There were 500 more fiction books than non-fiction books in a bookshop. When more books were added to the bookshop, the number of fiction books increased by 1/4, while the number of non-fiction books increased by 2/3. The total number of books became 3145. How many non-fiction books were there after the increase?
f = n+500
5/4 f + 5/3 (f-500) = 3145
f = 1364
so there were 864 nonfiction books at first, which increased to 1440
those, added to the 5/4 * 1364 = 1705 fiction books made 3145
Let's start by assigning some variables to the unknown quantities in the problem:
Let's call the initial number of non-fiction books "x".
Then, the initial number of fiction books will be "x + 500".
The problem states that the numbers increased by different fractions. The new number of fiction books increased by 1/4, while the new number of non-fiction books increased by 2/3.
So, the new number of fiction books after the increase will be "(x + 500) + 1/4(x + 500)", which simplifies to "5/4(x + 500)".
Similarly, the new number of non-fiction books after the increase will be "x + 2/3x", which simplifies to "5/3x".
According to the problem, the total number of books in the bookshop after the increase is 3145. So, we can set up an equation:
(5/4(x + 500)) + (5/3x) = 3145
Let's solve this equation step by step:
First, let's get rid of the fractions by multiplying all terms by 12 (the least common multiple of 4 and 3):
12[(5/4(x + 500)) + (5/3x)] = 12 * 3145
Simplifying this equation gives us:
15(x + 500) + 20x = 12 * 3145
Expanding the brackets further:
15x + 7500 + 20x = 12 * 3145
Combining like terms:
35x + 7500 = 12 * 3145
Now, let's simplify the right side:
35x + 7500 = 37740
Moving 7500 to the right side:
35x = 37740 - 7500
35x = 30240
Dividing both sides by 35:
x = 30240 / 35
x = 864
Therefore, the initial number of non-fiction books was 864.
Now, we can find the number of non-fiction books after the increase:
New number of non-fiction books = x + 2/3x = 864 + 2/3 * 864
Calculating this gives us:
New number of non-fiction books = 864 + 2/3 * 864
New number of non-fiction books = 864 + 2/3 * 864
New number of non-fiction books = 864 + 576
New number of non-fiction books = 1440
So, after the increase, there were 1440 non-fiction books in the bookshop.
To solve this problem, we can set up a system of equations.
Let's represent the number of non-fiction books as "x", and the number of fiction books as "x + 500" since there were 500 more fiction books.
After the increase, the number of fiction books increased by 1/4, which means there is an additional 1/4 of (x + 500) fiction books. Therefore, the new number of fiction books is (x + 500) + 1/4(x + 500).
Similarly, the number of non-fiction books increased by 2/3, which means there is an additional 2/3 of x non-fiction books. Therefore, the new number of non-fiction books is x + 2/3 * x.
We can add up the number of fiction and non-fiction books, and set it equal to the total number of books:
[(x + 500) + 1/4(x + 500)] + [x + 2/3 * x] = 3145
Now, we can solve this equation to find the value of x, which represents the number of non-fiction books after the increase.