What is the value of b + c ?

Hey dave13

To begin with, \(b(a+d)+c(a+d)\) can also be written as \((b+c)(a+d)\)
because if you take (a+d) as common, you will get the expression (b+c)

Whenever you have trouble with such expression - put some random values for a,b,c, and d

If a=1,b=2,c=3, and d=4
The right-hand side \(b(a+d)+c(a+d)\) is \(2(1+4) + 3(1+4) = 2*5 + 3*5 = 25\)
Similarly,the left hand side \((b+c)(a+d)\) is \((2+3)(1+4) = 5*5 = 25\) and we can prove the same

Coming to the second part of your question, in Bunuel's solution it has been given
If \(a+d=1\) then \(b+c=6\) since in statement 1 it is written that \((a+d)(b+c) = 6\)
The reason it is not sufficient is there is a second possibility when \(a+d=6\) and \(b+c=1\)
So, (b+c) can have two values - 1 or 6

Hope this helps clear your confusion!

Target question: What is the value of b + c ?

Statement 1: ab + cd + ac + bd = 6
Rearrange terms: ab + ac + bd + cd = 6
Factor the a out of the first two terms to get: a(b + c) + bd + cd = 6
Factor the d out of the last two terms to get: a(b + c) + d(b + c) = 6
Rewrite as: (a + d)(b + c) = 6

There are infinitely many values of a, b, c and d that satisfy the equation (a + d)(b + c) = 6. Here are two:
Case a: a + d = 1 and b + c = 6. In this case, the answer to the target question is b + c = 6
Case b: a + d = 2 and b + c = 3. In this case, the answer to the target question is b + c = 3
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: a + d = 4
Since we're given NO INFORMATION about b and c, we cannot answer the target question with certainty.
So, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that (a + d)(b + c) = 6
Statement 2 tells us that a + d = 4

Take the first equation and replace a + d with 4
We get: (4)(b + c) = 6
So, is MUST be the case that b + c = 1.5
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer: C

Cheers,
Brent

What is the value of b + c?

(1) ab + cd + ac + bd = 6
Four variables, so we will not get our answer but let's simplify the equation..
\(ab+cd+ac+bd=6..........ab+ac+cd+bd=6.........a(b+c)+d (b+c)=6.........(a+d)(b+c)=6\)
Insufficient

(2) a + d = 4
Clearly insufficient

Combined
\((a+d)(b+c)=6\)
So if a+d=4, 4*(c+b)=6......c+b=6/4=3/2
Sufficient

C

Question: b+c = ?

Statement 1: ab + cd + ac + bd = 6
i.e. (a+d)*(b+c)=6
NOT SUFFICIENT

Statement 2: a + d = 4
NOT SUFFICIENT

Combining the two statements
(a+d)*(b+c)=6
i.e. b+c = 6/4
SUFFICIENT

Answer: Option C

What is the value of b + c ?

(1) ab + cd + ac + bd = 6
(a+d)(b+c) = 6
Since the value of (a+d) is unknown
NOT SUFFICIENT

(2) a + d = 4
NOT SUFFICIENT

(1) + (2)
(1) ab + cd + ac + bd = 6
(a+d)(b+c) = 6
(2) a + d = 4
b+c = 6/4 = 3/2 = 1.5
SUFFICIENT

IMO C

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