For real numbers a, b, and c, is ab = bc + 3?

Question

(1) ab = 3b
(2) b + 3 = c

udyans123 · Answer

(1) ab = 3b => It gives no information about 'c' --> Insufficient

(2) b + 3 = c => It gives no information about 'a' --> Insufficient

Both (1) and (2) together .. Lets substitute these two conditions (1) and (2) in the main eqn (to be proved)

ab = b.(b+3) + 3 = b^2 + 3b + 3

3b = b^2 + 3b + 3 => b^2 = -3 --> For this eqn to be equal, 'b' should be imaginary.

But, we are given that a, b and c are real numbers => These numbers cannot be imaginary ..

Hence, b^ 2 = -3 CANNOT be true for any values of a,b and c.. Option (C)

Senthil1981 · Answer

Consider 1)  ab = 3b 
there are 2 possibilities,  b = 0  or  b
eq{0}  which means  a = 3  or  a
eq{3}  and also  c  is not known.
 Not Sufficient , Eliminate A and D

Consider 2)  b + 3 = c 
Substituting on the given equation:  ab = b(b+3) + 3 = b^2 + 3b + 3  . Still  a  and  b  are not known.
 Not Sufficient , Eliminate B

Consider 1) and 2) together (include that 1) has 2 possibilities based on  b = 0  and  b
eq{0} )
when  b = 0 , then  c=3 and  ab = bc +3  becomes  0 = 0 + 3  and Hence  FAILS 
when  b
eq{0} , then  a = 3  and  ab = bc + 3  becomes  3b = b(b+3) + 3=> b^2 = -3 
giving values for b which are not real and the Hence  Fails ,

so answer is  C , since when 1) and 2) considered together, the equality  FAILS .

PS: this took more than 2 mins, anyone with faster approach please.

+1 for kudos.

JatinR · Answer

C is the correct option: Below is the solution

Statement 1: ab=3b i.e b(a-3)=0 . Either b=0 or a=3 : We leave this, as no solution we get
Statement 2: b+3=c No info about anything else, so leave this too

Combining both 1 & 2

If we take b=0 , then c=3 according to statement 2 . 
Try inputting this in ab= bc+3 . 0 is not equal to 3 Therefore ruled out

Now take a=3 and b+3 =c 
Input these in ab=bc+3 ; b(a-c)=3 ; (c-3)(3-b-3) =3 ; -bc+3b =3 ; b(3-c)=3 : This matches the required where 3=a.

Hence combining the statements gave the answer .

Therefore Option C :D

hellosanthosh2k2 · Answer

ab = bc + 3?
ab - bc = 3 ?
b(a-c) = 3?

Statement 1:
 ab - 3b = 0 
 => b = 0 OR a = 3
 if b = 0 => b(a-c) not equals to 3
 if a = 3 => we dont know the value of c to say if b(a-c) equals to 3

Statement 2:
 b + 3 = c
 b(a-c) = 3? => (c-3)(a-3)= 3 ? => don't know about a to conclude => insuff

1 + 2
 if b = 0, we saw from statement 1, b(a-c) is not equal to 3
 if a = 3 and b + 3 = c, b(a-c) = 3 ? => (c-3)(3-c) = 3? => (3-c)(3-c) = -3? => (3-c)^2 = -3? definitely not equal to -3, as squared values are >= 0
sufficient to answer the question as NO => (C)

MathRevolution · Answer

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Since we have 3 variables and 0 equations, E is most likely to be the answer. So, we should consider 1) & 2) first.

Conditions 1) & 2):
ab = 3b
⇔ ab - 3b = 0
⇔ (a-3)b = 0
⇔ a = 3 or b = 0

Case 1:  a = 3 
 ab = 3b  and  bc + 3 = b(b+3)+3 
 ab = bc + 3 
 ⇔ 3b = b(b+3)+3 
 ⇔ 3b = b^2 + 3b + 3 
 ⇔ 0 = b^2 + 3 , which is false.
The answer is "No".

Case 2:  b = 0 
 ab = bc + 3 
 ⇔ 0a = 0c + 3 
 ⇔ 0 = 0 + 3 , which is false.
The answer is "No".

The answer is always "No".

They are sufficient by CMT (Common Mistake Type) 1, since the answer "no" also means the condition is sufficient.

The answer is C.

In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2), when taken together. Obviously, there may be occasions on which the answer is A, B, C or D.

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For real numbers a, b, and c, is ab = bc + 3?

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For real numbers a, b, and c, is ab = bc + 3?

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