If a

Question

GMATH  practice exercise (Quant Class 16)

If  a<b<0  are integers, what is the value of  a^3-b^3  ?

(1)  a^2-b^2 = 65 
(2)  a^2 +b^2 < 100

chetan2u · Answer

If  a<b<0  are integers, what is the value of  a^3-b^3  ?

So we have to find the value of a and B...

(1)  a^2-b^2 = 65 
(a-b)(a+b)=65=(-1)(-65)...so a=-33 and b=-32
But (a-b)(a+b) can be (-5)(-13), so a=-9 and b=-4
Different possibilities..
Insufficient

(2)  a^2 +b^2 < 100 
a and B can take various values...
a=-9, b=-1 or a=-8, b=-2, or a=-8, b=-3 and so on
Insufficient

Combined..
 (-33)^2+(-32)^2>100 , so eliminate
 (-9)^2+(-4)^2=81+16=97<100 , so possible..
Thus  a^3-b^3=(-9)^3-(-4)^3 
Sufficient

C

Archit3110 · Answer

#1
 a^2-b^2 = 65 
(a+b)*(a-b)=65
a =-33,b=-32
a=-9, b=-4
in sufficeint
#2
 a^2 +b^2 < 100 
we can get many values of a & b to satisfy this relation , viz , 1,2 ; 3,4; 5,4... so on 
in sufficient
from 1 & 2
only a= -9 and b= -4 suffices both eqns 1&2
IMO C

fskilnik · Answer

? = {a^3} - {b^3}

\,a < b < 0\,\,\,{m{ints}}\,\,\,\,\left( * ight)

\left( 1 ight)\,\,\,\left\{ \matrix{
 \,\left( {a + b} ight)\left( {a - b} ight) = 65 = 1 \cdot 65 = 5 \cdot 13 \hfill \cr 
 \,\left( * ight)\,\,\,\,\, \Rightarrow \,\,\,\,\,a + b < a - b\,\,\, < 0\,\,\,{m{ints}} \hfill \cr} ight.\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,{m{I}}.\,\,\left\{ \matrix{
 \,a + b = - 65 \hfill \cr 
 \,a - b = - 1 \hfill \cr} ight.\,\,\,\,\,\,\,\,{m{or}}\,\,\,\,\,\,{m{II}}.\,\,\left\{ \matrix{
 \,a + b = - 13 \hfill \cr 
 \,a - b = - 5 \hfill \cr} ight.

{m{I}}.\,\,\left\{ \matrix{
 \,a + b = - 65 \hfill \cr 
 \,a - b = - 1 \hfill \cr} ight.\,\,\,\,\,\mathop \Rightarrow \limits^{\left( + ight)} \,\,\,\,\,\, \ldots \,\,\,\,\, \Rightarrow \,\,\,\,\left( {a,b} ight) = \left( { - 33, - 32} ight)\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = {\left( { - 33} ight)^3} - {\left( { - 32} ight)^3}

{m{II}}.\,\,\left\{ \matrix{
 \,a + b = - 13 \hfill \cr 
 \,a - b = - 5 \hfill \cr} ight.\,\,\,\,\,\mathop \Rightarrow \limits^{\left( + ight)} \,\,\,\,\,\, \ldots \,\,\,\,\, \Rightarrow \,\,\,\,\left( {a,b} ight) = \left( { - 9, - 4} ight)\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = {\left( { - 9} ight)^3} - {\left( { - 4} ight)^3} 
e {\left( { - 33} ight)^3} - {\left( { - 32} ight)^3}

\left( 2 ight)\,\,\left\{ \matrix{
 \,{m{Take}}\,\,\left( {a,b} ight) = \left( { - 2, - 1} ight)\,\,\,\, \Rightarrow \,\,\,{m{?}}\,\,{m{ = }}\,\,{\left( { - 2} ight)^{m{3}}} - {\left( { - 1} ight)^3}\,\, \hfill \cr 
 \,{m{Take}}\,\,\left( {a,b} ight) = \left( { - 3, - 1} ight)\,\,\,\, \Rightarrow \,\,\,{m{?}}\,\, = {\left( { - 3} ight)^{m{3}}} - {\left( { - 1} ight)^3}\,\, 
e \,{\left( { - 2} ight)^{m{3}}} - {\left( { - 1} ight)^3}\,\, \hfill \cr} ight.

\left( {1 + 2} ight)\,\,\,\,\, \Rightarrow \,\,\,\,{m{II}}.\,\,\,\,\,\, \Rightarrow \,\,\,\,{m{SUFF}}.

The correct answer is (C).

We follow the notations and rationale taught in the  GMATH  method.

Regards, 
Fabio.

If a<b<0 are integers, what is the value of a^3-b^3?

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