Let [tex]x[/tex] be the chocolate chip cookies, so the peanut butter treats will be [tex]144-x[/tex].
We know that the cookies and the treats are in a ratio of 5:3, so:[tex] \frac{cookies}{treats} = \frac{5}{3} = \frac{x}{144-x} [/tex]
Now we can solve for [tex]x[/tex]:
[tex] \frac{5}{3} = \frac{x}{144-x} [/tex]
[tex]5(144-x)=3x[/tex]
[tex]720-5x=3x[/tex]
[tex]8x=720[/tex]
[tex]x= \frac{720}{8} [/tex]
[tex]x=90[/tex]
We now know Lydia has 90 chocolate chip cookies, and [tex]144-x=144-90=54[/tex] peanut butter treats.
Then, Lydia's friend ate [tex] \frac{3}{5} [/tex] of her cookies, so her friend ate [tex] \frac{2}{5} (90)=36[/tex] cookies. Therefore, Lydia has [tex]90-36=54[/tex] cookies left.
Now we can calculate the total amount of baked goods after her friend ate the cookies:
[tex]144-54=90[/tex]
Therefore, our remainder treats will be:
[tex]90-x[/tex]
We also now that after her friend ate 54 cookies and some treats, the new ratio is 6:1, and that's all we need to set up our new equation and solve it to find how many treats she ate:
[tex] \frac{cookies}{treats} = \frac{6}{1} = \frac{54}{90-x} [/tex]
[tex]6(90-x)=54[/tex]
[tex]540-6x=54[/tex]
[tex]6x=486[/tex]
[tex]x= \frac{486}{6} [/tex]
[tex]x=81[/tex]
Finally, if she ate 81 out 90 treats, we can conclude the Lydia has left with 9 peanut butter treats.
