Question

Sets A and B, shown in the Venn diagram, are such that the total number of elements in set A is twice the total number of elements in set B. Altogether, there are 3011 elements in the union of A and B, and their intersection has 1000 elements. What is the total number of elements in set A?

Answer 1

Hi there,
Let's solve your problem step by step. First off, we need to assign variables to each set. Here's how you do that:

Let a represent set A and let b represent set B.

Now that we have that down, we can move on. Our next step is to translate our given information to numbers. We are given that set A has twice the total number of elements than in set B. This is what we get after the translation:


`a=2b`

We are also given that there are 1000 elements in the two sets' intersection. Hence, we get:

`a-1000` and
`b-1000`
The total number of elements combined in set A and set B can be represented as:

`(a-1000)+(b-1000)+1000`
The question gives us that there are 3011 total elements in the union of A and B, so we can equate the expression above to 3011. This is our resulting product:

`(a-1000)+(b-1000)+1000 = 3011`
We can simplify this equation to
`a+b=4011`. In the beginning, we found that a = 2b, or b = 1/2a, so we can substitute that into the equation. Here is the process:

`a+b=4011`

`a+ (1)/(2)a=4011`

`(3)/(2) a=4011`

`a=2674`
Therefore, the total number of elements in set a is 2674.