Fiona and camila took a load of peaches to the farmers market. During the first hour , they sold 1/2 of the peaches plus 1/2 of a peach. In the second hour, they sold 1/3 of the remaining peaches plus 1/3 of a peach. In the third hour , they sold 1/4 of what was left and 3/4 of a peach. During the final hour, they sold 1/5 of the peaches that were left plus 1/5 of a peach. Fiona and camila did not cut any peaches to make a sale, and they returned home with 19 whole peaches. How many peaches did they take to the market?

I will assume you have a typo and on the third day she sold 1/4 of what was left and 1/4 of a peach, or else it does not follow the pattern

let x be the number of peaches at the start.
number sold in 1st hour = x/2 + 1/2 = (x+1)/2
number left after 1 hour = x - (x+1)/2
= (2x - x - 1)/2 = (x-1)/2

number sold in 2nd hour = (x-1)/2÷3 + 1/3
= (x-1)/6 + 2/6) = (x+1)/6
number left after 2 hours = (x-1)/2 - (x+1)/6
= ( 3x-3 - x - 1)/6
= (2x - 4)/6
= (x-2)/3

number sold in 3rd hour = (x-2)/3 ÷ 4 +1/4
= (x-2)/12 + 3/12
= (x+1)/12
number left after 3 hrs = (x-2)/3 - (x+1)/12
= (4x-8 - x - 1)/12
= (3x-9)/12
= (x - 3)/4

MMMHH, I see a pattern, could the amount left after 4 hrs be ( x-4)/5 ??
let's see

number sold in 4th hr = (x-3)/4 ÷ 5 + 1/5
= (x-3)/20 + 4/20
= (x+1)/20
number left after 4 hrs = (x-3)/4 - (x+1)/20
= (5x-15 - x - 1)/20
= (4x-16)/20
= (x - 4)/5 , as predicted

so (x-4)/5 = 19
x - 4 = 95
x = 103

check:
day 1: sold 103/2 + 1/2 = 52
peaches left after day 1 = 103-52 = 51

day2: sold 51/3 + 1/3 = 17 1/3
left with 51 - 17 1/3 = 101/3

day 3: sold (1/4)(101/3) + 1/4 = 26/3
left with 101/3 - 26/3 = 25

day 4: sold 25/5 + 1/5 = 6 1/5
left with 25 - 6 1/5 or 18 4/5

ARgghhhh!!!!!

Where is my error ???

To solve this problem, we can work backwards by starting with the final number of peaches and figuring out how many peaches they had at each step.

Let's assume that the number of peaches they had at the beginning was 'x'.

During the first hour, they sold 1/2 of the peaches plus 1/2 of a peach. So, they had (x - 1/2 - 1/2) peaches remaining.

In the second hour, they sold 1/3 of the remaining peaches plus 1/3 of a peach. So, they had ((x - 1/2 - 1/2) - 1/3 - 1/3) peaches remaining.

In the third hour, they sold 1/4 of what was left and 3/4 of a peach. So, they had (((x - 1/2 - 1/2) - 1/3 - 1/3) - 1/4 - 3/4) peaches remaining.

Finally, during the last hour, they sold 1/5 of the peaches that were left plus 1/5 of a peach. So, they had ((((x - 1/2 - 1/2) - 1/3 - 1/3) - 1/4 - 3/4) - 1/5 - 1/5) peaches remaining.

According to the problem, they returned home with 19 whole peaches. So we can set up the equation:

((((x - 1/2 - 1/2) - 1/3 - 1/3) - 1/4 - 3/4) - 1/5 - 1/5) = 19

Now we can solve this equation to find the value of 'x'.