Of the 1500 books in the library, there are 3 times as many books in the fiction section as are in the nonfiction section. Also, there are 400 more books in the reference section than are in the nonfiction section. How many books are in each section?
Let x be the number of books in the nonfiction section.
The number of books in the fiction section is 3*x.
The number of books in the reference section is x+400.
In total, there are x+3*x+x+400=1500 books.
Combining like terms, we get 5*x+400=1500.
Subtracting 400 from both sides, we get 5*x=1100.
Dividing both sides by 5, we get x=<<220=220>>220 books in the nonfiction section.
There are 3*220=<<3*220=660>>660 books in the fiction section.
There are 220+400=<<220+400=620>>620 books in the reference section. Answer: \boxed{220, 660, 620}.
Let's solve the problem step by step.
Step 1: Assign variables.
Let's assign variables to represent the number of books in each section.
Let:
F = number of books in the fiction section
N = number of books in the nonfiction section
R = number of books in the reference section
Step 2: Translate the given information into equations.
From the problem statement, we have two pieces of information:
1) There are 3 times as many books in the fiction section as in the nonfiction section:
F = 3N
2) There are 400 more books in the reference section than in the nonfiction section:
R = N + 400
Step 3: Use the equations to solve the problem.
We know that the total number of books is 1500:
F + N + R = 1500
Now substitute the equations from Step 2 into this equation:
3N + N + N + 400 = 1500
Simplify the equation:
5N + 400 = 1500
Subtract 400 from both sides:
5N = 1100
Divide both sides by 5:
N = 220
Now, substitute the value of N back into the equations to find the values of F and R:
F = 3N = 3 * 220 = 660
R = N + 400 = 220 + 400 = 620
Step 4: Check the answer.
To check our solution, we can add up the number of books in each section:
660 + 220 + 620 = 1500
Since the sum is equal to the total number of books in the library, our solution is correct.
Therefore, there are 660 books in the fiction section, 220 books in the nonfiction section, and 620 books in the reference section.