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?
Well, well, well, it seems like the library has quite the collection! Let's solve this bookish puzzle, shall we?
Let's assume that the number of fiction books in the library is "F". According to the information given, the number of non-fiction books would then be "F - 240", and the number of picture books would be "F - 190".
Now let's add up the number of each type of book to find the total collection:
F + (F - 240) + (F - 190) = 2630
Simplifying this equation gives us:
3F - 430 = 2630
Now if we add 430 to both sides, we get:
3F = 3060
Finally, if we divide both sides by 3, we find:
F = 1020
So, there are 1020 fiction books, (1020 - 240) = 780 non-fiction books, and (1020 - 190) = 830 picture books in the library.
Voila! Happy reading!
To solve this problem, let's assign variables to represent the unknowns.
Let's assume:
F = Number of fiction books
N = Number of non-fiction books
P = Number of picture books
From the information given in the problem, we can derive the following relationships:
1. The number of non-fiction books (N) is 240 fewer than the number of fiction books (F).
N = F - 240
2. The number of non-fiction books (N) is 190 more than the number of picture books (P).
N = P + 190
Since we know that the total collection of books is 2630, we can write the equation:
F + N + P = 2630
Now, we can substitute the value of N in terms of F and P from equation (1) and (2) into the equation for the total collection of books:
F + (F - 240) + (P + 190) = 2630
Simplifying the equation:
2F + P = 2680
Now, we have two equations:
1. F + P = 2680 - 2F (obtained by subtracting 2F from both sides)
2. N = P + 190
Since we have three variables and two equations, we need an additional equation to solve for the values of F, N, and P.
Unfortunately, the given information doesn't provide us with another direct equation. Therefore, we can use trial and error or guess-and-check method to find the values that satisfy both equations.
Let's start with an assumption and see if it works.
Assume F = 800
Using equation 1:
800 + P = 2680 - 2(800)
800 + P = 2680 - 1600
P = 2680 - 1600 - 800
P = 280
Using equation 2:
N = P + 190
N = 280 + 190
N = 470
Now, we can check if the assumed values satisfy the total collection of books equation:
F + N + P = 800 + 470 + 280 = 1550
Since the assumed total of 1550 is different from the given total of 2630, we need to adjust the assumed values.
Let's try F = 1000
Using equation 1:
1000 + P = 2680 - 2(1000)
1000 + P = 2680 - 2000
P = 2680 - 2000 - 1000
P = 680
Using equation 2:
N = P + 190
N = 680 + 190
N = 870
Now, let's check if the calculated values satisfy the total collection of books equation:
1000 + 870 + 680 = 2550
Again, the calculated total of 2550 is different from the given total of 2630.
We can continue trying different values for F until we find the correct combination that satisfies all the given conditions and the total collection of books equation.