Problem Solving

[GMAT math practice question]


a, b and c are positive and a < 5. a, b and c satisfy a^2 - a - 2b - 2c = 0, and a + 2b - 2c + 3 = 0. What is the order among a, b and c?

A. c > a > b
B. a > c > b
C. a > b > c
D. c > b > a
E. b > c > a

Max@Math Revolution wrote:[GMAT math practice question]


a, b and c are positive and a < 5. a, b and c satisfy a^2 - a - 2b - 2c = 0, and a + 2b - 2c + 3 = 0. What is the order among a, b and c?

A. c > a > b
B. a > c > b
C. a > b > c
D. c > b > a
E. b > c > a

Let equations a^2 -a-2b-2c =0 - 1 and a+2b-2c+3=0 - 2
Adding 1 and 2. we get

a^2 = 4c-1

hence, for
a=1, c=1
a=2, c=7/4
a=3, c=3
a=4, c=19/4

Subtracting 2 from 1, we get
a(a-2) = 4b+3
Hence , for
a=1, b=-1
a=2, b=-3/4
a=3, b=0
a=4, b=5/4

Given a, b and c are positive and a<5 , so a=4 only. When a=1,2 and 3 . b is either negative or 0.

so c>a>b

The Answer is A

Would like to know if there is quicker method to solve this question.

=>

When we add the two equations a^2 - a - 2b - 2c = 0 and a + 2b - 2c + 3 = 0, we have a^2 - a - 2b - 2c + a + 2b - 2c + 3 = 0 + 0, a^2 - 4c + 3 = 0 or 4c = a^2 + 3.
When we subtract 4a from the both sides of the last equation, we have 4c - 4a = a^2 - 4a + 3, 4(c - a) = (a - 1)(a - 3).
Rearranging the equations given in the question gives us a^2 - a = 2(b + c) and a + 3 = 2(c - b) given in the question. When we subtract a + 3 = 2(c - b) from a^2 - a = 2(b + c), we have a^2 - a - (a + 3) = 2(b + c) - 2(c - b), a^2 - a - a - 3 = 2b + 2c - 2c +2b, 4b = a^2 - 2a - 3, 4b = (a - 3)(a + 1) > 0 and a > 3.
We have 4(c - a) = (a - 1)(a - 3) > 0 or c > a since a > 3 and (a - 1)(a - 3) > 0.

We have 4(b - a) = 4b - 4a and 4b - 4a = a^2 - 2a - 3 - 4a (because 4b = a^2 - 2a - 3). Then 4b - 4a = a^2 - 6a - 3 = a(a - 3) - 3(a - 3) - 12 = (a - 3)^2 - 12 < 0 or b < a since 3 < a < 5.
Thus, we have c > a > b.

Therefore, A is the answer.
Answer: A