The exercises
Fourteen problems from real tanks
Each one has the story first, then the numbers you are given, then the question. The formula or the constant you need is not printed here. Under every exercise is the page that carries it, so the first move is always the same: go and read, come back, then work.
The tap water figures are real. They come from a survey of Yerevan drinking water sampled district by district, so the tank in the story is a tank somebody in this city actually has.
Exercise 1 · one conversion
The same water, three different numbers
Your family lives in Davtashen. You buy a hardness test kit, and because most kits sold here are made in Germany it answers in German degrees. Your cousin is doing a school project with a water report that gives hardness in milligrams per liter as calcium carbonate, and your aunt has a laboratory printout in a third unit again. Everyone thinks they are arguing. They are all describing the same tap.
Given
- — Tap water in Davtashen: 7.7 German degrees of general hardness1
- — One German degree is a defined quantity, not a measurement, so the conversions follow from the definition
Task. Express that same water:
- in milligrams per liter as calcium carbonate, to the nearest whole number;
- in milliequivalents per liter, to two decimal places.
Read first: Chemistry, where the German degree is written out with every unit that hangs off it.
Exercise 2 · one conversion, backwards
A Soviet unit on an Armenian laboratory report
A neighbor has a laboratory analysis of the water in her building, done properly, with a stamp on it. It reports hardness in milligram-equivalents per liter, which is the unit Armenian and Russian laboratories have always used. She wants to know whether her water is the same as the water in the tap you have been testing with your German kit.
Given
- — The report reads 2.75 milligram-equivalents per liter
- — Three Yerevan figures to compare it against: Avan, Kanaker and Nork at 3.4 German degrees; the central districts at 6.4; Davtashen at 7.71
Task. Convert the report's number into German degrees, to one decimal place, and say which of the three districts her building is most likely in.
Read first: Chemistry, the table of every hardness unit converted. One row of it is the Soviet unit.
Exercise 3 · two steps, and a definition to look up
How big is the tank, and how wide is its lid of water?
A tank stands on a shelf in a flat in Yerevan. It is a plain rectangular box, and the water does not come to the top: there is a gap of a few centimeters between the surface and the rim, the way there always is. You want three numbers about it before you buy anything to put in it.
Given
- — Inside length 80 cm, inside width 35 cm
- — The water stands 40 cm deep
- — 1 liter is 1,000 cubic centimeters
Task. Work out:
- the volume of water, in liters;
- the area of the water surface, in square meters;
- the mean depth, using the definition the physics page gives for it rather than just reading 40 cm off the question;
- the surface area divided by the volume, in square meters per cubic meter.
Read first: Physics, where mean depth is defined out of volume and area.
Exercise 4 · one step
A quarter out, three quarters left
It is Saturday. The tank has been running for a few months, the plants are fine, and the nitrate test has gone a color you do not like. You take a bucket, siphon some water out, and put the same amount of tap water back. Yerevan tap water carries so little nitrate that for this problem you can call it none.
Given
- — Tank volume 80 liters
- — Nitrate before the change: 40 mg/L, as the test kit reads it
- — You remove and replace 24 liters
- — Nitrate in the tap water: treat as zero
Task. Answer both:
- What fraction of the water in the tank afterwards is still the old water? Give it as a percentage.
- What does the nitrate test read after the change?
Read first: Water changes, which turns on exactly this point: what a change of part of the water can and cannot do.
Exercise 5 · two divisions and a sum
Two metals, one hardness number
A bottle of Armenian spring water lists what is dissolved in it: calcium, magnesium, bicarbonate, a few others. It does not print a hardness figure, because bottle labels never do. Your kit is out of reagent and you want the hardness anyway. Everything you need is already on the label.
Given
- — Calcium 34.3 mg/L
- — Magnesium 7.2 mg/L
- — General hardness counts both metals, and one German degree is a different mass of each
Task. Work out the general hardness in German degrees, to one decimal place, and say which Yerevan district's tap water it most resembles.
Read first: Chemistry, which works one label conversion all the way through by three different routes, with the divisor for each metal.
Exercise 6 · one division, then a judgement
Two nitrates, and only one of them is on your kit
You read a scientific paper about how much nitrate fish and shrimp can live with. The paper gives a number. Your test kit gives a number. They look close enough to panic about. They are not measuring the same thing: a hobby kit almost always reports the whole nitrate ion, and a toxicology paper almost always reports only the nitrogen inside it.
Given
- — Your kit reads 35 mg/L of nitrate
- — A review of nitrate toxicity recommends staying at or below 2 mg/L of nitrate nitrogen for freshwater animals4
Task. Answer all three:
- Express your kit reading as nitrate nitrogen, to one decimal place.
- Express the recommendation the way your kit would print it, to one decimal place.
- How many times over the recommendation is the tank? Round to the nearest whole number.
Read first: Chemistry, which prints the factor between the two nitrates and where it comes from.
Exercise 7 · a short series
How many Saturdays?
Same tank as Exercise 4, same bucket, same habit. You decide to do the same water change every Saturday until the nitrate reading is where you want it. To keep the arithmetic clean, assume nothing new arrives in between, which is generous: in a real tank the fish are still eating all week.
Given
- — Tank 80 liters, nitrate 40 mg/L at the start
- — 24 liters changed every Saturday, tap nitrate treated as zero
- — Nothing adds nitrate between changes
Task. After how many Saturdays does the reading first fall below 10 mg/L? Write down the reading after each change on the way, to one decimal place.
Read first: the Lake Sevan biotope, which writes down what is left after one change and then after many.
Exercise 8 · powers of ten
Ten times, or a hundred times?
Two tanks in the same room. One has a lot of driftwood in soft water and reads acid; the other sits on limestone and reads alkaline. The pH numbers are only two apart, so it sounds like a small difference. It is not a small difference, and the reason is the shape of the scale rather than anything about the tanks.
Given
- — Tank A reads pH 6.3, tank B reads pH 8.3
- — Yerevan tap water runs from pH 6.01 to pH 7.30 depending on the district1
Task. Answer both:
- How many times more hydrogen ions are there in tank A than in tank B?
- How many times more are there in the most acid Yerevan tap water than in the least acid? Give this one to one decimal place.
Read first: Materials and water, where the pH scale is read as what it actually is.
Exercise 9 · geometry, then a rate
Same water, different shape
Two people buy tanks on the same day and both spend the same money on water. One buys a long low tank for a shelf. The other buys a tall narrow one for a corner. They hold exactly the same amount of water. One of them will spend the next two years wondering why his fish gasp at dawn in August and the other one's do not.
Given
- — Tank A: water surface 100 cm by 40 cm, water 30 cm deep
- — Tank B: water surface 60 cm by 40 cm, water 50 cm deep
- — Gas transfer velocity for both, the light air movement value used in the physics table: 0.5 meters per day
Task. Work out:
- the volume of each, in liters, to show they really are the same;
- the mean depth of each, in meters;
- the time constant of re-oxygenation for each, in hours;
- how many times faster tank A recovers, and which of the two sits inside the depth rule the physics page gives for a container that can refill its own oxygen overnight.
Read first: Physics, for the time constant, the depth rule and the published table you can check part 3 against.
Exercise 10 · three conversions in a row
What the flake leaves behind
A heavily planted tank reads zero nitrate every single week, and its owner is proud of that. Zero on the kit does not mean nothing arrived. It means the plants took up at least as much as arrived. Work out how much that actually is, and the pride is better earned.
Given
- — Tank 60 liters
- — 0.8 grams of dry flake food a day, every day
- — The food is 45 percent crude protein
- — Assume all the nitrogen in that protein ends up as nitrate
Task. How much nitrate arrives in the tank each week, in milligrams per liter, as a test kit would read it? Give it to the nearest whole number. Two conversions are needed and neither is printed here: protein to nitrogen, and nitrogen to nitrate.
Read first: Water changes for the protein-to-nitrogen step, and Chemistry for the nitrogen-to-nitrate one.
Exercise 11 · a measurement, then a rate, then a prediction
A strip of tape and seven days
There is one measurement this site asks you to make yourself, because nobody can look it up for you: how fast your own tank evaporates. It depends on your flat, your window, your lid and the month. You stick a strip of tape at the water line on a Sunday, do not top up all week, and come back the next Sunday with a ruler.
Given
- — The water was 38 cm deep when you put the tape on
- — Seven days later the level has fallen 1.9 cm
- — The tank was filled from a Davtashen tap, 305 mg/L of dissolved solids1, and is topped up with the same water
- — No water changes, only topping up
Task. Work out:
- your weekly evaporation rate, as a percentage;
- how many weeks until the dissolved solids in the tank are double what came out of the tap;
- what the dissolved solids read in that week.
Read first: The arithmetic, which gives the tape measurement, the evaporation line and the doubling time.
Exercise 12 · solve for the week
When does the shrimp tank leave its range?
Caridina shrimp are kept in soft water and they are not casual about it. A keeper in Avan sets a tank up straight from the tap, which in that district is soft enough to be inside the range these shrimp want. He tops it up faithfully every week, he never changes any water, and for a while everything is beautiful.
Given
- — Tap water in Avan, Kanaker and Nork: 3.4 German degrees1
- — Evaporation 5 percent of the tank volume a week, topped up with the same tap water
- — No water changes at all
- — The shrimp want 4 to 6 German degrees
Task. Answer both:
- In which week does the hardness first pass 6 German degrees? Give the week number, not a fraction of a week.
- Express that 6 degree ceiling in milligrams per liter as calcium carbonate, to the nearest whole number.
Read first: The arithmetic for the evaporation line, and Chemistry for the hardness unit.
Exercise 13 · a power, and a logarithm to undo it
How long does the lake water last?
A family drives to Lake Sevan for the day and comes home with a jar of lake water in the boot. The plan is a small tank that is really Sevan water, with Sevan stones in it. The jar fills the tank exactly once. After that, every water change is Yerevan tap water, and the lake in the tank starts quietly leaving.
Given
- — A 5 liter jar tank, filled entirely with carried lake water at the start
- — A quarter of the water changed every week, with tap water
- — Nothing is topped up with lake water ever again
Task. Answer both:
- After how many weekly changes is less than 5 percent of the original lake water left?
- If the family changed 40 percent every week instead, after how many changes?
Read first: the Lake Sevan biotope, which writes down what fraction survives after a given number of changes.
Exercise 14 · the whole model, three parts
Does the tank cross the line, and what stops it?
This is the problem the rest of the page has been building towards. A tank in Davtashen, set up from the tap, never water changed, topped up carefully every week by somebody who believes that topping up is the same thing as maintenance. Nothing is added. Nothing goes wrong. The dissolved solids climb anyway, and Armenia has a published limit for how mineral drinking water is allowed to be, which is a useful line to measure the tank against even though nobody is going to drink it.
Given
- — Davtashen tap water: 305 mg/L of dissolved solids1
- — Evaporation 5 percent of the tank volume a week, topped up with the same tap water
- — The Armenian drinking water standard sets total mineralisation at 1,000 mg/L2
Task. Answer all three:
- With no water changes at all, in which week does the tank first pass 1,000 mg/L?
- Suppose that instead the keeper changes 20 percent of the water every week from the start. What value do the dissolved solids settle at, and what is the highest they reach between one change and the next?
- What weekly change fraction would hold the tank within 10 percent of its own tap water?
Read first: The arithmetic, which has all three equations: the rising line, the ceiling a repeated change sets, and that ceiling turned around to give you the fraction.
