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:
`(cookies)/(treats) = (5)/(3) = (x)/(144-x)`
Now we can solve for
`x`:

`(5)/(3) = (x)/(144-x)`

`5(144-x)=3x`

`720-5x=3x`

`8x=720`

`x= (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
`(3)/(5)` of her cookies, so her friend ate
`(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:

`(cookies)/(treats) = (6)/(1) = (54)/(90-x)`

`6(90-x)=54`

`540-6x=54`

`6x=486`

`x= (486)/(6)`

`x=81`

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