Notebook Pallets: Rounding Estimates Versus Exact Bin Counts
The school store at Lincoln Elementary received 6 identical pallets of spiral notebooks, and the invoice states that each pallet holds 1,248 notebooks. Volunteers must repackage every notebook into clear bins that hold 36 notebooks each. A bin that is not full still needs its own lid and label.
The store manager has 180 empty bins in storage and tells the principal, 'We'll need about 200 bins total, so I'll buy just a few more.'
Before ordering, a volunteer recounts one pallet and finds it holds only 1,200 notebooks because 2 cartons were short; the other 5 pallets are correct.
The principal also wants a place-value label on each bin showing the number of notebooks written in expanded form, and she wants a quick estimate written on the purchase request so that anyone reviewing it can see the order size is sensible.
1. Use rounding to test the manager's 'about 200 bins' figure. Is it a sensible planning number? Explain how rounding the divisor can push an estimate in a misleading direction.
Read the worked answer
Round the number of notebooks: 6 pallets × about 1,250 = about 7,500 notebooks. Now round the bin size: 36 rounds to 40, and 7,500 ÷ 40 ≈ 187 - that lands below 200.
A more reliable check uses a landmark product instead of a rounded divisor: 36 × 200 = 7,200, and 7,200 is less than 7,500, so more than 200 bins would be needed. Using 30 as the rounded divisor gives 7,500 ÷ 30 = 250, which is a safe upper bound.
Judgment: 'about 200' is close and reasonable for a verbal estimate, but it is slightly low - and dangerous as an order number, because the estimate came from rounding the divisor up to 40 while the real divisor is 36. Rounding the divisor up always shrinks the quotient. A defensible written estimate is 'between 190 and 250 bins, closer to 200.'
2. Using the invoice figures (before the recount), find the exact number of bins required and the number the manager must buy. Show how the remainder is treated.
Read the worked answer
Total notebooks: 6 × 1,248. Using partial products: 6 × 1,000 = 6,000; 6 × 200 = 1,200; 6 × 48 = 288; total = 7,488.
Bins: 36 × 200 = 7,200 with 288 left over. Since 288 ÷ 36 = 8, the quotient is 200 + 8 = 208, and 208 × 36 = 7,488, which checks exactly.
Remainder: 0, so no bin is left partly full - every bin holds exactly 36 notebooks.
To buy: 208 needed − 180 on hand = 28 bins.
Judgment: because the remainder is zero, rounding up does not add a bin here. If there had been any remainder (say 24 notebooks), a whole extra bin would still be required, since 24 loose notebooks cannot be left unlabeled.
3. After the recount, recompute the bin count. The manager suggests, 'Just divide by 30 to be safe.' Evaluate both the new exact count and that shortcut.
Read the worked answer
New total: 5 correct pallets × 1,248 = 6,240, plus the short pallet's 1,200 = 7,440 notebooks.
Exact: 36 × 206 = 7,416, and 7,440 − 7,416 = 24. So there are 206 full bins plus 24 leftover notebooks. Those 24 still need a lid and label, so the store needs 207 bins. Against the 180 on hand, buy 27 bins.
The ÷30 shortcut: 7,440 ÷ 30 = 248. That is 41 bins more than needed - an over-order caused entirely by changing the divisor, not by any change in the notebooks. Dividing by a smaller number always inflates the quotient.
Judgment: estimates are for the purchase request, not the order form. Exact division, with the remainder rounded up for any container, is what determines how many bins and lids to buy.