docabuzar wrote:
Each of 435 bags contains at least one of the following three items: raisins, almonds, and peanuts. The number of bags that contain only raisins is 10 times the number of bags that contain only peanuts. The number of bags that contain only almonds is 20 times the number of bags that contain only raisins and peanuts. The number of bags that contain only peanuts is one-fifth the number of bags that contain only almonds. 210 bags contain almonds. How many bags contain only one kind of item?
A. 256
B. 260
C. 316
D. 320
E. 350
If you have no idea how to start setting up a solution for a problem, it's a worthwhile skill to develop to just brute force your way to a solution.
Let's try numbers. "Only peanuts" appears twice in our rules, so let's make something up for that.
Let's try 50. 50 only peanuts.
"The number of bags that contain only peanuts is one-fifth the number of bags that contain only almonds." So, 250 only almonds. Oops, that's too high: "210 bags contain almonds." We need to start with something smaller than 50 only peanuts.
Let's try 25. 25 only peanuts.
"The number of bags that contain only peanuts is one-fifth the number of bags that contain only almonds." So, 125 only almonds.
"The number of bags that contain only raisins is 10 times the number of bags that contain only peanuts." So, 250 only raisins.
"The number of bags that contain only almonds is 20 times the number of bags that contain only raisins and peanuts." So, 6.25 only raisins and peanuts.
We've accounted for 406.25 and there are 435 total, so we're missing 28.75.
We've yet to account for "only peanuts and almonds," "only raisins and almonds," and "all three." Each of those includes almonds. If we add 28.75 to 125, we get 153.75. We need 210. We needed more than 28.75, so we need lower the other things.
Let's try 20. 20 only peanuts.
"The number of bags that contain only peanuts is one-fifth the number of bags that contain only almonds." So, 100 only almonds.
"The number of bags that contain only raisins is 10 times the number of bags that contain only peanuts." So, 200 only raisins.
"The number of bags that contain only almonds is 20 times the number of bags that contain only raisins and peanuts." So, 5 only raisins and peanuts.
We've accounted for 325 and there are 435 total, so we're missing 110.
We've yet to account for "only peanuts and almonds," "only raisins and almonds," and "all three." Each of those includes almonds. If we add 110 to 100, we get 210. We need 210. Yay!
We are asked for the number of bags that contain only one item. 20 only peanuts + 100 only almonds + 200 only raisins = 320.
Answer choice D.