# Math Problem: Find # of Elements in 2nd/3rd Subsets

• MHB
• elimeli
In summary: So the total number of elements in either the second or third subset is 120, which is the union of the two subsets. So the answer is 120, not 60. Is that correct?Yes, that is correct. The answer is 120, which is the total number of elements in either the second or third subset. This is because the question is asking for the union of the two subsets, not just one of them.
elimeli
I am practicing for my math exam next week and I came across this problem:

A set has 200 elements in it. It is partitioned into three subsets so that the second and third subsets have the same number of elements. If four times the number of elements in the second subset is three times as many as in the first, how many elements are in either the second or third subset?

The answer is 120, but I don't understand how to get to that answer.

I equaled 4 nx(2) = 3 nx(1), and so I got that 4nx(2)/3 = nx(1). I then plugged that into 200 = nx(2) + nx(2) + 4nx(2)/3 and got that nx(2)=60. The answer is 120, so I would have to multiply my answer by 2 to get it and I do not understand why, since the answer is asking for either the second or third subset and not the addition of the two.

Can anybody explain?

I would let $A$ be the cardinality of the first subset, and $B$ be the cardinality of the second and third subsets, so that we have:

$$\displaystyle A+2B=200$$

$$\displaystyle 4B=3A\implies A=\frac{4}{3}B$$

Substituting for $A$ into the first equation we get:

$$\displaystyle \frac{4}{3}B+2B=200$$

Multiply through by $$\displaystyle \frac{3}{2}$$:

$$\displaystyle 2B+3B=300$$

$$\displaystyle 5B=300$$

$$\displaystyle B=60$$

Thus, the cardinality of the union of the second and third subsets is:

$$\displaystyle 2B=120$$

MarkFL said:
I would let $A$ be the cardinality of the first subset, and $B$ be the cardinality of the second and third subsets, so that we have:

$$\displaystyle A+2B=200$$

$$\displaystyle 4B=3A\implies A=\frac{4}{3}B$$

Substituting for $A$ into the first equation we get:

$$\displaystyle \frac{4}{3}B+2B=200$$

Multiply through by $$\displaystyle \frac{3}{2}$$:

$$\displaystyle 2B+3B=300$$

$$\displaystyle 5B=300$$

$$\displaystyle B=60$$

Thus, the cardinality of the union of the second and third subsets is:

$$\displaystyle 2B=120$$

But why is the answer the union of the second and third subsets if the question is asking for either one of them? Shouldn't it be 60? Or am I misinterpreting the question?

The question asks how many elements are in either the second or third subsets, so they are asking for the total number of elements in the union of the two subsets. If an element is in either the second or third subset, then it is to be counted. :)

MarkFL said:
The question asks how many elements are in either the second or third subsets, so they are asking for the total number of elements in the union of the two subsets. If an element is in either the second or third subset, then it is to be counted. :)

I see. Thank you so much for clarifying!

## 1. How do I find the number of elements in the 2nd and 3rd subsets?

To find the number of elements in the 2nd and 3rd subsets, you will need to know the total number of elements in the set and the number of elements in the 1st subset. You can then use the formula n - x, where n is the total number of elements and x is the number of elements in the 1st subset.

## 2. Can I use this formula for any set?

Yes, this formula can be used for any set as long as you have the required information, which is the total number of elements and the number of elements in the 1st subset.

## 3. What is the significance of finding the number of elements in the 2nd and 3rd subsets?

Finding the number of elements in the 2nd and 3rd subsets can help in analyzing the distribution of elements within a set. It can also be useful in solving probability problems or determining the size of a sample from a larger population.

## 4. Can I use this formula for infinite sets?

No, this formula can only be used for finite sets, meaning sets with a limited or countable number of elements. It cannot be applied to infinite sets.

## 5. Is there an alternative method for finding the number of elements in the 2nd and 3rd subsets?

Yes, there are other methods for finding the number of elements in subsets, such as using tree diagrams or counting techniques. However, the formula n - x is a straightforward and efficient method for finding the number of elements in the 2nd and 3rd subsets.

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