Question

If the arithmetic mean of $a$ and $b$ is double their geometric mean, with $a>b>0$, then a possible value for the ratio $\frac{a}{b}$, to the nearest integer, is

• A. 5
• B. 8
• C. 11
• D. 14

Answer 1

Answer: (D)

We have $\frac{a+b}{2}=2\sqrt{ab}\Rightarrow a+b=4\sqrt{ab}$
Squaring both sides and rearranging $a^2-14ab+b^2=0$
Dividing the equation through by $b^2$ gives ${\left(\frac{a}{b}\right)}^2-14\left(\frac{a}{b}\right)+1=0$.
Using the quadratic equation, the roots are $7-4\sqrt{3}$ and $7+4\sqrt{3}$ which are about 0 and 14