12 Days of Christmas GMAT Competition - Day 11: If a*b 0, is (a + 5)

a/b + 5/b > a/b ?

5/b > 0?

If b < 0 No
If b > 0 Yes

(1) Not sufficient
(2) Sufficient. (a+5)/b < a/b

B is the answer

Posted from my mobile device

Asked: If \(a*b ≠ 0\), is \(\frac{a + 5}{b} > \frac{a}{b}\) ?

Is \(\frac{a + 5}{b} > \frac{a}{b}\) ?
Is \(\frac{a + 5}{b} - \frac{a}{b} > 0?\)
Is \(\frac{5}{b} > 0?\)

(1) \(a < 0\)
Since \(\frac{5}{b}\) is independent of a.
NOT SUFFICIENT

(2) \(b < 0\)
\(\frac{5}{b} < 0\)
SUFFICIENT

IMO B
_________________

As a*b is not equal to 0, neither a=0 or b=0

Solving the equation: you get ab+5b<ab i.e b<0
(1) a<0 - Not sufficient
(2) b<0 - Sufficient

Ans: B

If a∗b≠0, is \((a+5)/b > a/b\)?

Simplify the question:
Is \((a+5)/b - (a/b) > 0\)?
Is \((a+5-a)/b > 0\)?
Is \(5/b > 0\)?
Is \(b > 0\)?

(1) a<0
Not sufficient.

(2) b<0
Sufficient.

Answer: B

Using backtracking approach:
S1)
a<0:
we know
=> 5>0
Adding a :
=>5+a > a

if b >0 => (5+a)/b > a/b
if b<0 => (5+a)/b < a/b



Not Suff

S2) b<0:
Multiplying by 5
=> 5b<0;
Adding ab both sides:
=> ab+5b<ab
=> b(a+5)<ab
Dividing by b^2
=> (a+5)/b < a/b

S2 is suff


Hence B)

given

\(a*b ≠ 0\)
is \(\frac{a + 5}{b} > \frac{a}{b}\)
#1
a<0
we get both yes & no to this statement
as value of b can be either +ve or -ve
insufficient
#2
b<0
same as #1
insufficient

from 1 &2
we can have a=b ; a>b ; a<b
RHS would always be +ve
where as for all values of a,b ; the LHS will always be -ve for cases when a>b
for cases when b>a also RHS value would be > than LHS as its Nr is reducing by 5

OPTION C is correct

The easiest way to deal with this task is to selectively look through some numbers for A and B, depending on the conditions.

This gives us four potential scenarios, based on whether \(a+5\) is still below zero or becomes positive:

• \(a<-5\) and \(b<0\), for instance \(a=-10\) and \(b=-1\)
  \(\frac{-5}{-1}\)<\(\frac{-10}{-1}\)
• \(-5<a<0\) and \(b<0\), for instance \(a=-3\) and \(b=-1\)
  \(\frac{2}{-1}\)<\(\frac{-3}{-1}\)
• \(a<-5\) and \(b>0\), for instance \(a=-10\) and \( b=1\)
  \(\frac{-5}{1}\)>\(\frac{-10}{1}\)
• \(-5<a<0\) and \(b>0\), for instance \(a=-3\) and \(b=1\)
  \(\frac{2}{1}\)>\(\frac{-3}{1}\)

So, as we can see, we have both options here - both 'more' and 'less', as highlighted in yellow. Therefore, Condition 1 by itself is not sufficient.


In this case, because we have sort of 'fixed' B, we need to look into three options for A - below -5, above -5 and above 0:

• \(a<-5\) and \(b<0\), for instance \(a=-10\) and \(b=-1\)
  \(\frac{-5}{-1}\)<\(\frac{-10}{-1}\)
• \(-5<a<0\) and \(b<0\), for instance \(a=-3\) and \(b=-1\)
  \(\frac{2}{-1}\)<\(\frac{-3}{-1}\)
• \(0<a\) and \(b<0\), for instance \(a=3\) and \(b=-1\)
  \(\frac{8}{-1}\)<\(\frac{3}{-1}\)

So, as can bee seen, ALL the signs here are 'less' which basically means that Condition 2 by itself is sufficient.

Therefore, the correct answer is B.

a+5/b>a/b

I) a>0
Take the case of a=-2
3/b>-2/b? If b>0, yes, if b<0, no. INS.

II)
b<0

If a=100
105/b>100/b? 100/b is less negative than 105/b, so no answer.
If a=-5
0>-5/b? b cannot be zero, so it will be some positive fraction or integer, once again no.
If a=-100
-95/b>-100/b? Once again, LHS will become positive and will thus be bigger than RHS (they have same base and sign, so bigger numerator is bigger).

B is sufficient.

(a+5)/b>a/b?

ab+5b>ab?
5b>0?

So we only need the sign of b

Option B)Statement II is sufficient

If a∗b≠0, is a+5/b>a/b ?

(1) a<0; We dont know if b < or > 0.
If b<0, then no because a/b will always be positive whereas LHS may or may not be.
If b>0 then yes. NS

(2) b<0. Sufficient because if a>0, then LHS will be more negative than RHS and hence LHS< RHS
If a<0 then RHS > LHS

Ans B

lets take below scenarios

1. when both a and b are negative

a/b will be positive and a+5/b will be positive for a<-5 and
a+5 < a (as a is negative)
a+5/b < a/b

2. when a is positive and b is negative
a+5 > a
since b is negative
a+5/b and a/b will be negative, and
a+5/b < a/b because a+5/b will be greater in value.

3. when both is positive
then a+5/b > a/b

now if only b is negative we know for sure that a+5/b > a/b

Kudos!!

Solution:

Given, \(a*b ≠ 0\) implies a ≠ 0 & b ≠ 0.
\(\frac{a + 5}{b} > \frac{a}{b}\) ?
or a/b+ 5/b> a/b?
or \(\frac{ 5}{b} > 0\) ?
or 5b > 0 ?
or b > 0?

So, the modified question stem simply asks if b>0?

(1) \(a < 0\)
This does not give any information about b. Insufficient.

(2) \(b < 0\)
This gives a clear answer, NO for b>0? Sufficient.

ANSWER B.

(1) a<0 means a is negative
b can be negative or positive

So there is 2 probabilities:
1. (-a+5)/b>-a/b Yes
2. (-a+5)/-b>-a/-b No (Not sufficient)

(2) b<0 means b is negative
a can be negative or positive
1. (a+5)/-b>a/-b No
2. (-a+5)/-b>-a/-b No (Sufficient)

IMO B

a*b 'not equal' to 0 implies

a 'not equal' to 0
b 'not equal' to 0

also, (a+5) / b = a/b + 5/b

So, a/b + 5/b > a/b ?

or, 5/b > 0

(1) a<0

Solution can't be found (as the solution depends on b, not a)

INSUFFICIENT

(2) b<0

Solution can be found (NO is the answer)

SUFFICIENT

IMO, (B) will be the answer

(a+5)/b>a/b
ab+5b>ab
b(a+5)>ab
a+5>a
We need only the value of a to solve.

Statement 1: a<0
-2+5<-2
3<-2(False) One unique answer, Sufficient

Statement 2: b<0
Not needed one, Insufficient


Answer: A

For statement (1),
Let a=-1, b=2
So,
-2>-1/2 which is false.

Now, let a=5, b=2

0>-5/2 which is true.

Hence, (1) is not enough.

For statement (2),
If b<0, it seems to be true for all values of a and b<0.
Hence, (2) is enough alone.

Hence, IMO B.

If a∗b≠0, is (a+5)/b > a/b?

Before looking at the answer options we must simplify the question as done below –
-> (a+5)/b > a/b
-> a/b + 5/b > a/b
-> 5/b > a/b – a/b
-> 5/b > 0

So, our question becomes, is 5/b > 0?

(1) a < 0

The information does not tell us any information about the value of b. Hence, this statement is insufficient to answer the question.

(2) b < 0

If b is negative, then 5/b is definitely less than 0. Hence, this statement is sufficient to answer the question.

Answer: Option B

Question stem evaluates to

Is 5/b > 0 ? (after cancelling a/b from both sides)

(1) a<0

No information about b;

YES/NO (can't be determined)

NOT SUFFICIENT


(2) b<0

Information available about b;

since, b<0
hence, 5/b < 0

NO is the answer

SUFFICIENT

Hence, (B) is the correct choice
_________________

Q: is (a+5)b > ab?
=> ab+5b-ab > 0?
=> b>0?

Statement 2 directly states that b<0. hence, answer to the ques is a definite No.
Hence, answer is B.
_________________