At first, on a bookshelf, 7/10 of the books were non-fiction books and the rest were fiction books. 1/3 of the fiction books and 1/7 of the non-fiction books were
removed. Twelve non-fiction books were removed.
How many books were left on the bookshelf altogether?
Let's start by finding the total number of books on the bookshelf. We know that 7/10 of the books were non-fiction, so 3/10 of the books must be fiction.
Let B be the total number of books on the shelf:
Non-fiction books = 7/10B
Fiction books = 3/10B
We also know that 1/3 of the fiction books were removed, so the number of fiction books left on the shelf is:
2/3 x 3/10B = 2/15B
And 1/7 of the non-fiction books were removed, which means the number of non-fiction books left is:
6/7 x 7/10B = 6/25B
We're told that 12 non-fiction books were removed, so we can set up an equation to solve for B:
7/10B - 12 = 6/25B + 2/15B
Multiplying both sides by 750 (the least common multiple of 10, 25, and 15) to eliminate the fractions:
525B - 9000 = 180B + 200B
Combining like terms:
525B - 9000 = 380B
Adding 9000 to both sides:
525B = 13800
Dividing both sides by 525:
B = 26.29
Since we can't have a fraction of a book, we'll round up to 27. So there were 27 books on the shelf to start with.
Now we can calculate how many books were left:
Non-fiction books left: 7/10B - 12 = (7/10)(27) - 12 = 14.9 (rounded to 15)
Fiction books left: 2/15B = (2/15)(27) = 3.6 (rounded to 4)
Altogether, there were 15 + 4 = 19 books left on the shelf.
Let's break down the problem step by step to determine the number of books left on the bookshelf.
1. Initially, 7/10 of the books were non-fiction books, and the rest were fiction books. Let's represent the total number of books on the bookshelf as "x."
Number of non-fiction books = (7/10) * x
Number of fiction books = (3/10) * x
2. It is given that 1/3 of the fiction books and 1/7 of the non-fiction books were removed. Let's calculate the number of books removed for each category.
Number of fiction books removed = (1/3) * (3/10) * x = (1/10) * x
Number of non-fiction books removed = (1/7) * (7/10) * x = (1/10) * x
3. It is also mentioned that twelve non-fiction books were removed. We can equate this to the number of non-fiction books removed and solve for "x."
(1/10) * x = 12
x = 120
4. Now that we know the total number of books initially, we can calculate the number of books left on the bookshelf after the removals.
Number of non-fiction books left = (7/10) * x - 12 = (7/10) * 120 - 12 = 84 - 12 = 72
Number of fiction books left = (3/10) * x - (1/10) * x = (2/10) * x = (2/10) * 120 = 24
5. Finally, to find the total number of books left on the bookshelf, we sum up the remaining non-fiction and fiction books.
Total number of books left = Number of non-fiction books left + Number of fiction books left
= 72 + 24
= 96
Therefore, the answer is 96 books left on the bookshelf altogether.
As an AI language model, I just want to make one correction: In step 4, the formula should be:
Number of fiction books left = (3/10) * x - (1/10) * x = (2/10) * x = (2/10) * 120 = 24
Other than that, your breakdown of the problem and solution is correct.
You are absolutely right, and I apologize for the mistake in step 4. The correct formula for calculating the number of fiction books left should be:
Number of fiction books left = (3/10) * x - (1/10) * x = (2/10) * x = (2/10) * 120 = 24
Thank you for pointing out the error, and I appreciate your correction. The final answer remains the same: there are 96 books left on the bookshelf altogether.