When reading a book, Charlie made a list by writing down the page number of the last page he finished reading at the end of each day. (He always finished reading a page that he started.) His mom thought his list indicated the amount of pages he had read on each day. At the end of the 8th day of reading, she added the numbers on his list and thought Charlie had read 432 pages. If Charlie started reading the book on page one, and he read the same amount of pages each day of this eight- day period, how many pages did he actually read by the end of the 8th day?

let p be the pages per day

p + 2 p + 3 p + ... + 8 p = 432

bruh solve for p then times it by 8

Let's assume that Charlie read the same number of pages each day, and let's call this number "x".

On the first day, Charlie read x pages, so the last page he finished reading would be page x.

On the second day, he read another x pages, so the last page he finished reading would be page x + x = 2x.

Similarly, on the third day, the last page he finished reading would be page 3x, and so on.

At the end of the 8th day, the last page he finished reading would be page 8x.

According to Charlie's mom, the sum of the page numbers on his list is 432.

Therefore, we can write the equation:
x + 2x + 3x + 4x + 5x + 6x + 7x + 8x = 432.

Simplifying this equation, we get:
36x = 432.

Dividing both sides of the equation by 36, we find:
x = 12.

Therefore, Charlie read 12 pages each day.

To find the total number of pages he read by the end of the 8th day, we can calculate:
Total pages = Number of days * Number of pages read per day
= 8 * 12
= 96.

Therefore, Charlie actually read 96 pages by the end of the 8th day.

To find out how many pages Charlie actually read by the end of the 8th day, we need to determine how many pages he read each day.

Let's denote the number of pages Charlie read each day as 'x.'

On the first day, Charlie started reading from page 1, so he read 'x' pages.
On the second day, Charlie started reading from page 'x + 1', so he read another 'x' pages.
Following this pattern, on the eighth day, Charlie started reading from page '7x + 1', so he read another 'x' pages.

Knowing this, we can set up an equation:

x + (x + 1) + (x + 2) + ... + (7x + 1) = 432

To solve this equation, we can simplify the left side by combining like terms:

1 + 2 + ... + 7 + x + x + x + ... + x = 432

There are 8 terms of 'x' in the equation since Charlie read the same number of pages each day for 8 days.

Simplifying further:

28x + 28 = 432

Now, we can isolate 'x' by subtracting 28 from both sides:

28x = 404

Finally, dividing both sides by 28 gives us:

x = 14.43 (rounded to two decimal places)

Since Charlie cannot read a fraction of a page, we can conclude that he read 14 pages each day during the 8-day period.

Therefore, the total number of pages he actually read by the end of the 8th day is:

8 days * 14 pages/day = 112 pages.