Out of 200 dog owners, is the number of owners who run their dogs

Creating a 3x3 matrix:

\begin{tabular}{|l|l|l|l|}
    \hline
        ~ & W & nW & T \\ \hline
        R & ~ & ~ & ~ \\ \hline
        nR & ~ & ~ & ~ \\ \hline
        T & ~ & ~ & 200 \\ \hline
    \end{tabular}
 ­
(1) All of the dog owners who run their dogs also walk their dogs

Let this be represented by x, which means

\begin{tabular}{|l|l|l|l|}
    \hline
        ~ & W & nW & T \\ \hline
        R & x & 0 & x \\ \hline
        nR & ~ & ~ & ~ \\ \hline
        T & ~ & ~ & 200 \\ \hline
    \end{tabular}

Looking at the table and the info provided by this stem, the number of owners who run their dogs is \(x\), while the number of owners who walk their dogs has a minimum value of \(x\) (if every person who walks their dog also runs their dog) and a maximum that will exceed \(x\) if there are people who walk but do not run their dogs.

Representing that algebraically, \(Walk ≥ Run\). In other words, the number who run their dogs cannot ever exceed the number who walk their dogs. This is enough to answer the question.

SUFFICIENT

(2) 75 dog owners walk their dogs

Putting this info into the table, and any values which can be workout:

\begin{tabular}{|l|l|l|l|}
    \hline
        ~ & W & nW & T \\ \hline
        R & ~ & ~ & ~ \\ \hline
        nR & ~ & ~ & ~ \\ \hline
        T & 75 & 125 & 200 \\ \hline
    \end{tabular}­

Without any further information, it is impossible to deduce whether number of owners who run their dogs greater than the number of owners who walk their dogs.

INSUFFICIENT


​​​​​​​ANSWER A­

I also assumed the same and answered E ? Could someone explain this ?

Yes, it is possible that some owners neither walk nor run their dogs - the question doesn’t say all 200 owners do one or the other.

But that doesn’t affect Statement (1).

Statement (1) says: "All of the dog owners who run their dogs also walk their dogs."

This means everyone in the "run" group is also in the "walk" group, so R ≤ W, no matter how many total owners actually walk or run their dogs. Therefore, R > W is not possible under this condition.

So even if many owners do neither, R can never be greater than W, making the answer to the question a definite "No" - which is why Statement (1) is still sufficient.

Hope it helps.