Felipe stores his contacts telephone numbers in two places: in the me

Question

Felipe stores his contacts’ telephone numbers in two places: in the memory of his cell phone, and in a notebook. If some numbers are stored both in his cell phone memory and in his notebook, does his notebook contain a greater number of telephone numbers than does his cell phone memory?

(1)  \frac{2}{3}  of all the distinct telephone numbers Felipe has recorded are recorded in his notebook.

(2)  \frac{4}{7}  of the telephone numbers stored in Felipe’s cell phone memory are also recorded in his notebook.

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Purnank · Answer

Given - Felipe stores his contacts’ telephone numbers in two places: in the memory of his cell phone, and in a notebook. If some numbers are stored both in his cell phone memory and in his notebook.
To find - does his notebook contain a greater number of telephone numbers than does his cell phone memory?
A=Cell phone, B = Notebook, Is n(B) > n(A) ?
n(A) = total number of contacts in cellphone,
n(B) = total number of contacts in notebook,
n(A∩B) = total number of contacts in both
n(AUB) = n(A) + n(B) - n(A∩B)

1st - 2/3 of all the distinct telephone numbers Felipe has recorded are recorded in his notebook.
A distinct = n(A) - n(A∩B), B distinct = n(B) - n(A∩B) Total distinct = n(A) + n(B) - 2*n(A∩B)
n(B) - n(A∩B) = 2*((n(A) + n(B) - 2*n(A∩B))/3
3n(B) - 3n(A∩B) = 2n(A) + 2n(B) -4n(A∩B)
n(B) = 2n(A) - n(A∩B)
n(B) = n(A) + n(A) - n(A∩B)
This shows n(B) > n(A) (As n(A) - n(A∩B) is all the distinct telephone numbers Felipe has recorded are recorded in his cell phone)
Sufficient.

2nd - 4/7 of the telephone numbers stored in Felipe’s cell phone memory are also recorded in his notebook.
Assuming total of telephone numbers n(AUB) = 7k, n(A∩B) = 4k,
n(A) + n(B) = n(AUB) + n(A∩B) = 11k
But nothing mentioned about distribution in between cell phone and notebook so we cant conclude anything. Not sufficient
Answer A.

tgsankar10 · Answer

All numbers = Distinct in Memory + Distinct in Notebook + Common in Memory & Notebook

Numbers in Memory = Distinct in Memory + Common in Memory & Notebook

Numbers in Notebook = Distinct in Notebook + Common in Memory & Notebook

Question:  Numbers in Notebook > Numbers in Memory

Distinct in Notebook > Distinct in Memory ?

Statement 1:   \frac{2}{3}  of all the distinct telephone numbers Felipe has recorded are recorded in his notebook

Distinct Numbers = Distinct in Memory + Distinct in Notebook =  \frac{x}{3}+\frac{2x}{3}

Distinct in Notebook > Distinct in Memory

Sufficient

Statement 2:   \frac{4}{7}  of the telephone numbers stored in Felipe’s cell phone memory are also recorded in his notebook.

No info about numbers in notebook alone.

Not Sufficient

Hence,  Answer is A

Nidzo · Answer

(1) 2/3 of all the distinct telephone numbers Felipe has recorded are recorded in his notebook.

As the statement is given as a fraction, let the total saved numbers be  1 .
We know that there are no numbers which are saved elsewhere.
We are told that  \frac{2}{3}  of the distinct numbers are saved on his notebook

Filling in these three pieces of information into a 3x3 matrix:

\begin{tabular}{|l|l|l|l|}
    \hline
        ~ & N & nN & Total \ \hline
        P & ~ & ~ & ~ \ \hline
        nP & \frac{2}{3} & 0 & ~ \ \hline
        Total & ~ & ~ & 1 \ \hline
    \end{tabular}

From here one can solve that the total for  nP = \frac{2}{3} , and thus the total for  P = \frac{1}{3}

\begin{tabular}{|l|l|l|l|}
    \hline
        ~ & N & nN & Total \ \hline
        P & ~ & ~ & \frac{1}{3} \ \hline
        nP & \frac{2}{3} & 0 & \frac{2}{3} \ \hline
        Total & ~ & ~ & 1 \ \hline
    \end{tabular}

One knows that  \frac{1}{3}^{_rd}  of his numbers are saved on his phone. While one cannot solve for an answer for the total of  N , it is evident that the total will either be  1  (ie. all of the numbers) or  \frac{2}{3}^{_rds}  both of which are greater than a  \frac{1}{3}^{_rd} .

SUFFICIENT

(2) 4/7 of the telephone numbers stored in Felipe’s cell phone memory are also recorded in his notebook.

Let the total numbers on his phone =  x 
Therefore the total numbers which are on both =  \frac{4}{7}x 
Solving for the ones only on the Phone =  \frac{3}{7}x .

\begin{tabular}{|l|l|l|l|}
    \hline
        ~ & N & nN & Total \ \hline
        P & \frac{4}{7}x & \frac{3}{7}x & x \ \hline
        nP & ~ & 0 & ~ \ \hline
        Total & ~ & \frac{3}{7}x & ~ \ \hline
    \end{tabular}
 
From here it is easy to assume that then the total numbers in the Notebook will be  \frac{4}{7}x  and that the very total numbers is  x , however,  x  denotes only how many numbers are saved on the Phone. Without knowing how this value relates to the full amount of numbers saved by Felipe, one cannot solve for more.

INSUFFICIENT

ANSWER A

Purnank · Answer

Answer C? How?

sherlocked221B · Answer

Could you please verify Answer A?
Or report the question for right solution?

ISHAAN96 · Answer

Hello when will the official explanation be posted? 
@chetan2u @karishmaB could you please explain.

Thanks

Harsh_Belani · Answer

Bunuel , Can you please post the official solution?

Felipe stores his contacts telephone numbers in two places: in the me

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