As with the other narratives I have written, this is based on a real event. Fortunately, the family is pretty good at maths!
Hoagy and Poonam think the time has come to redecorate the bathroom. Poonam insists that at least one wall shall be tiled and she has the idea that using two patterns of tile will be much more interesting than using only one. She assumes that Hoagy is on board, as he has suggested they go buy some tiles; for the purposes of our story they have just arrived at the tile shop and have been browsing the displays.
It occurs to Hoagy that the plain tiles are quite a lot cheaper than the patterned ones. In one display (that Poonam seems attracted to) the patterned ones are distinctly different from each other, but all use the same colours, rather as dining place mats with pictures on them can make a set. He begins to wonder how many of the fancy tiles they need to gain the sort of effect Poonam has in mind. Fortunately, he has realised his wife is expecting him to know just how big the wall is, but he won’t tell her he knows the figures just yet.
Poonam, meanwhile, is getting all excited by the prospect of mixing the fancy tiles with the plain ones – the effect will look much more expensive than using a whole load of the fancy (and much more expensive) ones. She has decided that she won’t tell Hoagy how little money they have left for the month (they take their budgeting very seriously).
The young couple spend a happy time dreaming how the bathroom will look once they have finished the tiling. Eventually, the room complete in their minds, they return to the practical parts, which begin with buying the requisite materials. To make this problem easier for you, I have put all the numerical information together.
Tiles are 10cm square. The wall to be covered is 180cm x 150cm. Start with the assumption that plain tiles cost 40p (£4 for a box of 10) and fancy ones cost £2. They think they want somewhere between 1:5 and 1:8 fancy tiles to plain ones (patterns of sixes or nines).
You are set some of the questions they would be considering:
1 Let p be the number of plain tiles and f be the number of fancy ones. Explain why p + f = 270, and write a similar statement for the cost of the tiles.
2 If p = 0, what is the cost? If p = 270, what is the cost?
3 Draw a graph, putting p on the horizontal axis and Cost vertically. Join the two points you have worked out so far with a straight line. Explain why this line shouldn’t really be a straight line (and what sort of shape it should be).
4 The ratios of fancy tiles to plain give you two more straight lines. 10 fancy tiles and 50 plain tiles give you a cost. Plot this point, and some other points fitting that ratio – and draw the line. Repeat this for the ratio 1:8 and draw this line too.
5 What is the smallest number of fancy tiles you can have? How many plain ones should there be? What will this cost? Have you plotted this point yet?
What is the biggest number of fancy tiles we could have? What will this cost? Plot the two lines on your graph. If you’re stuck, go straight to the next question.
6 Actually, plain tiles are £4 a box of ten and fancy tiles are £20 for a boxed set of ten, or £2.50 each (they’re cheaper by the box). Work out the revised costs for 30, 40 and 45 fancy tiles. I think the last of these totals £188.50. You should redraw the part of your graph where 30 < f < 50 and £155 < Cost < £190. Draw it sensibly large.
7 How many fancy tiles (more than three boxes) would they sensibly buy before buying four boxes instead of singles? What will this make the total cost?
8 As ever with problems of this type, the practical issues change the question. It dawns on Hoagy that some tiles will have to be cut, and that often means a tile is broken before he gets the cutting right. He can’t imagine a pattern where the fancy tiles end up needing to be cut, so he will only want some spare plain ones. Which combinations would you now exclude (because they have become significantly more expensive) ?
9 Which combinations now look the most attractive? Poonam is sure to want the biggest number of fancy tiles for the money they spend.
10 If they have £180 available. What do you recommend they do? What would you suggest for £10 less?
This is an example of Decision Mathematics (real maths?). It uses many of the skills we learn, all at once. While I have written the story for H & P, I have actually faced up to the problem, with the lady wife and in the shop, confornted with tiles, before it occurred to me just how difficult the problem could be. I solved it by drawing on some packaging and we found a satisfactory solution without making a second visit — well, not a visit for more tiles. I moved the prices to something roughly modern. I’d love to have response to this sort of ‘real-life’ maths.
Q1-5 C=540-1.6p, C=0.8p; C=0.65p are the three lines.
Q6 45 fancy tiles is 4 boxes and 5 singles, £(80+12.50)=£92.50. 235 plain tiles is 24 boxes, £96. So this bundle is £188.50 with perhaps 5 plain tiles remaining. Q7 38 fancy tiles cost the same as 40, £80. So you can argue that at 34 fancy tiles, £70 and no breakages, to 36 fancies (£75), then they are making a good decision. In this region they must buy 240 plain tiles, £96, So the total is around £170.
Q8-10 Now we know they must buy at least 24 boxes of plain tiles. 230 plain won’t allow for breakages and mistakes. 240 plain or 250 plain looks like the choice that gives the most fancy tiles. At £180 they are very limited in choices: 240 plain, £96, allow £84 on fancies, which will be 40 or 41 and for £170, £74 on fancies means up to 35. At 250 plain, £100, their choices are up to 40 fancy at £180 budget, and up to 34 fancy at £170. The majority students are going to say £180 buys 25 boxes of plain and 40 of fancy and that this will satisfy Poonam best. £170 will buy 250 plain and 34 fancy. They have forgotten to allow for fixing materials, but that will be the same fixed amount whatever combination they choose; this is probably why the £180 dropped at the end of the calculation.
