# Common ratio problems

• Mar 2nd 2010, 04:50 PM
nightrider456
Common ratio problems
Been having problems with the solutions to the following problems dealing with finding the common ratio.

Solve for the missing variable and find the common ratio
1.) $
\sqrt{x},3,3\sqrt{x}
$

2.) $
m+2,m+4,2m+11
$
• Mar 2nd 2010, 08:05 PM
Soroban
Hello, nightrider456!

If we are given three terms $a,b,c$ with a common ratio,

. . then: . $\frac{a}{b} \:=\:\frac{b}{c}$

Quote:

Solve for the variable and find the common ratio.

. . $(1)\;\;\sqrt{x},\;3,\;3\sqrt{x}$

We have: . $\frac{\sqrt{x}}{3} \;=\;\frac{3}{3\sqrt{x}} \quad\Rightarrow\quad 3x \:=\:9 \quad\Rightarrow\quad x \:=\:3$

The three terms are: . $\sqrt{3},\;3,\;3\sqrt{3}$

. . The common ratio is: . $\sqrt{3}$

Quote:

$(2)\;\;m+2,\;m+4,\;2m+11$

We have: . $\frac{m+2}{m+4} \:=\:\frac{m+4}{2m+11} \quad\Rightarrow\quad 2m^2 + 15m + 22 \:=\:m^2 + 8m + 16$

. . $m^2 + 7m + 6 \:=\:0 \quad\Rightarrow\quad (m+1)(m+6) \:=\:0$

Hence: . $m \;=\;-1,\:-6$ . . . two solutions

If $m = -1$, the three terms are: . $1,\:3,\:9$
. . The common ratio is: . $3$

If $m = -6$, the three terms are: . $-4,\:-2,\:-1$
. . The common ratio is: . $\tfrac{1}{2}$