Final answer:
To find the value of k, we can set up equations based on the given information. By solving these equations, we can determine that the value of k is 5.
Step-by-step explanation:
To find the value of k, we need to set up equations based on the given information.
From the first investment, Lana invested $40 and received a return of k times that amount. So, her return was $40 x k.
From Betty's investment, she initially invested $160 and received a return of k times that amount. So, her total return was $160 x k.
In the second investment, Betty invested her $160 plus her earnings ($160 x k). So, her total investment for the second round was $160 + ($160 x k).
Lana invested her initial investment ($40) plus her earnings ($40 x k) and the remaining $120. So, her total investment for the second round was $40 + ($40 x k) + $120.
According to the given information, in the end, Betty had twice the amount Lana had. So, we can set up an equation:
$160 x k = 2($40 + ($40 x k) + $120)
Simplifying, we get:
$160 x k = 2($160 + ($40 x k) + $120)
Now we can solve for k.
Dividing both sides by $160, we get:
k = 2($160 + ($40 x k) + $120) / $160
Further simplifying, we have:
k = 2 + 0.5k + 0.75
Subtracting 0.5k from both sides, we get:
k - 0.5k = 2 + 0.75
0.5k = 2.75
Dividing by 0.5, we get:
k = 2.75 / 0.5
Therefore, the value of k is 5.