Calculating material quantities and costs

How many two-metre aluminium tubes do you need to cut thirty-six plant-stand legs, each seven hundred millimetres long? The tempting sum adds every leg together: twenty-five point two metres, or twelve point six tubes, so you buy thirteen. Order thirteen and you are ten legs short, because a leg cannot be pieced together from offcuts. Counting parts properly starts with a sheet of MDF. We will settle those thirteen tubes once the cutting method is in hand.

A workshop cuts clock faces, each two hundred and thirty by one hundred and sixty millimetres, from MDF sheets one thousand two hundred and twenty by six hundred and seven. Ignore the width of the saw cut. Fit the faces in rows. Along the long edge, one thousand two hundred and twenty divided by two hundred and thirty is five point three. Part of a face is no use, so round down to five. Across the short edge, it is six hundred and seven divided by one hundred and sixty. Rounding down, how many faces fit across, and so how many on one sheet? Three across, so fifteen faces on one sheet. Six hundred and seven divided by one hundred and sixty is three point seven nine, which rounds down to three. Five along times three across is fifteen. Now turn the face round and fit it again. Along the long edge, one thousand two hundred and twenty divided by one hundred and sixty is seven point six, so seven. Across, six hundred and seven divided by two hundred and thirty is two point six, so two. Turned round, how many faces fit, and which layout should the workshop cut? Fourteen, seven times two. That is one fewer, so keep the first layout and its fifteen faces. That is the method for any sheet. Fit the part along each edge, rounding down each time. Multiply the two counts. Then turn the part, do it again, and keep the better answer. One student divides areas instead. The sheet is seven hundred and forty thousand five hundred and forty square millimetres, one face is thirty-six thousand eight hundred, and dividing gives twenty point one. Why can twenty faces never come from one sheet? Because every face has to be one whole piece. The area sum counts the scraps round the edge as if they could be glued into faces. On the sheet of fifteen, an offcut seventy millimetres wide is left along one end, and another a hundred and twenty-seven wide along the side. Both are narrower than a face's short side of one hundred and sixty, so not one more face fits. If a question gives an allowance, such as a saw cut or a seam allowance, add it to each part's size before you fit. That holds whether the part comes from a sheet or from a length of stock. A seam allowance is the extra fabric stitched into the seam, and it uses up material like the part itself. One examiner's report, on cutting pattern pieces from a length of fabric, says most attempted this question dividing the area of the rib into the area of the material length, which is an incorrect method of calculation. The rib there is one of the pattern pieces. Dividing one area by another is what went wrong. The fix is to fit the part along each edge, round down, and multiply.

One sheet gives fifteen faces, and the workshop needs a batch of one hundred. How many sheets must the workshop buy for a batch of one hundred faces? Seven sheets. One hundred divided by fifteen is six point six seven, and part of a sheet cannot be bought, so round up. The two roundings go in opposite directions. Parts per sheet round down, because a part cannot hang off the edge. Sheets per batch round up, because nobody sells part of a sheet. Sheets cost nine pounds sixty each. Seven sheets cost seven times nine pounds sixty, which is sixty-seven pounds twenty. Per face, divide by the one hundred faces. Sixty-seven pounds twenty divided by one hundred is nought point six seven two pounds, which is sixty-seven pence to the nearest penny. The supplier has a bulk price: eight pounds forty a sheet for orders of ten or more. Bulk buying means a lower price per item for a bigger order. At eight pounds forty for ten or more, should the workshop buy ten sheets or seven? Seven. Ten at eight pounds forty is eighty-four pounds, which is sixteen pounds eighty more than the sixty-seven pounds twenty for seven. The bulk price is lower per sheet, but the workshop would pay for three sheets it does not need. Ten is only worth it if those spares will be used.

Back to the plant stands. A batch of twelve stands needs three legs each, and every leg is seven hundred millimetres long. The tube is sold in two-metre lengths at six pounds eighty each. Put both sizes in one unit first: two metres is two thousand millimetres. How many seven hundred millimetre legs can be cut from one length? Two legs. Two thousand divided by seven hundred is two point eight six, and you round down. Two legs use one thousand four hundred millimetres, which leaves six hundred. A third leg needs seven hundred, so that offcut can never become a leg. How many lengths of tube does the batch of twelve stands need? Eighteen. Twelve stands times three is thirty-six legs, and at two a length that is eighteen. Eighteen lengths at six pounds eighty is one hundred and twenty-two pounds forty for the batch. The thirteen tubes came from adding every leg's length together, twenty-five point two metres. That treats the offcuts as if they could be joined into legs, and thirteen tubes give only twenty-six legs, ten short. One examiner's report puts it like this: Far too often, candidates added the whole lengths of the material together. Those answers summed the lengths instead of counting pieces. The fix is to count whole pieces per stock length first, rounding down, and only then divide the batch by it. And when a price is per metre, turn millimetres into metres before you multiply.

Labour costs money too. A small firm makes a batch of eight oak desk organisers, with two methods to choose from. Method A is one maker, taking three hours per organiser at eighteen pounds an hour, with four pounds of materials per organiser. Build a per-item line first: materials plus time times rate. For Method A, three hours times eighteen pounds is fifty-four pounds, plus four pounds of materials, which makes fifty-eight pounds an organiser. What does Method A cost, in pounds, for the batch of eight organisers? Four hundred and sixty-four pounds. Eight times fifty-eight pounds is four hundred and sixty-four. The line is multiplied as one, materials and labour together. Adding the materials once for the batch is the slip to avoid: four pounds of oak is for one organiser, and eight of them need thirty-two pounds. One examiner's report, on costing a batch of six weather vanes, notes: A common error present in students' working was neglecting to take into account that cost of six weathervanes was in fact thirty pounds and not five pounds. The same report also records this: Many students were able to correctly give the required answer of five hundred and fifty-eight pounds. Where an error in arriving at the correct answer was seen many other students still gained a mark for having evidence of using a correct method in their response. Both are true. Many got the batch cost right, and the ones who slipped still had a method written down. Five pounds was the cost for one vane, so the fix is the per-item line, multiplied as one, so nothing in it is costed only once. Method B is a CNC router, a computer-controlled cutting machine, with an operator. It takes nought point seven five hours per organiser, at twenty-six pounds an hour for machine and operator. Materials are four pounds sixty, because the router's holding tabs waste a little more oak. Cost Method B's batch the same way, line first, then the multiply. Have a go at this one before I show you. One hundred and ninety-two pounds eighty. Nought point seven five times twenty-six pounds is nineteen pounds fifty. Add the four pounds sixty of oak, and it is twenty-four pounds ten an organiser. Eight times twenty-four pounds ten is one hundred and ninety-two pounds eighty. The firm asks which method is more cost effective. One student writes a single line: Method B is more cost effective. What is missing from that student's answer? The figures. It repeats the question's own words, with nothing to back them up. A full answer reads like this. Method B is more cost effective, at one hundred and ninety-two pounds eighty against four hundred and sixty-four pounds for Method A. That saves two hundred and seventy-one pounds twenty on the batch. It uses more oak, but it takes a quarter of the time. A further line from that report reads: Some students just repeated the question stem saying method two was cost effective. This did not gain credit. The question stem is the question's own wording. The habit that earns the credit is a full sentence: the method, both figures, the difference, and the reason.

Four checks to finish, on an acrylic sheet and what a batch from it costs.

Parts three hundred by two hundred, sheet one thousand by six hundred: how many fit? Ten, with the part turned. One way, three along times three across is nine. Turned, five along times two across is ten. Now a batch of forty-five of those parts: how many sheets? Five sheets. Forty-five divided by ten is four point five, and you round up. Here's the cost: five sheets at twelve pounds, for forty-five parts. Cost per part? One pound thirty-three. Five sheets cost sixty pounds, and sixty over forty-five is one pound thirty-three, to the nearest penny. One more: why do parts per sheet round down, but sheets per batch round up? A part hanging off the edge is no part at all, so parts go down. Nobody sells two thirds of a sheet, so sheets go up.

That is why the plant stands need eighteen tubes, not thirteen. Each tube gives two whole legs and a six hundred millimetre offcut that no leg can use.

If the sheet and the tubes both made sense, there is no need to see this twice, so mark it with a thumb and move on. If it went by quickly, give it a day and look again, because the method settles once it has had some time.

That completes the chapter on maths skills for angles, area, volume and material costs. Next chapter: Materials and their properties, starting with physical properties. Hopefully that's next in your feed, but if not, search for ScholaFly D T oh three, oh oh.

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