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

GMAT CLUB Official Explanation:

If \(a + b + c + d = 12\), what is the value of \(\sqrt{a^2+b^2+c^2+d^2}\) ?

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

It's so tempting to think that \(a=b=c=d=3\). In this case \(\sqrt{a^2+b^2+c^2+d^2}=\sqrt{36}=6\) It's certainly a possibility but is it the only one? Let's check:

Well, notice that if any three of the unknowns is 0, then \(ab = cd = ad\) holds and from \(a + b + c + d = 12\) it would follow that the fourth unknown must be 12. For example, \(a=b=c=0\) and \(d=12\). In this case \(\sqrt{a^2+b^2+c^2+d^2}=\sqrt{12^2}=12\)

So, this statement is not sufficient.

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

We can have the following two ases:

I. If all four of the unknowns are positive, then we'd have \(a = b = c = d\). In this case \(a + b + c + d = 4a=12\), which gives \(a = b = c = d=3\) and thus \(\sqrt{a^2+b^2+c^2+d^2}=\sqrt{36}=6\).

II. If three of the unknowns are positive and the fourth one is negative, then we'd have \(a = b = c = -d\) (it does not matter which three are positive, so we can assign any). In this case \(a + b + c + d = 2a=12\), which gives \(a = b = c = -d=6\) and thus \(\sqrt{a^2+b^2+c^2+d^2}=\sqrt{4*36}=12\).

Two different answers. Not sufficient.

(1)+(2) Case II from (2) cannot be true because if one of the unknowns is negative and the other three are positive, then \(ab = cd = ad\) won't hold true: in this case \(ab\), \(cd\), or \(ad\) will be negative and the remaining two expressions will be positive. Thus, we have case I from (2), which means that \(a = b = c = d=3\) and thus \(\sqrt{a^2+b^2+c^2+d^2}=\sqrt{36}=6\). Sufficient.


Answer: C