Push real numbers through the relationships you have been learning and watch what moves. Each one shows its working, names the normal range, and says what the result means. Open any panel to start; every calculation runs in your browser.
Three numbers, three questions, in order. Is the pH acidic or alkaline? Which value — carbon dioxide or bicarbonate — moved in the direction that would cause it? And has the other system moved to compensate? Compensation never fully corrects the pH, so whichever side still explains the pH is the primary problem.
Cardiac output is just rate multiplied by how much leaves on each beat. Mean arterial pressure is weighted toward diastole because the heart spends about two thirds of each cycle filling — so the average pressure organs actually see sits nearer the diastolic number than halfway between the two.
The alveolar equation asks what the oxygen pressure in the alveolus ought to be. Subtract what actually reached the artery and the difference tells you whether the problem is getting oxygen into the alveolus or getting it across the membrane. A normal gradient with a low PaO₂ points at hypoventilation; a wide gradient points at the membrane, the blood flow, or a shunt.
Charge has to balance. The measured cations exceed the measured anions because some anions — proteins, phosphate, lactate, ketones — are not on the panel. A widened gap means an unmeasured acid has arrived. Albumin is itself one of those unmeasured anions, so a low albumin hides a gap that is really there.
Clearance is a volume, not an amount: the millilitres of plasma completely scrubbed of a substance each minute. Compare it with the filtration rate and the nephron tells on itself — clearance above GFR means the tubule secreted the substance, below GFR means it reabsorbed some back.
Glucose that cannot enter cells stays in the plasma and pulls water out with it, diluting the sodium. The sodium reading falls even though total body sodium has not changed. Correcting for glucose tells you what the sodium would read once the glucose is brought down — which is why a "low" sodium in severe hyperglycaemia often needs no sodium at all.
This is the calculation that explains why a patient can have a perfect saturation and still be starved of oxygen. Almost all the oxygen is carried on haemoglobin; the amount dissolved in plasma is tiny. Halve the haemoglobin and you halve the delivery, whatever the pulse oximeter says.
Body surface area matters more than weight for anything distributed through the circulation, which is why cardiac index and many drug doses are expressed per square metre. Resting energy expenditure is what the body spends doing nothing at all — roughly two thirds of a sedentary day's total.