GMAT Club World Cup 2022 (DAY 3): If a + b + c + d = 12, what is the

Statement 1
ab = cd = ad
=> ab = cd ---- (eq i)
=> b = d ---- (eq ii)
=> a = c ---- (eq iii)

Now we can try a few values, lets say
a = c = 3
b = d = 3
Also ab = cd = 9 is satisfied
Then we have \(\sqrt{a^2+b^2+c^2+d^2}\) = \(\sqrt{3^2 * 4}\) = 6

Plugging different values we get
a = c = 6
b = d = 0
Also ab = cd = 0 is satisfied
Then we have \(\sqrt{a^2+b^2+c^2+d^2}\) = \(\sqrt{6^2 * 2}\) = \(\sqrt{72}\)

Getting different values hence Insufficient

Statement 2
We already know that a + b + c + d = 12, then we have |a| = |b| = |c| = |d|
The only value that satisfies these two equations is 3
Hence we know the value of
\(\sqrt{a^2+b^2+c^2+d^2}\) = \(\sqrt{3^2 * 4}\) = 6
Sufficient

IMHO Option B

Given information: a+b+c+d=12
To determine value of: \(\sqrt{a^2+b^2+c^2+d^2}\)

(1) \(ab = cd = ad\)

Best way to go about this is to test values

a b c d ab cd ad \(\sqrt{a^2+b^2+c^2+d^2}\)
3 3 3 3 9 9 9 6
4 2 4 2 8 8 8 Root(40)

NOT SUFFICIENT

(2) \(|a| = |b| = |c| = |d|\)


a b c d \(\sqrt{a^2+b^2+c^2+d^2}\)
3 3 3 3 6
6 6 6 -6 12

NOT SUFFICIENT

(1) + (2) together:

Now combining all information together, we have:
a) ab=cd=ad
b) a + b + c + d=12
c) |a|=|b|=|c|=|d|

From the above, we can infer the following cases
a) a, b, c and d all have the same absolute value
b) Either they are all negative (NOT POSSIBLE because then original sum=12 condition will not satisfy)
c) a and c are negative (NOT POSSIBLE because all same absolute value and 2 negative will lead to sum as 0, not 12)
d) b and d are negative (NOT POSSIBLE same reason as c)
e) All positive and equal in value (Only possible and viable option)

So based on this, we can say that the only values that fit are when each a, b, c and d are all equal to 3, which will give us 1 definite answer

SUFFICIENT

Answer - C

we have a+b+c+d=12
Let's elevate to square2 : ( a+b+c+d=12)^2=144
=> (a^2+b^2+c^2+d^2) +(2ab+2ac+2bc+2ad+2bd+2cd)=144

(1) ab=cd=ad
if a is not Zero then :b=d and a=c
=>(a^2+b^2+c^2+d^2) + (8ab+4a^2+4b^2)=144
=>(a^2+b^2+c^2+d^2) + 4(a+b)^2=144 (*)
No info is available about a and b to conclude the value of sqrt of (a^2+b^2+c^2+d^2
statement 1 is not sufficient

(2) |a|=|b|=|c|=|d|
since (|a|)^2 = a^2
from (*) we can conclude that : 4a^2+16a^2=144 => we can conclude the value of a and then conclude the value of the other b, c and d
and hence conclude the value of sqrt of (a^2+b^2+c^2+d^2

Second statement is sufficiant

Answer is B

My answer is B.
The attached file indicates the solution

IMO E

Given a+b+c+d=12 --- (1)

Statement 1: ab = cd=ad
Take ab= ad, which gives either a=0 or b=d
Similarly take cd=ad, which gives either d=0 or c=a
Insufficient in all ways.

Statement 2: |a|=|b|=|c|=|d|

Using the property |a| = root(a^2) and statement 2, the original question is reduced to Root(4a^2)
Using equation (1) we can not find the value of a as we know that the magnitude is equal for all but not sign.
Insufficient.

Combining also doesn't give any specific value of a.

If a+b+c+d = 12 , what is the value of √a2+b2+c2+d2 ?

(1) ab=cd=ad

(2) |a|=|b|=|c|=|d|


correct ans is c

statement 2 states that a2 = b2= c2 = d2
hence the expression becomes = 2a

not sufficient

statement 1 states
product of 2 element is constant

insufficient


staement 1 and 2 to0gether

sufficient

I think answer is C.

Given that a+b+c+d=12.

Statement 1: ab=cd=ad

ab=ad means either a=0 or b=d or both.
cd=ad means either d=0 or a=c or both.
There are multiple options for a, b, c and d and so there is no unique solution.

Statement 2: |a|=|b|=|c|=|d|

Possibility 1- 6+6+6-6=12
Possibility 2- 3+3+3+3=12
Two different answers. Hence, statement 2 alone not sufficient.

On combining both statements, b=d and a=c. Only unique answer will be 3+3+3+3=12
Hence, together sufficient and answer will be C.
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Bunuel please explain why the OA should not be A. Where I was wrong?

For (1) we can have many other cases. For example, a = c = d = 0 and b = 12, gives \(\sqrt{a^2+b^2+c^2+d^2}=12\). Or, a = d = 0 and b = c = 6, gives \(\sqrt{a^2+b^2+c^2+d^2}=6\sqrt{2}\).
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Bunuel please see the highlighted portion in green.