What is the value of |a| - |b| ?

OFFICIAL SOLUTION:

What is the value of |a| - |b| ?

(1) |a/b| = 1. This is the same as |a|/|b| = 1 (must know properties: |xy| = |x|*|y| and |x/y| = |x|/|y|). Thus, |a| = |b|, so |a| - |b| = 0. Sufficient.

(2) |a - b| = 0. This means that the distance between a and b is 0, hence a and b are the same number: |a| - |b| = 0. Sufficient.

Answer: D.
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Hello all

My attempt:

Statement I:
By this we can conclude that the absolute value is the same for \(a\) and \(b\). Therefore it is sufficient to answer \(|a|-|b|\) which is \(zero\).

Statement II:
By this statement we conclude that not only the absolute vale but the signs are also same for \(a\) and \(b\). Therefore this statement too is sufficient to answer \(|a|-|b|\)which is \(zero\).

Hence I will go with option \(D\)

In first statement ,if we remove modulus , a/b=1 or a/b=-1 .. then how can we say absoulate value is same for both of them.
and in second statement can you please explain how signs are also same ?

I am sorry but i am badly struggling with these absoulate value type questions

Hi adityadon,

In first statement ,if we remove modulus , a/b=1 or a/b=-1 .. then how can we say absoulate value is same for both of them.
so in two cases you have mentioned a=b or a=-b... if you take absolute value(lal=lbl=l-bl=l-al) , which means the digit without the -ive sign a=b

and in second statement can you please explain how signs are also same ?
it is given la-bl=0... so a-b=0 or a-b=-0,(which is not true).. so a-b=0 or a=b same sign
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It's always true that |xy| = |x| * |y|, and that |x/y| = |x| / |y| .

So S1 tells us:

\(\begin{align}\\
\left| \frac{a}{b} \right| &= 1 \\\\
\frac{|a|}{|b|} &= 1 \\ \\
|a| &= |b| \\\\
|a| - |b| &= 0\\
\end{align}\)

so is sufficient.

For S2, the only number with an absolute value of 0 is 0 itself. So if |a-b| = 0 then a - b must equal 0. But then a = b, and |a| - |b| = 0, so S2 is also sufficient.

Or, for S2, if you know that |a-b| always means "the distance between a and b on the number line", then S2 tells us "the distance between a and b is 0", which means a and b are the same number.
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What is the value of |a| - |b|

(1) |a/b| = 1
A and b are the same number with different or same signs. The sign of a and the sign of b does not matter.
|a| - |b| still equals 0
sufficient

(2) |a - b| = 0
a and b are the same number, both pos or both neg. still equals 0
Sufficient

Answer: D

Hi adityadon,

Instead of trying to deal with these situations using a layered-'math' approach, you might find it easier to try thinking about real-world 'examples' to prove what's actually going on (TESTing VALUES):

We're asked for the value of |A| - |B|.

Fact 1: |A/B| = 1

Since this is an absolute value, we know that A/B = 1 OR A/B = -1

IF....
A = 2
B = 2
The answer to the question is |2| - |2| = 0

IF....
A = -2
B = 2
The answer to the question is |-2| - |2| = 0

Notice how the answer stays the SAME? As long as you're selecting values for A and B that fit the 'restrictions' in Fact 1, the answer will ALWAYS be 0.
Fact 1 is SUFFICIENT

Fact 2: |A-B| = 0

Since the result of this calculation is 0, the absolute value has NO impact on the math. To end up with 0, since B is subtracted from A, the ONLY possible situation that fits is that A=B (you can try TESTing VALUES here too....1 and 1, 2 and 2, 0 and 0, -3 and -3, etc.). Since the two variables are equal to one another, subtracting one from the other will ALWAYS = 0.
Fact 2 is SUFFICIENT

Final Answer:

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

Hi Rich,
Your approach is fantastic. However, I can't understand How |A/B| = 1. It can not be A/B= -1. I I think it should be only 1 ( and always positive).

Can you explain please in some detail? the modulus problems confuse me

Thanks

What are the values of |a/b| and a/b if a = 1 and b = -1?

a/b = -1, while |a/b| is still 1.
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Hi Mo2men,

The absolute value symbol turns negative "results" into positive results; the symbol does nothing to 0s and positive results.

For example:
|2| = 2
|0| = 0
|-2| = 2

When you have any type of calculation inside of an absolute value, then you have to do the calculation FIRST.... then if the result is negative then the absolute value symbol turns that result positive.

For example:

|X - Y|

If....X = 1 and Y = 4, we have....

|1-4| = |-3| = 3

In this prompt, we're dealing with an equation with an absolute value symbol in it:

|A/B| = 1

Since |1| = 1 AND |-1| = 1, we have two possibilities that we have to consider for A/B....

A/B = 1

OR

A/B = -1

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

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