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?

Question

Reiny · Answer

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 ???