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.