, 2021

We work backwards through the given information.

At the end, there is 1 candy remaining.

Since $\frac{5}{6}$ of the candies are removed on the fifth day, this 1 candy represents $\frac{1}{6}$ of the candies left at the end of the fourth day.

Thus, there were $6 \times 1 = 6$ candies left at the end of the fourth day.

Since $\frac{4}{5}$ of the candies are removed on the fourth day, these 6 candies represent $\frac{1}{5}$ of the candies left at the end of the third day.

Thus, there were $5 \times 6 = 30$ candies left at the end of the third day.

Since $\frac{3}{4}$ of the candies are removed on the third day, these 30 candies represent $\frac{1}{4}$ of the candies left at the end of the second day.

Thus, there were $4 \times 30 = 120$ candies left at the end of the second day.

Since $\frac{2}{3}$ of the candies are removed on the second day, these 120 candies represent $\frac{1}{3}$ of the candies left at the end of the first day.

Thus, there were $3 \times 120 = 360$ candies left at the end of the first day.

Since $\frac{1}{2}$ of the candies are removed on the first day, these 360 candies represent $\frac{1}{2}$ of the candies initially in the bag.

Thus, there were $2 \times 360 = 720$ in the bag at the beginning.

Solution 1

We work backwards through the given information.

At the end, there is 1 candy remaining.

Since $\frac{5}{6}$ of the candies are removed on the fifth day, this 1 candy represents $\frac{1}{6}$ of the candies left at the end of the fourth day.

Thus, there were $6 \times 1 = 6$ candies left at the end of the fourth day.

Since $\frac{4}{5}$ of the candies are removed on the fourth day, these 6 candies represent $\frac{1}{5}$ of the candies left at the end of the third day.

Thus, there were $5 \times 6 = 30$ candies left at the end of the third day.

Since $\frac{3}{4}$ of the candies are removed on the third day, these 30 candies represent $\frac{1}{4}$ of the candies left at the end of the second day.

Thus, there were $4 \times 30 = 120$ candies left at the end of the second day.

Since $\frac{2}{3}$ of the candies are removed on the second day, these 120 candies represent $\frac{1}{3}$ of the candies left at the end of the first day.

Thus, there were $3 \times 120 = 360$ candies left at the end of the first day.

Since $\frac{1}{2}$ of the candies are removed on the first day, these 360 candies represent $\frac{1}{2}$ of the candies initially in the bag.

Thus, there were $2 \times 360 = 720$ in the bag at the beginning.

Solution 2

Suppose that there were $x$ candies in the bag at the beginning.

On the first day, $\frac{1}{2}$ of the candies are eaten, which means that $1 - \frac{1}{2} = \frac{1}{2}$ of the candies remain.

Since there were $x$ candies at the beginning of the first day, there are $\frac{1}{2}x$ candies at the end of the first day.

On the second day, $\frac{2}{3}$ of the remaining candies are eaten, which means that $1 - \frac{2}{3} = \frac{1}{3}$ of the candies from the beginning of the day remain at the end of the day.

Since there were $\frac{1}{2}x$ candies at the beginning of the second day, there are $\frac{1}{3} \times \frac{1}{2}x = \frac{1}{6}x$ candies at the end of the second day.

On the third day, $\frac{3}{4}$ of the remaining candies are eaten, which means that $1 - \frac{3}{4} = \frac{1}{4}$ of the candies from the beginning of the day remain at the end of the day.

Since there were $\frac{1}{6}x$ candies at the beginning of the third day, there are $\frac{1}{4} \times \frac{1}{6}x = \frac{1}{24}x$ candies at the end of the third day.

On the fourth day, $\frac{4}{5}$ of the remaining candies are eaten, which means that $1 - \frac{4}{5} = \frac{1}{5}$ of the candies from the beginning of the day remain at the end of the day.

Since there were $\frac{1}{24}x$ candies at the beginning of the fourth day, there are $\frac{1}{5} \times \frac{1}{24}x = \frac{1}{120}x$ candies at the end of the fourth day.

On the fifth day, $\frac{5}{6}$ of the remaining candies are eaten, which means that $1 - \frac{5}{6} = \frac{1}{6}$ of the candies from the beginning of the day remain at the end of the day.

Since there were $\frac{1}{120}x$ candies at the beginning of the fifth day, there are $\frac{1}{6} \times \frac{1}{120}x = \frac{1}{720}x$ candies at the end of the fifth day.

Since 1 candy remains, then $\frac{1}{720}x = 1$ which gives $x = 720$, and so there were 720 candies in the bag before the first day.