divide by (1/2) or, which is the same, multiply by 2
p = 48
Thus the number of all cups n in Nancy's storeroom equals:
n = p + k + w = 48 + 20 + 4 = 72.
Ooops! I found a little nasty error in my solution. Well, I'll do it once again:
Solution:
p:= (2/3) * (p + k + w) = 7 w | times 3; Insert known conditions
2 (7 w + 20 + w) = 21w | expand
14w + 40 + 2w = 21w
16w + 40 = 21w
40 = 5w
w = 8
Now we know k = 20 and w = 8. So, what's the number of p?
p = (2/3) * (p + k + w) | insert values of w and k
p = (2/3) * (p + 28) | divide by (2/3) or, which is the same, multiply by (3/2)
(3/2)p = p + 28 | minus p [that would be here (2/2)p]
(1/2)p = 28 | divide by (1/2) or, which is the same, multiply by 2
p = 56
Thus the number of all cups n in Nancy's storeroom equals:
n = p + k + w = 56 + 20 + 8 = 86.
So, now it's correct, I Hope ;-)