I had a number problem fighting with a horrible binary paint, in tins far too big for the work I am prepared to do in a day, and other issues like fumes). I need to put the paint in smaller pots. This is the maths I ended up having to do.

Part of the task here is to extract from the text what it is you need to know so as to tackle the maths (which is mostly arithmetic; there is one formula for volume you are expected to know). The idea is to let you see that the maths might assist with the problem as presented, not that I'm making the maths morre difficult by hiding what you need to know!

The paint comes in A and B and is to be mixed on the day of use. The tin containing A is where B is to be put—though that is far from obvious—and the instructions say 1:1 where what that means is one of B goes into one of A, the plastic bottle of B is poured oh so carefully into tin A. The stuff in B has a high content of phosphoric acid, quite scary; A and B mixed puts out fumes that make one wish to do the painting outside; the paint takes an age to dry and then needs a sealing coat of varnish applied within 3-7 days or the bas layer comes off.

The paint is a fire-retardant; it raises the fire resistance of an existing door significantly and as such allows me to keep the original doors in the house rather than have to replace the (very nice) wooden doors with relatively uninteresting modern ones. The paint is very expensive; I could 'send the doors away' to be painted at around £200 per door and have a week or two without doors. Additionally, each door has to have an intumescent strip inserted in a groove around three sides (not the bottom). This will seal the door into the frame in a fire, which reduces spread of fire so that residents can escape. Quite why I am required to do this in my own home I do not undestand, but the regulations are there to be followed. The required groove is made with a tool called a router (rowter, not rooter) and I can do that myself at about an hour per door. The elapsed time quite a bit longer, since both the tool and the user need to cool down; I need coffee at frequent intervals too.

I mixed A and B is per the instructions and discovered (only some of) the problems: the lengthy difficulty with the fumes; the drying time; that the doors really needed to be out of the frames and level; that the brushes would have to be written off afterwards. The experience was so bad, I have left the doors alone and the base coat has delaminated—come unstuck from the door, with what looks like air underneath the finish. Clearly the doors have to be done all over again. Based on the previous experience I think it would be sensible to do a door or at most two doors at a time; to keep the doors off the hinges while they are worked on.

So I need to know the mixture by volume, or in some other way to copy the proportions provided. Tin A is the usual cylinder and I have measured the depth of paint as 65 or 66 mm, the internal diameter as being between 145 and 148mm.

1. What volume of A is in the tin? You will have a range of results.

2. Bottle B is labelled as being 1.25 litres. What proportion do you think is intended for A:B?

3. Tin A has a total height of 185mm. At what integer number of litres is the tin rated? What depth of paint would that imply when A and B are mixed?

4. If one assumes that the amount of A intended to be in the tin is correct to somewhere between an eighth and a twelfth of a litre, what proportion do you now think is appropriate?

A full tin of A&B combined painted all ten doors adequately on one side each. I think I can face doing a door a day. After a very long time runnig scared at the prospect of doing all this work again I decided that the way forwards was to do a single door to see if that made the whole thing a doable task. Since the paints have to be mixed on the day, I need a suitable container.

5. What size container do I need to paint one side of one door?

I discovered that a paint supplier will not provide me with empty tins; it is 'not allowed'. I have not kept any small tins clean. I thought of a syrup tin and a jam jar for holding the single-door's worth of paint, but I think I need to be able to measure reasonably accurate volumes into whatever container I use. Eventually (I really am avoiding this task) it dawns on me that I have some paint sample pots that are now finished (either I've used the paint or discovered this is not what we want), so I go use one up and wash it. I recognise that in all probability whatever I use for measuring cannot be used again for anything but this task—so everything in the kitchen is immediately removed from consideration; I looked for but failed to find any cheap measuring spoons. I will accept that I'm going to, probably, consume a brush per day spent with this paint. Significantly, the nasty B is added to A, not A added to B. That is, whatever is used for painting from (dipping the brush into) starts with A at the correct amount.

The sample pots I have are identical, but labelled 50ml and 75ml. I measure the internal diameter as 49mm and the internal height of the pot as 60mm.

6. a) What is the volume of the pot to the nearest millilitre? What volume could you put in there to the nearest 25ml?

7. The '50ml' pot has what depth of paint in it? Express this also as a percentage of the pot volume.

Given the work done above, in questions 1-5, I think I need 225-250ml of mixed A&B for a door, and I think I can use multiples of 25ml, (25, 50, 75, 100) to measure A from the tin to the container of the mixed paint for one door.

8. What do you suggest is a reasonable way to use the sample pot as a measuring device? Is it big enough for a whole door's-worth of paint?

If a sample pot is too small for a single door's worth, is a glass jamjar big enough? A jamjar holds 1 pound of jam, 454g. The dimensions of a jamjar are height 125mm and lid diameter 63mm, external diameter 70mm. I think this is 380ml capacity in practice.

9. Check I have the right size of jamjar in mind. Specifically, check that the jamjar volume is a little bigger than 380ml.

I will probably try to finish one door at a time, but I accept that this may become two doors at a time and somehow cope with having two doors missing for most of a week at a time. There's no way the wife will accept having doors missing when she's at home over the weekend, so don't come up with models that cause marital conflict; it's bad enough just doing this painting without adding stresses.

10. Are you confident enough in your answers enough to go start the work? Might one route to this to divide tin A into say five jamjars equally filled, to then divide container B into five equal amounts and to store these ten containers separately until use? If so, I think I'll prefer the smaller 125ml jam jar. But I need sets of five or a better way of measuring B (the horrid, dangerous stuff).

Partial answers: I think the volume of A is from 1.07 to 1.135 ml. I used radius of 72.5, 73 and 74 and a depth of paint as 65 and 66mm.

I am sure this is a nominally 3 litre tin (tin A), so that radius and 'full' depth pairs off as (r,h) being (72.5, 182), (73, 179) or (74,174), all of which fit what I have measureed.

I thiink probably the intended volume of A is 1.1 or 1.125litres. Given that vol B is 1.25 litres, that suggests a ratio of A:B as between 1:1.36 and 1:1.111; writing these as integer combinations, between 22:25 and 10:9. This is *not* 1:1 by volume, which was evident as soon as I opened the first tin.

I thought about working on the assumption of 4:5, a litre of A to 1.25 litres of B. I think I need to have a little more of A than that. Somewhere between 40 and 45 ml of A for each 50 ml of B will work out. If all the tin of A and B does ten doors, then each door needs (definitely) 125ml of B and therefore between 100 and 112.5 ml of A. If I could measure 125ml of B accurately (in a dedicated container never used for food in the future) then I could recognise the right amount of A to use, probably just by eye.

I put the story without the nunbers on this page.