Question
A caterer is required to buy two small sacks of flour and one large sack for a set of baking orders. A small sack holds 3 kg and costs $12; a large sack holds 8 kg and costs $24. All the flour remains usable, even if some is stored for the next baking day. Sacks cannot be split at the supplier.
What is the caterer's effective purchase price per kilogram for the combined order? A colleague takes the two displayed prices per kilogram, $4/kg and $3/kg, and averages them to get $3.50/kg. Explain precisely why that averaging method does not answer the question.
Get hint
Hint 1
Calculate the dollars paid for the complete order before computing any average price.
Check your answer
Solution
Two small sacks contain \(2\times3=6\) kg and cost \(2\times\$12=\$24\). The large sack contains 8 kg and costs $24. Altogether, the caterer buys \(6+8=14\) usable kilograms and spends $48. Therefore
\[
\text{combined price per kg}=\frac{\$48}{14\text{ kg}}
=\boxed{\frac{24}{7}\approx\$3.4286\text{ per kg}}.
\]
The individual rates are indeed \(\$12/3=\$4/\mathrm{kg}\) and \(\$24/8=\$3/\mathrm{kg}\), but the caterer did not buy equal kilogram quantities at those rates. The correct quantity-weighted rate is
\[
\frac{6}{14}(\$4/\text{kg})+\frac{8}{14}(\$3/\text{kg})
=\frac{24+24}{14}=\$\frac{24}{7}/\text{kg}.
\]
The arithmetic average \((4+3)/2=3.50\) treats the two sack types as equally represented by quantity. Even averaging across the three individual sacks would be wrong without kilogram weights: the large sack supplies 8 kg, much more than either small sack. Because all purchased flour remains usable, there is no need to remove discarded kilograms from the denominator.
Assumptions
The prescribed purchase is exactly two 3-kg sacks and one 8-kg sack; supplier pricing is per complete sack.
There are no delivery charges, spoilage or discounts, and unused flour remains usable and owned by the caterer.
The requested rate is per kilogram purchased, not per sack or per kilogram consumed today.
Final Answer
Total cost $48, total quantity 14 kg, combined rate \(48/14=24/7\approx\$3.43/\text{kg}\). The simple $3.50 average gives equal weight to package rates despite their different purchased quantities.