A stationery order's cost, \(C\) pounds, is directly proportional to the number of packs, \(n\), ordered, for orders up to 50 packs; 20 packs cost \(\pounds...
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A stationery order's cost, \(C\) pounds, is directly proportional to the number of packs, \(n\), ordered, for orders up to 50 packs; 20 packs cost \(\pounds 70\). Above 50 packs, the rate is \(\pounds 3\) per pack.
- Find the cost per pack for orders up to 50 packs. (1)
- Work out the cost of ordering 80 packs at the bulk rate. (1)
- Compare the cost of 80 packs at the bulk rate with 80 packs at the rate from part (a), and state which is cheaper. (2)
- The school's budget is \(\pounds 230\). Working backwards from this, find the maximum number of packs it can buy at the bulk rate, and state whether this meets its need for 80. (2)
Answer Details
Below the \(50\)-pack threshold, cost is directly proportional to the number of packs, so a rate per pack is found from the known order and used for any order in that range; above the threshold, the flat bulk rate applies instead, and the two rates are compared directly for a specific order size before checking what a fixed budget can afford at the bulk rate.
- \[ 70\div20 = \pounds3.50 \text{ per pack} \] [1 mark]
- \[ 80\times3 = \pounds240 \] [1 mark]
- At the standard rate of \(\pounds3.50\) per pack, \(80\) packs would cost \(80\times3.50=\pounds280\); since \(\pounds240\lt\pounds280\), the bulk rate is cheaper for an order of \(80\) packs. [2 marks]
- At the bulk rate: \[ 230\div3 = 76.67 \] so the budget buys a maximum of \(76\) complete packs, which does not meet the school's need for \(80\) packs. [2 marks]
The bulk rate exists precisely so that a large order costs less per pack than the standard rate would suggest, which is why it is cheaper here even though it kicks in only above \(50\) packs; but a fixed budget still has a hard limit, and \(76.67\) packs must be rounded down, not up, since only whole packs can be bought.
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