the library has a total collection of 2630 books. the number of non-fiction books is 240 fewer than the number of fiction books but 190 more than the number of picture books. how many books of each type are there in the library?

n=f-240

n=p+190

n+f+p = 2630
(f-240) + f + (f-240-190) = 2630
f = 1100
n = 860
p = 670

To solve this problem, let's assign variables to represent the number of each type of book.

Let:
F = Number of fiction books
NF = Number of non-fiction books
P = Number of picture books

We are given three pieces of information:

1. The total collection of books is 2630.
This can be expressed as an equation:
F + NF + P = 2630

2. The number of non-fiction books is 240 fewer than the number of fiction books.
This can be expressed as an equation:
NF = F - 240

3. The number of non-fiction books is 190 more than the number of picture books.
This can be expressed as an equation:
NF = P + 190

Now, we can use these equations to solve the problem.

Start by substituting equation 2 into equation 3:
F - 240 = P + 190

Next, rearrange equation 3 to isolate P:
P = F - 430

Now, substitute this value of P into equation 1:
F + NF + (F - 430) = 2630

Combine like terms:
2F + NF - 430 = 2630

Rearrange the equation to solve for NF:
NF = 2630 - 2F + 430

Substitute this value of NF into equation 2:
2630 - 2F + 430 = F - 240

Combine like terms:
3F = 2840

Divide both sides by 3:
F = 947

Substitute this value of F into equation 2 to find NF:
NF = 947 - 240
NF = 707

Finally, substitute the values of F and NF into equation 1 to find P:
947 + 707 + P = 2630
P = 2630 - 1654
P = 976

Therefore, the number of fiction books is 947, the number of non-fiction books is 707, and the number of picture books is 976 in the library.