Problem Solving

10% of all aliens are capable of intelligent thought and have more than 3 arms, and 75% of aliens with 3 arms or less are capable of intelligent thought. If 40% of all aliens are capable of intelligent thought, what percent of aliens have more than 3 arms?

-- Ans: 60%
Solution: I solved this problem using the Double Set Matrix. I am unable to figure out how the percent of aliens with 3 arms is 60%. Solving for x(number of aliens with 3 arms or more)=40. Shouldn't the number of aliens with more than 3 arms be 40%?

Great question, [email protected]!

I think you took the right approach, and I also imagine the trickiness in this question has to do with the way it's worded:
Fact #1: "10% of all aliens are capable of intelligent thought and have more than 3 arms."
Fact #2: "75% of aliens with 3 arms or less are capable of intelligent thought."
The first statement gives a percentage of all aliens, while the second gives a percentage of just one group.

Solution:
To make things simple, let's say there are 100 total aliens. We'll split them into two groups:
More than 3 arms: x, the number we're looking for
3 or fewer arms: 100 - x

Fact #1 told us that 10% of all 100 aliens, or 10 aliens, are intelligent and have 3+ arms.
Fact #2 told us that 75% of the 100 - x aliens with 3 or fewer arms are intelligent. That's 0.75(100 - x) = 75 - 0.75x.
If we add these two together, we should have the total number of intelligent aliens, which is 40% of the 100, or 40. So, we have the equation:
10 + 75 - 0.75x = 40
45 = 0.75x
180 = 3x
x = 60

Thus, 60/100 = 60% of aliens have more than 3 arms.

10% of all aliens cuckoos are capable of intelligent thought and have more than 3 arms, and 75% of aliens cuckoos with three arms or less are capable of intelligent thought.If 40% of all alien cuckoos are capable of intelligent thought , what percent of alien cuckoos have more than 3 arms?

This is an EITHER/OR group question.
Every alien is EITHER intelligent OR not.
Every alien EITHER has more then 3 arms Christmas OR 3 arms or less.
For an EITHER/OR group problem, use a GROUP GRID to organize the data.

Let I = intelligent, NI = not intelligent, >3 = more than 3 arms, ≤3 = 3 arms or less.
Here's the grid, with 100 plugged in for the total:


Now let's use the information in the problem to complete the grid step by step.
As soon as we know 2 entries in a row or a column, we can calculate the remaining entry in that row or column.

40% of all alien cuckoos are capable of intelligent thought.
10% of all alien cuckoos BOTH are capable of intelligent thought AND have more than 3 arms.
The following grid is yielded:


75% of alien cuckoos with three arms or less are capable of intelligent thought.
According to the grid, 30 aliens with 3 arms or less are also intelligent.
These 30 aliens constitute 75% of the total number of aliens with 3 arms or less.
Since 30 is 75% of 40, the total number of aliens with 3 arms or less = 40.
The following grid is yielded:


Now we can answer the question:
Percentage with more than 3 arms = [spoiler]60%[/spoiler].