Answer:
we will need four cases and a bookend
Explanation:
STEP 1 - How many books fit on 1 shelf?
We divide 2%263%2F4 feet by 1%2F4 feet.
We could use fractions, or decimals.
We could even transform both measurements to inches.
The result is that 2%263%2F4 feet divided by 1%2F4 is 11 exactly.
It divides evenly, so we can fit 11 books per shelf, tightly packed, with no room to spare.
Calculation using fractions:
2%263%2F4=2%2B3%2F4=8%2F4%2B3%2F4=11%2F4 (transforming mixed number into improper fraction)
11%2F4�1%2F4=11%2F4�4%2F1=11
Calculation using decimals:
If we know that 2%263%2F4=2.75
and we know that 1%2F4=0.25
and we can use a calculator, it is to divide
2.75%2F0.25=11
If we prefer to measure in inches we would think that
the shelf width is 33 inches
and each book is 3 inches wide,
so we divide 33%2F3=11 to find that 11 books fit on each shelf.
STEP 2 -How many shelves will be used?
We divide the 35 books into piles of 11 books.
We find that we can make 3 piles of 11 books, and we will have 2 books left over.
We will need highlight%284%29 shelves (and a bookend).
Three (3) shelves could have 11 books (full shelves),
but you need at least one more shelf, for a total of 4 shelves.
If the first 3 shelves are filled with 11 books each,
the fourth shelf could hold the remaining 2 books.
Or you could arrange the books differently, but you need at least 4 shelves.
The calculation:
1 pile = 1 x 11 books = 11 books
2 piles = 2 x 11 books = 22 books
3 piles = 3 x 11 books = 33 books
4 piles = 4 x 11 books = 44 books
Obviously, we can make 3 piles of 11 books, and we will have 2 books left over.
35 - 33 = 2
The teacher would say that 35 divided by 11 is 3 with a remainder of 2.
When we do long division, we do exactly the calculations above.
THE WRONG WAY:
Someone could figure that 35 books at 1%2F4 foot per book require
35%281%2F4%29=35%2F4 feet of shelf space.
Then, dividing the space needed by the space per shelf,
35/4 feet divided by 2%263%2F4 feet,
that person could find that 3&2/11 shelves are needed,
and round up to 4 shelves.
The result is the same in this case, but the approach is wrong,
because if the measures of book and shelf width were different,
you might find a number of shelves that could not hold the books,
unless we cut some of the books in pieces.
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3x-x+2=4