your school librarian has asked your class for some help on the purchase of some new bookcases with two shelves each. She has 300 new books she needs to shelve. One half of the books are half an inch thick. Ones third of them are one fourth of an inch thick, and the rest are three fourths of an inch thick. Her shelves are 30 inches long. How many bookcases should she buy? Show your work
150*1/2 + 100*1/4 + 50*3/4 = 276"
She needs 9+ shelves, or 5 bookcases
Please show how to solve the above question?
To determine the number of bookcases she should buy, let's calculate the total thickness of the books first.
For books that are half an inch thick:
Number of books = 1/2 * 300 = 150 books
Total thickness = 150 * 1/2 inch = 75 inches
For books that are one fourth of an inch thick:
Number of books = 1/3 * 300 = 100 books
Total thickness = 100 * 1/4 inch = 25 inches
For books that are three fourths of an inch thick:
Number of books = 300 - (150 + 100) = 50 books
Total thickness = 50 * 3/4 inch = 37.5 inches
Now let's calculate the total thickness of books on each shelf:
Total thickness per shelf = 75 + 25 + 37.5 = 137.5 inches
Since each shelf is 30 inches long, the number of bookshelves needed can be calculated as:
Number of bookshelves = Total thickness / Length of each shelf
Number of bookshelves = 137.5 / 30 = 4.58 (rounded up to the nearest whole number)
Therefore, she should buy 5 bookcases to accommodate all the books.
To solve this problem, we need to determine how many books of each thickness can fit on a single shelf. Then we can calculate how many shelves are needed to accommodate all the books. Here's how we can approach it step by step:
1. Calculate the maximum number of books that can fit on each shelf:
- Since the bookcases have two shelves each, we will divide the total length of the shelf (30 inches) by 2 to determine the length available for each shelf: 30 inches / 2 = 15 inches.
2. Determine the number of books that are half an inch thick:
- Given that one-half of the books are half an inch thick, we can calculate the number of books as follows:
Number of half-inch books = (1/2) * 300 = 150 books.
3. Calculate the number of books that are one-fourth of an inch thick:
- Given that one-third of the books are one-fourth of an inch thick, we can calculate the number of books as follows:
Number of one-fourth-inch books = (1/3) * 300 = 100 books.
4. Calculate the number of books that are three-fourths of an inch thick:
- Since the remaining books are three-fourths of an inch thick, we can calculate the number of books as follows:
Number of three-fourths-inch books = Total books - (half-inch books + one-fourth-inch books) = 300 - (150 + 100) = 50 books.
5. Calculate the number of shelves needed for each type of book:
- To determine the number of shelves needed for each type of book, divide the number of books by the maximum number of books that can fit on a single shelf.
For half-inch books:
Number of shelves for half-inch books = Half-inch books / Maximum books per shelf
Number of shelves for half-inch books = 150 books / (15 inches / 0.5 inches) = 150 / 30 = 5 shelves
For one-fourth-inch books:
Number of shelves for one-fourth-inch books = One-fourth-inch books / Maximum books per shelf
Number of shelves for one-fourth-inch books = 100 books / (15 inches / 0.25 inches) = 100 / 60 = 1.67 shelves
For three-fourths-inch books:
Number of shelves for three-fourths-inch books = Three-fourths-inch books / Maximum books per shelf
Number of shelves for three-fourths-inch books = 50 books / (15 inches / 0.75 inches) = 50 / 20 = 2.5 shelves
6. Round up the fractional shelves to the nearest whole number:
Since we cannot purchase a fractional number of shelves, we need to round up the fractional shelves to the nearest whole number.
Number of shelves needed for half-inch books = 5 shelves
Number of shelves needed for one-fourth-inch books = 2 shelves
Number of shelves needed for three-fourths-inch books = 3 shelves
7. Sum up the total number of shelves needed:
Total shelves needed = Shelves for half-inch books + Shelves for one-fourth-inch books + Shelves for three-fourths-inch books
Total shelves needed = 5 + 2 + 3 = 10 shelves
Therefore, the librarian should buy 10 bookcases to shelve her 300 new books.