Verified answer

45

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Catch the Idea!

Two sets are equal if and only if when all elements coincide.

[tex]\;[/tex]

What Is the Solution?

First, let's check set A.

There needs a total of [tex]\text{$150$}[/tex] overlapping elements.

[tex]\text{$\Longrightarrow \underbrace{x_{1},x_{1},x_{1}\cdots}_\textrm{10 times},\cdots,\underbrace{x_{15},x_{15},x_{15},\cdots}_\textrm{10 times}$}[/tex]

The diagram shows [tex]\text{$15$}[/tex] distinct elements.

(That is, the union of set A is [tex]\text{$S=\{x_{1},x_{2},\cdots,x_{15}\}$}[/tex].)

[tex]\;[/tex]

Lastly, let's check set B.

There needs a total of [tex]\text{$3n$}[/tex] overlapping elements.

[tex]\text{$\Longrightarrow \underbrace{x_{1},x_{1},x_{1},\cdots }_\textrm{9 times},\cdots,\underbrace{x_{\frac{n}{3}},x_{\frac{n}{3}},x_{\frac{n}{3}},\cdots }_\textrm{9 times}$}[/tex]

The diagram shows [tex]\text{$\dfrac{n}{3}$}[/tex] distinct elements.

(That is, the union of set B is [tex]\text{$S=\{x_{1},x_{2},\cdots,x_{\frac{n}{3}}\}$}[/tex].)

[tex]\;[/tex]

Now, all we need is to compare the elements.

All distinct elements need to coincide.

[tex]\text{$\Longrightarrow \dfrac{n}{3}=15$}[/tex]

[tex]\text{$\Longrightarrow \underline{\underline{n=45}}$}[/tex]

So, the value [tex]\text{$n=\boxed{45}$}[/tex].

[This is the required answer.]

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