Sets M and N contain exactly m and n elements, respectively.

Question

Sets M and N contain exactly m and n elements, respectively. What is the value of n?

(1) 7m=8n
(2) The intersection of M and N contains exactly 0.4m elements.

chetan2u · Accepted Answer

Hi,

the Q asks us a numeric value for n..

However I gives us the ratio of m and n and II gives us the common elements in term of m again..
so NO numeric value of any kind to work on..
E

But lets see what each statements tells us.. 
May be helpful in some other Q..
 (1) 7m=8n 
 m is a multiple of 8 and n is a multiple of 7

(2) The intersection of M and N contains exactly 0.4m elements. 
this means common numbers are 0.4m..
so,
m has to be a multiple of 5, otherwise .4m will be decimal..

combined we can say  
 so both m and n are multiples of 5.. 
m is a multiple of 8*5=40
and n is a multiple of = 7*5=35

if we say we were told m <75.. we would have got our answer

chetan2u · Answer

sure..
there are m and n elements..
Also there are 0.4m common elements...
 Now these elements have to be integer.. MEANING we can't have 4.5 or 5.6 elements common, it has to be an integer say 3,4, or 5 etc..
When will 0.4*m be an integer, ONLY when m is a multiple of 5, say 5x.. 
so 0.4 *m = 0.4*5*x = 2x, hence an integer..
m could be anything 5,10,35 or 100 but it will surely be a multiple of 5..
Hope it helps

jpe7856 · Answer

I'm a little confused - can you please explain what "the intersection of M and N contains exactly 0.4m elements" translates to algebraically? Would that mean N=0.4M? Since these are sets of numbers, it's also inherently assumed that the answer will be an integer, correct?

chetan2u · Answer

Hi,

Intersection of M and N consists of 0.4m elements in SET would be written as -- 
M \cap N = 0.4M..
 so there are 0.4M elements common to M and N.. 
Since M and N are integers and common elements too should be integer, 0.4M should be integer..
 so M has to be a multiple of 5 for 0.4M to be an integer..

broilerc · Answer

Why do both m and n need to be multiples of 5 ? Why also n ? Could you explain the "multiple of 5" part again please?

chetan2u · Answer

hi 
it may have been written in statement 2, But it is meant when both statements are combined..
As 8n = 7m.. so if m is multiple of 5 n has to be a multiple of 5..

broilerc · Answer

thanks for the quick help! can you explain again why m is a multiple of 5 because of 0.4m ?

prakhar4 · Answer

Sir,

I have a question regarding the below solution by you. Specifically in the last line where if m,75 was mentioned we would have gotten our answer. How so?

chetan2u · Answer

Hi Prakhar,

we have got m as a multiple of 40 and n as a multiple of 35...
since intersection is 0.4m, we can say that m and n are  
eq{0} ..

so if we were given m<75, ONLY 40 would have fit in as next multiple of 40 is 80, which would have >75.. 
and 7m = 8n....
so 7*40 = 8n......n =35

that is why suff..

Storm9 · Answer

The question is asking for a specific value of n.
Statement 1: This is a ratio, so clearly insufficient.
Statement 2: Another ratio, you don't know m and this statement says nothing about n.

1+2: From statement 1:  m = \frac{8}{7}n  and using statement 2: Intersection contains:  0.4(\frac{8}{7}n)  elements. Doesn't really tell you anything about n, so answer is E.
If you look at it from a higher level both statements give you some sort of ratio info, while the question is asking for a value. You can't get a value from just ratio info.

NYCBS19 · Answer

Great explanation, thank you! However, I have one question about the above quoted bit... After combining both statements, we know that M---> Multiple of 7 AND a multiple of 5 N---> Multiple of 8 M/N are in the ratio of 8/7... So for ex if M = 7 x 5, then N= 8 x 5 (giving us values of M=35 and N=40) Using this logic, we can have M= 7 x 10 , then N= 8 x 10 (giving us values of M=70 and N=80) Therefore even if we were given m<75, we will still have 2 values..?? I think I am misunderstanding something maybe, can you please help to point out my misunderstanding? Thanks

chetan2u · Answer

Hi, 
you have understood the method but going wrong on inference from 7m=8n...
This tells us that n and NOT m, is multiple of7.....
M is s multiple of 8, as m= n*8/7....so n is multiple of 7

rcaraujo20 · Answer

Hello,

Another way to see the second statement:
(1) m : n = 8 : 7
(2) Think about groups, so: m + (n - 0.4m) = T (total of elements) => 0.6m + n = T

(1-2) Substitute (1) in (2): 0.6 x 8n/7 = T
n and T are variables, but we have just one equation to solve it. Hence, insufficient.

Best,

minustark · Answer

Together, when m has to be a multiple of 5 and n has to produce an integer when 7m is divided by 8. Again many answers are possible for this condition, like m = 40, n= 35. or m =80, n=70.

E is the answer.

Kritisood · Answer

Could please explain this point further?

Why if m is a multiple of 5 ALSO n must be a multiple of 5?  VeritasKarishma

KarishmaB · Answer

Given m and n are integers,

If 7m = 8n, 
7 * m = 2^3 * n

m must have 2^3 in it and n must have 7. Now, if m is a multiple of any other prime number, n must be a multiple of that prime number too.

If m = 2^3 * 5,
n = 7 * 5

If m = 2^3 * 11
n = 7 * 11

Only then can both sides of the equations be same.

Gmatboat · Answer

But lets see what each statements tells us.. 
May be helpful in some other Q..
  (1) 7m=8n 
 m is a multiple of 8 and n is a multiple of 7

Could any experts please explain the highlighted part? I am confused. I understand that if we were told m= 8*(some integer n), then we can deduce that m is a multiple of '8'. However, in this case, how can we deduce that 'm' is a multiple of 8? if we re-arrange the formula, we get m = 8/7 * n (how is this a multiple of 8?). At best, the equation looks like m = multiple of 8/7.

ScottTargetTestPrep   JeffTargetTestPrep   BrentGMATPrepNow

BrentGMATPrepNow · Answer

The key piece of information here is that m and n are  integers .
If 7m = 8n, then the prime factorization of both sides should be equal. 
That is: (7)(m) = (2)(2)(2)(n)
We can see that, in order for the equation to hold true, m must have at least three 2's in its prime factorization, and n must have at least one 7 in its prime factorization. 
If m has at least three 2's in its prime factorization, then m is a multiple of 8
If n has at least one 7 in its prime factorization, then n is a multiple of 7

Does that help?

ShubhangiGoel · Answer

can we solve it using below approach

M U N= m+ 7m/8-0.4m +neither

insufficient

Sets M and N contain exactly m and n elements, respectively.

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Data Sufficiency (DS) Questions

Sets M and N contain exactly m and n elements, respectively.

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards