What is the value of |a + b|?

Statement 1: (a + b + c + d) (a + b – c – d) = 16

i.e. [(a + b) + (c + d)] [(a + b) – (c + d)] = 16

i.e. \((a + b)^2 – (c + d)^2 = 16\)

No information about (c+d) hence we can't infer the value of (a+b)

Hence, NOT SUFFICIENT

Statement 2: c + d = 3

No information about (a+b) hence we can't infer the value of (a+b)

Hence, NOT SUFFICIENT

Combining the two statements:

i.e. \((a + b)^2 – (c + d)^2 = 16\) AND c + d = 3

i.e. \((a + b)^2 – (3)^2 = 16\)

i.e. \((a + b)^2 = 16+9\)
i.e. \((a + b)^2 = 25\)

i.e. \((a + b) = +5\)

But anyways, |a + b| = 5

Hence, SUFFICIENT

Answer: Option

1) Insufficient because there is a lot of variants possible
2) Insufficient because we know nothing about a and b

1+2) if c + d = 3 than -c-d = -3
so we can rewrite first statement as
\((a+b+3)(a+b-3)=16\)
\((a+b)^2-9=16\)
\((a+b)^2=25\)
\(|a+b|=5\)
Sufficient
Answer is C
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Hi All,

This is a layered, scary-"looking" DS question, but if you organize the given information in a certain way, you can take advantage of the built-in patterns and get to the correct answer without too much trouble (you will need to take a bunch of notes though and TESTing VALUES will help).

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

Fact 1: (A + B + C + D) (A + B – C – D) = 16

The equation we have here is certainly complicated, but there are some patterns in it worth noting:
1) (A+B) exists in both parentheses
2) (C+D) exists in the both parentheses (in the 1st, it's added; in the 2nd, it's subtracted)
3) The product is +16, so the parentheses are either BOTH positive OR BOTH negative.

IF...
A = 4
B=C=D=0
(4+0+0+0)(4+0+0+0) = 16
The answer to the question is |4+0| = 4

For the second TEST, I'm going to actually consider the information in Fact 2....what IF... C+D = 3....how would that impact the equation in Fact 1.....

IF...
A = 4
B = 1
C+D = 3
(4+1+3)(4+1-3) = (8)(2) = 16
The answer to the question is |4+1| = 5
Fact 1 is INSUFFICIENT

Fact 2: C+D = 3

This tells us NOTHING about A or B.
Fact 2 is INSUFFICIENT

Combined, we know....
(A + B + C + D) (A + B – C – D) = 16
C+D = 3

Substituting in, we have...
(A+B+3)(A+B-3) = 16

There aren't that many ways to end up with a product of 16 under these circumstances. From our prior TESTs, we know that A+B = 5 is one possibility....

To find the other possibility, I'm going to refer to (A+B) as X....now the given equation is...
(X+3)(X-3) = 16

FOILing this out gives us...
X^2 - 9 = 16
X^2 = 25
X = 5 or -5

Thus the only OTHER possibility is if A+B = -5.
|5| = 5
|-5| = 5
The answer to the question is ALWAYS 5.
Combined, SUFFICIENT

Final Answer:

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

Whenever we see a or b, we always see "a+b", and whenever we see c or d, we see "c+d" (or "-c-d" which is just -(c+d) ). So we can make everything look a lot simpler if we just let k = a+b and m = c+d and rephrase the entire question:

What is |k| ?

1. (k + m)(k - m) = 16
2. m = 3

Then each statement is clearly insufficient alone, and using both statements, noticing the factors on the left side of S1 are in the difference of squares pattern, we find k^2 - 9 = 16, so k^2 = 25 and |k| = 5, so the answer is C.

1. \((a + b + c + d) (a + b – c – d) = 16\) --> \((a + b + c + d) (a + b - (c + d)) = 16\). Not sufficient since c+d could be 0, meaning |a+b| is 4. If c+d is 1, we have \((a + b + 1) (a + b - (1)) = 16\) --> \((a + b)^2 - 1 = 16\) --> \((a + b)^2= 17\) so not sufficient.
2. Nothing about a or b so not sufficient.

Together: \((a + b + 3) (a + b - 3) = 16\) --> \((a + b)^2 - 9 = 16\) --> \((a + b)^2= 25\) so sufficient, the answer is C.

ST-1
(A+B+C+D)(A+B-C-D)=16
Now we can see the above expression can be written as (A+B)+(C+D) X (A+B)-(C+D) = (A+B)^2 - (C+D)^2
So we got to this expression but cannot solve it further, hence NS

ST-2
C+D=3
NS

Combining ST1 AND ST2 we get
(A+B)^2 - 9 =16
(A+B)^2 = 25
(A+B) = +/-5
|A+B|=5

Answer C

MANHATTAN GMAT OFFICIAL SOLUTION:

We are asked for the absolute value of a + b, so we will try to manipulate the statements to isolate that combination of variables, a + b. We will start with the easier statement, which in this case is Statement (2).

(2) INSUFFICIENT: This gives us information about c and d, and the relationship between them, but no information about a or b.

(1) INSUFFICIENT: We can manipulate the equation to group (a + b) and (c + d) terms:
(a + b + c + d) (a + b – c – d) = 16
[(a + b) + (c + d)][(a + b) – (c + d)] = 16

Note that this is of the form (x + y)(x – y), where x = (a + b) and y = (c + d). We recognize this as the “difference of two squares” special product, (x + y)(x – y) = x^2 – y^2. Thus, we may transform this expression:
[(a + b) + (c + d)][(a + b) – (c + d)] = 16
(a + b)^2 – (c + d)^2 = 16
(a + b)^2 = 16 + (c + d)^2

We don't know the value of c + d, so we cannot determine the value of (a + b)^2.

(1) AND (2) SUFFICIENT: From statement (2), we know that (a + b)^2 = 16 + (c + d)^2. From statement (1) we know that c + d = 3. Substituting for c + d:
(a + b)^2 = 16 + (c + d)^2
(a + b)^2 = 16 + 3^2
(a + b)^2 = 16 + 9
(a + b)^2 = 25
(a + b) = 5 or –5
|a + b| = 5

The correct answer is C.
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Good question.
replacing c+d by 3 in both the brackets solves this.
I. (a+b+c+d) (a+b-(c+d)) = 16
let a+b=x.
II. c+d = 3
(x + 3) ((x-3) = 16. therefore, x = 5 = (a+b)
suffient both. C

Good question.
replacing c+d by 3 in both the brackets solves this.
I. (a+b+c+d) (a+b-(c+d)) = 16
let a+b=x.
II. c+d = 3
(x + 3) ((x-3) = 16. therefore, x = 5 = (a+b)
suffient both. C

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 2 variables ( a and b ) and 0 equations, C is most likely to be the answer.
Thus, we need to consider both conditions together first.

\((a+b+c+d) (a+b–c–d) = 16\)
\({(a+b)+(c+d)}{(a+b)–(c+d)} = 16\)
\((a+b)^2 - (c+d)^2 = 16\)
\((a+b)^2 = 25\) since \(c + d = 3\)
\(|a+b| = 5\)

They are sufficient.

By CMT(Common Mistake Type)4, we need to check A or B and consider each condition only.

Condition 1)
\((a+b+c+d) (a+b–c–d) = 16\)
\({(a+b)+(c+d)}{(a+b)–(c+d)} = 16\)
\((a+b)^2 - (c+d)^2 = 16\)
This is not sufficient, since we don't know \(c+d\).

Condition 2)
We don't have anything about \(a\) and \(b\).
This is not sufficient either.

Therefore, the answer is C.

Normally, in problems which require 2 or more additional equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E).

Let a+b = x and c+d = y Then, |a+b|

(1) (a+b+c+d)(a+b-c-d) = 16

so (x+y)(x-y) = 16 2 variables one equation NS

(2) c+d = 3 so y = 3 What about x? NS

(1) and (2) y =3 and (x+y)(x-y)=16 so (x+3)(x-3) = 16 so x^2 - 9=16 and x^2 = 25 so x=+-5 and |x|=5 so |a+b| =5 sufficient. OA is C

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