danzig wrote:

The ratio of \(2 + \sqrt{6}\) to \(3\) is approximately equal to which of the following ratios?

A. 2 to 1

B. 3 to 2

C. 4 to 3

D. 7 to 6

E. 13 to 12

A fast method to solve it? Thanks!

fast method revolves around estimation : 2*2 = 4 and 3*3 = 9 so sq root of 6 will fall between 2 and 3. To further gauze the same : 2.4*2.4=5.76 while 2.5*2.5 = 6.25 so we know we are looking for ratio slightly greater than 2+2.4/3 i.e. 4.4/3<x<4.5/3 fortunately 4.5 is a multiple of 3 so we can write 4.4/3<x<1.5 i.e. very close to 1.5

Now A is too big, B is close we know our ratio is near1.5 i.e 3 to 2 so wait until we find a better fit, C is definitely far away than is 1.5 so c is out.

D: its basically 3.5 to 3 and so this too is far wide than B.

E: denomitar is a multiplied by 4 will be numerator, it means we are talking about 13/4 i.e. 3.something so E is out.

SO B stands out