Is mn

Question

Is mn < 10?

(1) m < 5 and n < 2
(2) 1 < m < 3 and n^2<25

CrackverbalGMAT · Accepted Answer

Hi Srav,

The best thing to do while solving questions of this type is to make use of the  Max/Min Concept of Inequalities.

If we are given two ranges say  -11 < x < 9  and  -9 < y < 2  and the question asks us to find the max and min values of x * y then we can place the ranges one below the other and find the values for x * y for the 4 extreme values i.e. -11, 9, -9 and 2.

The four extreme values of x * y are  99, 18, -22 and -81 . So  -81 < xy < 99 .

To apply the max min concept we need to keep in mind that the inequalities of the two ranges are the same. If they are not then we manipulate the numbers or use the properties of inequalities to make them the same. This same procedure can also be used to find the extreme values of x + y and x - y.

So where do we use the max min concept?

Whenever we have been given two ranges, one for x and the other for y (with the same inequality sign) and the question asks us to find the value of x + y, x - y and x * y.

Now coming to the question given

Is mn < 10?

(1) m < 5 and n < 2
(2) 1 < m < 3 and n^2 < 25

Statement 1  : Clearly  insufficient  as m can be 4 and n can be 1, which gives us a YES for mn < 10 or m can be -5 and n can be -2 which gives us a NO for mn < 10.

Statement 2 : 1 < m < 3 and n^2 < 25

Now here we can apply the max min concept, since 1 < m < 3 and -5 < n < 5. Multiplying the extreme values we get -5, 5, 15 and -15. So the range here is  -15 < mn < 15 .  Insufficient .

Combining statements 1 and 2  :  The range for m is 1 < m < 3 and the range for n is -5 < n < 2.

Again applying the max min concept we get  -15 < mn < 6.  Since mn here will always be less than 10.  Sufficient .

Hope this helps!

For some more strategies on Inequalities, you can download a free Inequality eBook from the link given below

https://gmat.crackverbal.com/free-resources/ebook-library/

CrackVerbal Academics Team

ENGRTOMBA2018 · Answer

The question asks mn<10 ? Per statement 1, m<5 and n<2 ---> no if m=-10, n = -5 but yes if m=1 and n =1 . Not sufficient Per statement 2, 1 1

krisnaren2010 · Answer

unless it is stated that m, n are integers shouldnt the answer be E?

m=2.99 n=4 mn> 10,
so shouldnt the answer be E?

ganand · Answer

Hi krisnaren2010,

By combining statement 1 and 2 we have following:

1<m<3 and -5<n<2 .

So, you can't take  n = 4.

Hope it helps.
Thanks.

Nunuboy1994 · Answer

Statement 1 
m < 5 and n < 2

m could be (-7) and n could be (-6) which would result in a number greater than ten; however, 
m could be (-7) and n could be (1) which would result in a number less than ten.

Insufficient.

Statement 2

1<2.7<3 and (4)^2 equals a number greater than ten

1<2<3 and (4)^2 equals a number less than ten
 
 Statement 1 and Statement 2

1<m<4 subsumes a value less than five

(-4)<n<2 contains a number that when squared is less than 25 and less than 2

The product of each and every combination that can be formed from both sets is less than ten (e.g the product of 2.7 and 1 is less than 10)

Sufficient.

ashikaverma13 · Answer

What is the source? Good question.

Bunuel · Answer

Check source in the tags above the original post. It's Veritas Prep.

sashiim20 · Answer

(1) m < 5 and n < 2 Lets try some numbers. m = 4; n = 1 mn = (4*1) = 4 < 10 m = -6; n = -2 mn = (-6*-2) = 12 > 10 Hence I is Not Sufficient. (2) 1 < m < 3 and n^2<25 m = 2; n = 4 mn = (2*4) = 8 < 10 m = 2.9; n = 4 mn = (2.9*4) = 11.6 > 10 Hence II is Not Sufficient. Combining I and II Value of m is between => 1 < m < 3 n^2 < 25 -5 < n < +5 From (1) n < 2 Therefore; Value of n is between => -5 < n < 2 Checking minimum and maximum value of mn ; Minimum value of mn > (3*-5) = -15 Maximum value of mn < (3*2) = 6 -15

EMPOWERgmatRichC · Answer

Hi All,

We're asked if (M)(N) is less than 10. This is a YES/NO question. We can solve it by TESTing VALUES.

1) M < 5 and N < 2

IF..... 
M=4 and N=1 then the product is 4 and the answer to the question is YES. 
M= -6 and N= -2 then the product is 12 and the answer to the question is NO. 
Fact 1 is INSUFFICIENT

2) 1 < M < 3 and N^2 <25

IF..... 
M=2 and N=1 then the product is 2 and the answer to the question is YES. 
M= 2.9999 and N=4.9999 then the product is really close to 15 and the answer to the question is NO. 
Fact 2 is INSUFFICIENT

Combined, the various inequalities 'limit' us down to: 
1 < M < 3 
-5 < N < 2 
At MOST, the product (M)(N) would get really close to 6 - but no higher. Thus, the answer to the question is ALWAYS YES. 
Combined, SUFFICIENT

Final Answer:  C

GMAT assassins aren't born, they're made, 
Rich

BrentGMATPrepNow · Answer

Target question :    Is mn < 10?

Statement 1 : m < 5 and n < 2  
Let's TEST some values.
There are several values of m and n that satisfy statement 1. Here are two: 
 Case a : m = 0 and n = 0. In this case mn = (0)(0) = 0. So, the answer to the target question is  YES, mn IS less than 10 
 Case b : m = -5 and n = -5. In this case mn = (-5)(-5) = 25. So, the answer to the target question is  NO, mn is NOT less than 10 
Since we cannot answer the  target question  with certainty, statement 1 is NOT SUFFICIENT

Statement 2 : 1 < m < 3 and n² < 25 
Let's TEST some values again.
There are several values of m and n that satisfy statement 2. Here are two: 
 Case a : m = 1 and n = 0. In this case mn = (1)(0) = 0. So, the answer to the target question is  YES, mn IS less than 10 
 Case b : m = 2.9 and n = 4. In this case mn = (2.9)(4) = 11.6. So, the answer to the target question is  NO, mn is NOT less than 10 
Since we cannot answer the  target question  with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined    
IMPORTANT: When we combine the statements, we must adhere to the most RESTRICTIVE case for m and n each
For example, statement 1 tells us that m < 5, and statement 2 tells us that 1 < m < 3
Since both statements are TRUE, we can conclude that  1 < m < 3  since those possible values of m also satisfy the condition that m < 5

Likewise, for the value of n, statement 1 tells us that n < 2, and statement 2 tells us that n² < 25 (which is the SAME as saying  -5 < n < 5 )
Since both statements are TRUE, we can conclude that  -5 < n < 2

So, we know the following about m and n: 
 1 < m < 3 
 -5 < n < 2

So, one possible case is that m = 2 and n = 0. In this case mn = (2)(0) = 0. So, the answer to the target question is  YES, mn IS less than 10

IMPORTANT: Since we've already shown that mn can be LESS THAN 10 (yielding a YES answer to the target question)
Is it possible for mn to be GREATER THAN or equal to 10 (which would yielding a NO answer to the target question)?

To find out, let's maximize the values of m and n. 
So, m COULD equal 2.9999999... and n COULD equal 1.9999999....., in which case mn is LESS THAN 10
This tells us that, if  1 < m < 3  and  -5 < n < 2 , then it's IMPOSSIBLE for mn to be GREATER THAN or equal to 10

So, the answer to the target question is  YES, mn IS less than 10 
Since we can answer the  target question  with certainty, the combined statements are SUFFICIENT

Answer: C

Cheers,
Brent

Basshead · Answer

(1) If m = -10 and n = -20 then mn = 200; NO
if m = -6 and n = 1; YES

INSUFFICIENT.

(2) 1 < m < 3 and -5 < n < 5

If m = 1.01 and n = -4; YES
If m = 2.99 and n = 4.99; NO

INSUFFICIENT.

(1&2) Combined, 1 < m < 3 and -5 < n < 2

Given these ranges, mn < 10. SUFFICIENT.

Answer is C.

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Is mn<10?

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