danzig wrote:

Every member of Meg’s immediate family agrees to share equally the cost of her wedding, which is $18,000 in total. How many people are in Meg’s immediate family?

(1) Everyone in Meg’s immediate family will pay $1,500.

(2) If 4 immediate family members do not contribute their share, each of the other family members will have to contribute $750 more than if everyone had contributed.

I agree with the OA. However, the statement # 2 requires to solve a quadratic equation, which requires more time. Is there a way to know that I will get just one valid answer in that quadratic equation? That would save time instead of solving the equation. Thanks!

Wedding cost = 18000

Number of family members = N

Share of each member=S

(18000/N)=S

Question is What is N? So we need value of either S or N.

S1) S=1500 so N=12 Sufficient.

S2) (18000/(N-4))-750=18000/N ------> 3000N-750N(sq)+72000=0 ------> N(sq)-4N-96=0 ------> (N-12)(N+8)=0 ------> N=12 (N can not be negative) sufficient.

as for your query about

Is there a way to know that I will get just one valid answer in that quadratic equation? consider the following characteristics of quadratic equation.

In any quadratic equation ax(sq)+bx+c=0

if b(sq)-4ac=0 roots are real and equal.

if b(sq)-4ac>0 roots are real and distinct.

if b(sq)-4ac<0 roots are imaginary.

further if the sign of C is negative then the roots will have opposite signs.

so in our equation N(sq)-4N-96=0 C is negative and b(sq)-4ac>0 so you can conclude at this point - without solving the equation - that you will get real and distinct roots of opposite signs which ultimately means N has one positive & one negative value. Now here negative value has no any meaning because number of family members can not be negative. So with positive value of N we can reach to definite answer and hence statement is sufficient.

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