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Introduction — the most improvable domain on the exam

This is the chapter where the arithmetic lives, and arithmetic is the one part of the PTCE you can reliably move from weak to strong in a fortnight. The reason is structural: the exam changes the numbers, not the methods. There are perhaps twenty genuinely different calculation skills in pharmacy practice. Once you can do each one without looking anything up, a new question is a new set of digits in a shape you already recognise.

Three rules before any of it.

Write the units down and carry them through. Most wrong answers in pharmacy maths are not arithmetic failures — they are unit failures. Milligrams got divided by millilitres and nobody noticed. If your units cancel to the unit the question asked for, you are probably right; if they do not, you are certainly wrong.

Convert before you calculate. Pounds to kilograms, teaspoonfuls to millilitres, grams to milligrams — do all of it first, in a separate line, before any division happens.

Sanity-check the size of the answer. A child's dose that came out larger than an adult's, an infusion that will finish in four minutes, a days supply of 900 — these are not close calls. Ask whether the number could be true before you write it down.

And one rule that is about safety rather than arithmetic: a calculation that produces a surprising number is escalated, not trusted. If you work a dose and it looks wrong, you do not adjust it to look right. You take it to the pharmacist.

4.1a Sig codes and prescription abbreviations

Reading a sig

A prescriber compresses directions into abbreviations, most of them from Latin. The technician's job is to expand them into plain patient language stating how much, by what route, how often, and for what, written at a reading level the patient can actually use.

Category	Code	Meaning
Frequency	QD / q.d.	once daily — banned abbreviation, spell out "daily"
	BID	twice daily
	TID	three times daily
	QID	four times daily
	QAM	every morning
	QPM	every evening
	QHS	at bedtime
	q4h, q6h, q8h, q12h	every 4, 6, 8, 12 hours
	QOD	every other day — banned, spell out
	TIW	three times a week — banned, write "3 times weekly"
	PRN	as needed
	STAT	immediately
Route	PO	by mouth
	SL	sublingual
	PR	per rectum
	IV / IM / SUBQ	intravenous / intramuscular / subcutaneous
	OD / OS / OU	right eye / left eye / each eye — banned, spell out
	AD / AS / AU	right ear / left ear / each ear — banned, spell out
	NGT	nasogastric tube
Quantity and form	tab / cap	tablet / capsule
	gtt	drop
	tsp / tbsp	teaspoonful (5 mL) / tablespoonful (15 mL)
	ss	one half
	ut dict / UD	as directed
	aa	of each
	q.s.	a sufficient quantity
	q.s. ad	add a sufficient quantity to make
Timing and condition	AC	before meals
	PC	after meals
	HS	at bedtime
	NPO	nothing by mouth
	disp	dispense
	sig	write on the label

Worked expansion

The sig i tab PO BID PRN pain, disp #20 expands to:

• i tab — one tablet
• PO — by mouth
• BID — twice daily
• PRN pain — as needed for pain
• disp #20 — dispense 20 tablets

The patient label reads: "Take one tablet by mouth twice daily as needed for pain." Not "1 tab PO BID PRN," and not "take as directed" — the patient has to be able to follow it.

Roman numerals

Quantities and doses still appear in Roman numerals, and misreading one is a quantity error.

Symbol	Value
ss	½
i / I	1
v / V	5
x / X	10
l / L	50
c / C	100
d / D	500
m / M	1,000

The rule: a smaller numeral after a larger one is added; a smaller numeral before a larger one is subtracted. So XXX is 30, XL is 40, LX is 60, XC is 90, CX is 110, and iss is 1½.

The wrong turn: reading XL as 60 instead of 40, or XC as 110 instead of 90. Whenever a smaller symbol sits to the left, subtract.

Medical terminology and symbols

A handful of terms and symbols recur in orders and profiles:

Term or symbol	Meaning
Rx	prescription, "recipe"
℞ only	prescription-only (legend) product
c̄	with
s̄	without
> / <	greater than / less than
↑ / ↓	increase / decrease
Δ	change
@	at — avoid in orders; mistaken for "2"
HTN	hypertension
DM	diabetes mellitus
CHF	congestive heart failure
COPD	chronic obstructive pulmonary disease
GERD	gastro-oesophageal reflux disease
UTI	urinary tract infection
URI	upper respiratory infection
RA	rheumatoid arthritis
CKD	chronic kidney disease
NKDA	no known drug allergies
hypo- / hyper-	below normal / above normal
-itis	inflammation
-emia	in the blood
-algia	pain
brady- / tachy-	slow / fast

### Key numbers & deadlines — reading orders 5 mL per teaspoonful. 15 mL per tablespoonful. These two conversions appear in more days-supply questions than any others. Banned in orders: U, IU, QD, QOD, cc, µg, MS, D/C, TIW, SC/SQ, and the eye/ear abbreviations. Roman numerals: smaller before larger subtracts. XL = 40, not 60. iss = 1½. The patient label is written in plain language, never in the prescriber's shorthand.

4.1b The conversions everything else rests on

The metric ladder

Every metric conversion in pharmacy is a factor of 1,000, and every error is a decimal point that moved the wrong number of places.

From	To	Multiply by
kilogram (kg)	gram (g)	1,000
gram (g)	milligram (mg)	1,000
milligram (mg)	microgram (mcg)	1,000
litre (L)	millilitre (mL)	1,000

Worked example — grams to milligrams.

An order calls for 0.25 g of a drug. The stock is labelled in milligrams. - Step 1. Identify the factor: g → mg is × 1,000. - Step 2. 0.25 g × 1,000 = 250 mg. Answer: 250 mg.

The wrong turn: moving the decimal point two places instead of three, which produces 25 mg — one tenth of the dose. This is the single most common metric error in pharmacy, and it produces exactly the tenfold discrepancy that the leading-zero and trailing-zero rules in Chapter 3 exist to prevent. If a conversion produces an answer that is off by a factor of ten, count the places again.

Household and apothecary conversions

These are conventional equivalences. A well-written exam question states the factor it wants you to use, and you should use the stated factor rather than a more precise one.

Measure	Conventional metric equivalent
1 teaspoonful (tsp)	5 mL
1 tablespoonful (tbsp)	15 mL
1 fluid ounce (fl oz)	30 mL
1 cup	240 mL
1 pint	480 mL
1 quart	960 mL
1 gallon	3,840 mL
1 pound (lb)	454 g
1 kilogram (kg)	2.2 lb
1 inch	2.54 cm

Worked example — fluid ounces to millilitres.

A prescription directs the patient to take 8 fluid ounces of a solution. - Step 1. Conventional factor: 1 fl oz = 30 mL. - Step 2. 8 × 30 = 240 mL. Answer: 240 mL.

The wrong turn: using the weight ounce (about 28.35 g) instead of the fluid ounce. That produces 226.8, and more importantly it produces grams where the question asked for millilitres. The unit check catches it: you were asked for a volume and you produced a mass.

Temperature conversion

°C = (°F − 32) × 5/9 · °F = (°C × 9/5) + 32

Worked example — Fahrenheit to Celsius.

A refrigerator log reads 41 °F. Is the unit within the 2–8 °C vaccine range? - Step 1. Subtract 32: 41 − 32 = 9. - Step 2. Multiply by 5/9: 9 × 5 ÷ 9 = 5. - Step 3. Compare: 5 °C is inside 2–8 °C. In range. Answer: 5 °C, in range.

Worked example — Celsius to Fahrenheit.

A label says store at 25 °C. What is that in Fahrenheit? - Step 1. Multiply by 9/5: 25 × 9 ÷ 5 = 45. - Step 2. Add 32: 45 + 32 = 77. Answer: 77 °F — which is the upper end of controlled room temperature, 20–25 °C / 68–77 °F.

The wrong turn: doing the operations in the wrong order — multiplying before subtracting when converting °F to °C, or adding 32 before multiplying when going the other way. Converting 41 °F by multiplying first gives 41 × 5/9 = 22.8, then − 32 = −9.2, which is nonsense but not obviously nonsense on a stressful exam. Subtract first going down; add last coming up.

Ratio and proportion — the workhorse

Most pharmacy calculations reduce to one setup: a known relationship on one side, the unknown on the other, in the same units, cross-multiplied.

Worked example — how much liquid contains the dose.

Stock is 250 mg per 5 mL. The dose ordered is 400 mg. How many mL? - Step 1. Write the known: 250 mg / 5 mL. - Step 2. Write the unknown in the same orientation: 400 mg / x mL. - Step 3. Cross-multiply: 250 × x = 400 × 5 → 250x = 2,000. - Step 4. Divide: x = 2,000 ÷ 250 = 8. Answer: 8 mL.

The wrong turn: setting the proportion up inverted — 250/400 = 5/x worked the wrong way round gives 3.125 mL, which is less than the volume that holds the smaller 250 mg dose. That is the sanity check: a bigger dose needs a bigger volume. If your answer moved the wrong direction, you inverted the ratio.

4.1c Days supply — six different products, six different setups

Days supply = total quantity dispensed ÷ amount used per day. The formula never changes. What changes is what counts as "amount used per day," and that is where the errors are.

Solid oral dosage forms

Worked example.

90 tablets, sig one tablet three times daily. - Step 1. Amount per day = 1 × 3 = 3 tablets. - Step 2. 90 ÷ 3 = 30. Answer: a 30-day supply.

The wrong turn: dividing by the number of tablets per dose rather than per day, or ignoring the frequency entirely and answering 90 days. A quantity divided by a dose is not a days supply.

Split and uneven daily doses

Worked example.

90 tablets, sig one tablet in the morning and two at bedtime. - Step 1. Add the day's doses: 1 + 2 = 3 tablets per day. - Step 2. 90 ÷ 3 = 30. Answer: a 30-day supply.

The wrong turn: treating "twice daily" as the daily total and dividing by 2, which gives 45 days. Count tablets per day, not administrations per day. These are only the same number when every dose is one unit.

Liquids

Worked example.

A 300 mL bottle of suspension, sig one teaspoonful three times daily. - Step 1. Convert the dose: 1 tsp = 5 mL. - Step 2. Amount per day = 5 mL × 3 = 15 mL. - Step 3. 300 ÷ 15 = 20. Answer: a 20-day supply.

The wrong turn: dividing the bottle by the number of doses per day instead of the millilitres per day — 300 ÷ 3 = 100 days, which would be five times the true figure and a guaranteed insurance rejection. For liquids the denominator is always mL per day.

Insulin

Worked example.

One 10 mL vial of U-100 insulin, patient uses 40 units per day. - Step 1. Units in the vial: 100 units/mL × 10 mL = 1,000 units. - Step 2. 1,000 ÷ 40 = 25. Answer: a 25-day supply.

The wrong turn: dividing the millilitres by the units — 10 ÷ 40 = 0.25 — which produces a quarter of a day. The unit check kills it instantly: mL ÷ units does not give days. Convert the vial to units first, every time.

The second wrong turn: assuming 100 units/mL when the product is concentrated. A U-300 pen holds three times the units of a U-100 pen of the same volume, and a days-supply calculation that assumes U-100 will be a third of the true figure. Read the strength.

Eye drops

Drop counts depend on the dropper, so the question must give you the drops-per-millilitre figure. Use the figure you are given.

Worked example.

A 5 mL bottle, sig one drop in each eye twice daily. The question states 20 drops per mL. - Step 1. Drops in the bottle: 5 mL × 20 = 100 drops. - Step 2. Drops per day: 1 drop × 2 eyes × 2 times daily = 4 drops. - Step 3. 100 ÷ 4 = 25. Answer: a 25-day supply.

The wrong turn: forgetting that "each eye" doubles the drops. Counting 2 drops per day instead of 4 gives 50 days — exactly double, and exactly the sort of error that lets a glaucoma patient run out two weeks early.

Inhalers and metered devices

Worked example.

An inhaler labelled 200 actuations, sig two puffs four times daily. - Step 1. Actuations per day: 2 × 4 = 8. - Step 2. 200 ÷ 8 = 25. Answer: a 25-day supply.

The wrong turn: dividing by the number of administrations (4) rather than actuations (8), giving 50 days. Two puffs is two actuations.

Topicals

Worked example.

A 30 g tube, sig apply 1 gram to the affected area twice daily. - Step 1. Grams per day: 1 × 2 = 2 g. - Step 2. 30 ÷ 2 = 15. Answer: a 15-day supply.

The wrong turn: treating the tube size as a volume in millilitres. A cream is dispensed by weight; 30 g is not 30 mL, and the two are only equal when the specific gravity is exactly 1.

### Key numbers & deadlines — days supply Days supply = total quantity ÷ amount used per day. The denominator is in the same unit as the quantity. Liquids: convert to mL per day first. 1 tsp = 5 mL; 1 tbsp = 15 mL. U-100 insulin: 100 units/mL, so a 10 mL vial is 1,000 units. Check for U-200, U-300 and U-500 products. "Each eye" doubles the drops. So does "each ear" and "each nostril." Two puffs four times daily is eight actuations a day, not four. Creams and ointments are dispensed by weight, not volume. A days-supply error is both a billing problem and a safety problem. Too long and the patient runs out early with no refill; too short and a controlled substance accumulates.