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

[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?

 

[MarkFL]

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:

\(\displaystyle 2B=120\)

 

[elimeli]

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?

 

[MarkFL]

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. :)

 

[elimeli]

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!