Question

Lydia baked a total of 144 chocolate chip cookies and peanut butter treats for Valentine's Day. Initially, the ratio of chocolate chip cookies to peanut butter treats was 5:3. After Lydia's friends ate 2/5 of her chocolate chip cookies and some of her peanut butter treats, the cookies outnumbered the treats 6 to 1. How many peanut butter treats did she have left?
Please show your work with your answer!

Answer 1

Let:
 x: chip cookies
 y: peanut butter
 We have the following system of equations:
 x + y = 144
 x / y = 5/3
 We solve the system of equations:
 Step 1:
 x + y = 144
 y = (3/5) x
 Step 2:
 x + (3/5) x = 144
 (8/5) x = 144
 x = (5/8) * (144) = 90
 Step 3:
 y = (3/5) x
 y = (3/5) * (90) = 54
 Lydia's friends ate 2/5 of her chocolate chip cookies:
 x '= (2/5) * (90) = 36
 the cookies outnumbered the treats 6 to 1
 x '/ y' = 6/1
 Clearing y '
 y '= (1/6) * x'
 y '= (1/6) * 36
 y '= 6
 Answer:
 she had left 6 butter treats

Answer 2

Let $x$ be the chocolate chip cookies, so the peanut butter treats will be $144-x$.
We know that the cookies and the treats are in a ratio of 5:3, so:$\frac{cookies}{treats} = \frac{5}{3} = \frac{x}{144-x}$
Now we can solve for $x$:
$\frac{5}{3} = \frac{x}{144-x}$
$5(144-x)=3x$
$720-5x=3x$
$8x=720$
$x= \frac{720}{8}$
$x=90$

We now know Lydia has 90 chocolate chip cookies, and $144-x=144-90=54$ peanut butter treats.

Then, Lydia's friend ate $\frac{3}{5}$ of her cookies, so her friend ate $\frac{2}{5} (90)=36$ cookies. Therefore, Lydia has $90-36=54$ cookies left.
Now we can calculate the total amount of baked goods after her friend ate the cookies:
$144-54=90$
Therefore, our remainder treats will be:
$90-x$

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:
$\frac{cookies}{treats} = \frac{6}{1} = \frac{54}{90-x}$
$6(90-x)=54$
$540-6x=54$
$6x=486$
$x= \frac{486}{6}$
$x=81$

Finally, if she ate 81 out 90 treats, we can conclude the Lydia has left with 9 peanut butter treats.