BMS 1110 · Common Core (Tronc commun)Section 1.3 · Number foundations: algebra and fractions

Section 3 of 30

Fractions: parts of a whole

Read the numerator and denominator of any fraction, convert between improper fractions and mixed numbers, and add, subtract, multiply and divide fractions and mixed numbers — always reducing the answer to lowest terms.

New to this course? How each section works

Learn gives you one short idea at a time. Each idea starts with a single sentence, then explains it, gives an example, and asks you a small question you can answer straight away.
Explore shows a diagram of the same ideas. Click the numbered parts to read what each one is.
Check asks a few questions. Answer them all correctly to finish the section and unlock the next one.
“Go deeper” is optional. It adds mechanism and extra detail for when you are ready — nothing you need for the checks is hidden there.

Why this matters

Fractions are the everyday language of the ward. An order for “1/2 tablet”, a solution at “3/4 strength” or a dose of “2/3 of the 24-hour total” is a fraction, and the patient must receive that exact amount. Every dilution, drip rate and paediatric calculation later in this course is built on multiplying and dividing fractions correctly, so the rule you learn here is used again in almost every section that follows.

What you need to know first

You need the whole-number multiplication and division you already know, plus the multiplication tables. Everything in this section is built from two pictures: cutting a whole into equal parts, and finding a number that two denominators both divide into evenly.

Words to know (6)

Numerator: The top number of a fraction. It counts the parts you have, and it is the dividend because the fraction bar means “divided by”. In 3/4 of a tablet, the numerator 3 means three of the four equal pieces.
Denominator: The bottom number of a fraction. It names the size of each part and states how many parts make one whole, so it is the divisor. In 3/4 of a tablet, the denominator 4 says the tablet is cut into four equal pieces.
Equivalent fractions: Fractions that look different but name the same amount, because the numerator and the denominator were both multiplied or divided by the same number. 6/8 and 3/4 are equivalent: divide both 6 and 8 by 2 and you are back to 3/4 of a tablet.
Improper fraction: A fraction whose numerator is equal to or larger than its denominator, so its value is one or more than one. It is the easiest form to calculate with. 7/4 tablets is improper; it equals 1 3/4 tablets, the mixed form of the same dose.
Mixed number: A whole number written beside a proper fraction, such as 1 3/4. It reads easily in the ward, but must be changed to an improper fraction before multiplying or dividing. An order for 2 1/2 tablets means 2 whole tablets plus half of another one.
Complex fraction: A fraction that contains a fraction in its numerator, in its denominator, or in both, such as (3/4)/(1/2). It is only a division written vertically. (3/4)/(1/2) = 3/4 ÷ 1/2 = 1 1/2, so one and a half half-tablet doses fit into three quarters of a tablet.

One idea at a timeIdea 1 / 6

In one sentence The denominator names how many equal parts make the whole; the numerator counts how many of those parts you have.

A fraction is part of a whole — and also a division

A fraction represents a part of a whole. If a scored tablet is cut into 4 equal pieces and the order asks for 3 of them, the dose is 3/4 of a tablet. The bottom number, 4, is the denominator: it tells how many equal parts it takes to make one whole, and it names the size of each part. The top number, 3, is the numerator: it counts the actual number of those parts you are taking. This is why the numerator is the dividend and the denominator is the divisor — the bar of a fraction means “divided by”, so 3/4 is another way of writing 3 ÷ 4, and 3 ÷ 4 = 0.75. The same dose can therefore appear as a fraction now and as a decimal in section 1.4. Keep one condition in view: the whole must be cut into equal parts. Three unequal slices of a tablet are not 3/4 of anything.

For example

A 500 mL infusion bag is half empty: 1/2 of the bag has gone in, which is 1/2 × 500 = 250 mL.

Pause and check yourself

An order reads 2/3 of the 24-hour dose. Which number is the numerator, and what does the denominator tell you?

Show the answer

The numerator is 2: it counts that two of the three equal parts of the daily dose are being given. The denominator 3 says the whole daily dose is divided into three equal parts, so each part is one third of the total.

Read detailed explanations

Why inverting the divisor works: A division asks a question: how many copies of the divisor fit into the dividend? Six divided by two is three because two fits into six three times. Put the same question to fractions and inverting stops looking like a trick. Take 3/4 ÷ 1/8 and read it aloud: “how many eighths fit into three quarters?” Rewrite both quantities with a common denominator of eighths: 3/4 becomes 6/8, so the question is how many times 1/8 fits into 6/8, which is plainly 6. Algebraically, a/b ÷ c/d equals (ad)/(bd) ÷ (cb)/(bd); because both terms now carry the same denominator, the denominators cancel and only ad ÷ cb survives, which is exactly a/b × d/c. The second argument is the missing-factor argument. Division is the inverse of multiplication, so a/b ÷ c/d = x means x × c/d = a/b. Multiply both sides by d/c, and the divisor on the left is undone because c/d × d/c = 1; what remains is x = a/b × d/c. Inverting is not an arbitrary decree, it is the only move that undoes the divisor. Test it on tablets: 5/8 ÷ 1/16 asks how many sixteenth tablets fit into five eighths of a tablet, and since 5/8 = 10/16, ten sixteenths fit. A nurse who can say that sentence out loud can sanity-check any division without trusting the flip.
Complex fractions: a division written vertically: A complex fraction contains a fraction somewhere above or below its main bar, such as (3/4)/(1/8) or (1/2)/(3/4). It looks alarming and is nothing more than a division written vertically, so it obeys the same rule as any other division: invert the divisor and multiply. (3/4)/(1/8) = 3/4 ÷ 1/8 = 3/4 × 8/1 = 6. A second, often safer method clears the small denominators first: multiply the top and the bottom of the complex fraction by the lowest common denominator of all the fractions involved, here 8. The numerator becomes 3/4 × 8 = 6 and the denominator becomes 1/8 × 8 = 1, so the whole expression equals 6/1 = 6. Both routes give the same answer, and doing both is a legitimate cross-check. The clinical version of this shape is common: a half-tablet order against a quarter-tablet stock is (1/2)/(1/4) = 1/2 × 4/1 = 2, so two quarter-tablet pieces make the ordered half tablet. The supplied notes list “reducing a complex fraction” as an objective but give no procedure, so use the two methods above; whichever you choose, keep the units attached and reduce the final quotient to lowest terms before you write it on the chart.
Reducing with the greatest common factor: Reducing to lowest terms is not cosmetic. A fraction written with the smallest numbers is less likely to be misread, and the arithmetic that follows uses smaller values. The mathematical reason reduction is allowed is that dividing the numerator and the denominator by the same non-zero number multiplies the fraction by k/k, which equals 1, and multiplying by 1 changes nothing: 6/8 = (6 ÷ 2)/(8 ÷ 2) = 3/4. A fraction is in lowest terms when the greatest common factor of its numerator and denominator is 1. Two reliable ways to find that factor exist. The first is prime factorisation: write each number as a product of primes, 24 = 2³ × 3 and 36 = 2² × 3², then multiply the primes the two lists share, here 2² × 3 = 12, so 24/36 = 2/3. The second is repeated division by any common factor you can see: 24/36 ÷ 2 = 12/18, ÷ 2 = 6/9, ÷ 3 = 2/3, and the method stops when no common factor remains. When you multiply a chain of fractions, cancel before multiplying instead of reducing a large product afterwards: in 4/9 × 3/8, divide 4 and 8 by 4 and divide 3 and 9 by 3, leaving 1/3 × 1/2 = 1/6. The ward version of the same habit is to leave an answer as a mixed number only after it has been reduced, so 26/8 becomes 13/4 and then 3 1/4 tablets.

Explore the diagram

Look at the whole drawing first. Then select one numbered part and connect it to the idea you just read.

Fractions in the ward: one rule for all four operations

Four teaching panels. Panel 1 cuts a tablet into four equal parts to name the numerator and the denominator. Panel 2 compares addition and subtraction, which need a common denominator, with multiplication and division, which do not. Panel 3 flows between the improper form 7/4 and the mixed form 1 3/4 and shows reduction. Panel 4 divides fractions, inverting the divisor, to count how many 1/4-tablet doses fit into 3 1/2 tablets, and multiplies 2/3 of a 1,200 mg daily dose.

New to diagrams? Read this first

Look at the whole drawing once without clicking anything. Ask yourself only: what kind of picture is this (a comparison, a route, a sequence, a table)?
Press Play, or use the arrows, to move through the numbered parts in order. The drawing stays still; only the highlight moves.
Click a number, or its name in the list below, to read what that part is and why it matters. Use “Hide labels” when you want to test your memory.

More controls

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01 / 4Parts of a whole: numerator and denominator

If the drawing is wider than this panel, scroll it sideways. Labels stay readable; the numbered explanations below show the same ideas.

Parts of a whole: numerator and denominator

The tablet is cut into four equal pieces, and three of them are shaded, so the dose is 3/4 of a tablet. The bottom number, 4, is the denominator: it names the size of each part and is the divisor. The top number, 3, is the numerator: it counts the parts taken and is the dividend, because the bar means divided by, and 3 ÷ 4 = 0.75. Unequal pieces would not be a fraction of anything.

Schematic teaching plate: tablet quarters are drawn at equal size for the arithmetic, and the doses shown are illustrative exactly as written. Not to scale.

Key points to remember

The numerator counts the parts taken and is the dividend; the denominator names the size of the parts and is the divisor, so 3/4 is also 3 ÷ 4 = 0.75.
Multiplying the numerator and the denominator by the same number gives an equivalent fraction, and the lowest common denominator is the smallest number every denominator divides into evenly.
Fractions must share a denominator before they are added or subtracted, and must never be forced to share one before they are multiplied or divided.
To multiply, multiply numerator by numerator and denominator by denominator; a whole number goes over 1 and a mixed number becomes improper first.
To divide, invert the divisor and multiply, because a/b × b/a = 1; then reduce every answer to lowest terms so it is safe to read aloud.

Go deeper

Worked example

Reading a 7/4-tablet order and finding what is left. A ward order is written as 7/4 tablets of a drug, to be given in two parts. The first part, 1 1/2 tablets, has already been given. Before preparing the second part, the nurse converts the order into a form that can be measured and works out how much is still owed to the patient.

Read the improper fraction as a mixed number: divide 7 by 4. Four goes into seven once, with 3 left over, so 7/4 = 1 3/4 tablets. This is the amount the ward can actually measure.
Bring both amounts to the same denominator before subtracting. The denominators are 4 and 2, and 4 is the lowest common denominator, so 1 1/2 = 3/2 = (3 × 2)/(2 × 2) = 6/4 tablets.
Subtract only the numerators and keep the common denominator: 7/4 − 6/4 = 1/4 tablet.
Reduce the answer. The numerator 1 and the denominator 4 share no common factor other than 1, so 1/4 is already in lowest terms.
Check against the decimal form: 7/4 = 1.75 and 1 1/2 = 1.5, and 1.75 − 1.5 = 0.25, which is 1/4. The fraction answer and the decimal answer agree.

The second part of the dose is 1/4 tablet. Notice the order of the moves: convert the improper fraction, match the denominators, subtract the numerators, reduce, then check against a decimal. Matching the denominators was necessary because the operation was subtraction — had the order asked for a multiplication, the denominators would have stayed as they were.

Tutorial 1 — adding unlike denominators for a total dose

Over one 24-hour period a patient receives the same drug twice: 1 1/2 tablets at 08:00 and 3/4 tablet at 20:00. The chart asks for the total number of tablets given during the day, written as a mixed number.

Rule used: Fractions must share a denominator before they are added; once the denominators match, add only the numerators and keep the common denominator. A mixed number is first changed to an improper fraction: parts = whole × denominator + numerator, placed over the denominator. The unit throughout is tablets.

Change the mixed number 1 1/2 into an improper fraction, writing every move of the rule.
Reveal this step
1 1/2 = (1 × 2 + 1)/2 = 3/2 tablets. The whole tablet is two halves, and the extra half makes three halves; 3 ÷ 2 = 1.5 confirms the value.
Why can 3/2 and 3/4 not be added exactly as they stand? What has to be true before the numerators can be added?
Reveal the answer
The parts are different sizes: one fraction counts halves and the other counts quarters. Adding counts parts, so the parts must be the same size first. Both fractions must be rewritten over the lowest common denominator, which here is 4.
Find the lowest common denominator of 2 and 4, then rewrite 3/2 over it.
Reveal this step
4 is a multiple of 2, so 4 itself is the lowest common denominator. 3/2 = (3 × 2)/(2 × 2) = 6/4. Check: 6 ÷ 4 = 1.5, exactly the value of 3/2.
Add the two fractions, keeping the common denominator.
Reveal this step
6/4 + 3/4 = 9/4 tablets. Only the numerators 6 and 3 are added; the denominator 4 stays unchanged because it names the size of the parts, and the parts did not change size.
The chart wants a mixed number. Is 9/4 already in lowest terms, and what is the total?
Reveal the answer
9 and 4 share no common factor other than 1, so 9/4 is already in lowest terms. As a mixed number, 9 ÷ 4 = 2 with remainder 1, so 9/4 = 2 1/4 tablets: two whole tablets and one quarter.
Check the total against the decimals of the two original amounts.
Reveal this step
3/2 = 1.5 and 3/4 = 0.75; 1.5 + 0.75 = 2.25, which is 2 1/4. The fraction total is larger than the first dose alone and a little more than two tablets, exactly as expected.

Answer: The patient received 9/4 = 2 1/4 tablets of the drug during the 24 hours.

Keep this habit: Convert mixed numbers first, then match the denominators, then add the numerators only, then reduce. Writing the unit “tablets” on every line is what stops a total dose from being mistaken for a count of doses, and the decimal check at the end catches any slip in the common denominator.

Tutorial 2 — how many quarter-tablet doses fit the order

A ward must give 1/4 tablet of a drug with each dose. The stock box holds 3 1/2 tablets of that strength. Work out how many complete doses can be prepared from what is left, without opening a new box.

Rule used: To divide by a fraction, invert the divisor — write it upside down — and multiply: a/b ÷ c/d = a/b × d/c. Dividing by c/d and multiplying by d/c are inverse operations, because c/d × d/c = 1. Mixed numbers must be changed to improper fractions first, and the answer is a count of doses, with no unit of volume.

Change the stock, 3 1/2 tablets, into an improper fraction.
Reveal this step
3 1/2 = (3 × 2 + 1)/2 = 7/2 tablets. Three whole tablets are six halves, plus the extra half, giving seven halves of a tablet.
Read 7/2 ÷ 1/4 as a sentence about tablets. What is the question really asking?
Reveal the answer
It asks how many quarter-tablet pieces fit into seven halves, that is, into three and a half tablets. Because both quantities are parts of the same tablet, the answer will be a whole number — a count of 1/4-tablet doses.
Invert the divisor and rewrite the calculation as a multiplication.
Reveal this step
The divisor is 1/4, so inverting it gives 4/1. The calculation becomes 7/2 ÷ 1/4 = 7/2 × 4/1. The dividend, 7/2, is not inverted; only the fraction you divide by is turned over.
Why is multiplying by 4/1 allowed? What does 1/4 × 4/1 equal, and what does that prove?
Reveal the answer
1/4 × 4/1 = 4/4 = 1. Multiplying by 4/1 therefore undoes dividing by 1/4: the divisor is cancelled and the original quantity is recovered. This is why inverting the divisor works — division asks what value, times the divisor, gives the dividend, and d/c is precisely that value.
Multiply the fractions, reduce, and state the number of doses with its unit.
Reveal this step
7/2 × 4/1 = (7 × 4)/(2 × 1) = 28/2 = 14. Reducing 28/2 gives 14, so 14 doses of 1/4 tablet can be prepared. The answer is a count of doses, not a volume.
A new box holds 3 3/4 tablets. How many 1/4-tablet doses does it give, and what does the change tell you about the division?
Reveal the answer
3 3/4 = 15/4, so 15/4 ÷ 1/4 = 15/4 × 4/1 = 60/4 = 15 doses. Each extra 1/4 tablet adds exactly one dose, because the divisor is 1/4: dividing a number of quarters by one quarter simply counts the quarters available.

Answer: 3 1/2 ÷ 1/4 = 7/2 × 4/1 = 14, so 14 doses of 1/4 tablet can be prepared from the stock box.

Keep this habit: Change mixed numbers to improper fractions, invert the divisor, multiply, then reduce. Read the division aloud as “how many of these fit into that?” before you calculate, and check that the answer is a count of doses — if your result carries millilitres or milligrams, you have divided the wrong way round.

Tutorial 3 — 2/3 of a 24-hour dose, and multiplying mixed numbers

A 24-hour order for an antibiotic is 1,200 mg, given as two divided doses. The evening dose is 2/3 of the daily total. Calculate the evening dose in milligrams, then find the morning dose and check that the two parts make the whole day.

Rule used: The word “of” between a fraction and a quantity means multiply. To multiply fractions, multiply numerator by numerator and denominator by denominator; a whole number is written over 1, and a mixed number is first changed to an improper fraction: parts = whole × denominator + numerator. The product is then reduced to lowest terms, and the unit of the answer is milligrams.

Write the evening dose as a multiplication, putting the whole number over 1.
Reveal this step
The evening dose is 2/3 of 1,200 mg, that is 2/3 × 1,200, and 1,200 written over 1 is 1,200/1. The calculation is 2/3 × 1,200/1 mg.
Multiply numerator by numerator and denominator by denominator.
Reveal this step
2 × 1,200 = 2,400 and 3 × 1 = 3, so the product is 2,400/3 mg. The denominators were simply multiplied; no common denominator was needed.
Why was no common denominator needed here, when the earlier tutorial needed one?
Reveal the answer
Because the operation is multiplication, which takes a part of a quantity rather than counting equal parts. Denominators are multiplied, not compared, so no rewriting is required. Only addition and subtraction need matching denominators, which is the single rule of this section.
Divide and reduce to give the evening dose in milligrams.
Reveal this step
2,400 ÷ 3 = 800 mg. Check for reduction: 800/1,200 = 2/3, since dividing both 800 and 1,200 by 400 gives 2 and 3. The evening dose is 800 mg.
If the two doses together make the whole day, what fraction is left for the morning, and how many milligrams is it?
Reveal the answer
The whole day is 3/3 and the evening took 2/3, so the morning dose is 3/3 − 2/3 = 1/3 of the total: 1/3 × 1,200 = 400 mg. The check is 800 + 400 = 1,200 mg, the full 24-hour order.
A second order is written 1 1/2 × 3/4 tablet. Convert the mixed number, multiply and reduce.
Reveal this step
1 1/2 = (1 × 2 + 1)/2 = 3/2. Then 3/2 × 3/4 = 9/8 = 1 1/8 tablets. Nine and eight share no common factor other than 1, so 9/8 is already in lowest terms.

Answer: The evening dose is 2/3 × 1,200 = 800 mg and the morning dose is 400 mg; together they return the full 1,200 mg order.

Keep this habit: Read “of” as multiply, put whole numbers over 1, convert mixed numbers to improper fractions, multiply across, then reduce. Finish by adding the parts back together: if the two fractions of the day do not rebuild the whole, one of the conversions is wrong.

Now you try

Attempt each one on paper first. Write the formula, substitute the numbers, then check the unit of your answer.

A patient is ordered 1/2 tablet in the morning and 3/4 tablet at night of the same drug. How many tablets is that in total, and is the total more than one tablet?
Hint
The denominators are 2 and 4. Match them before adding, and remember that the numerator only is added.
Show the worked answer
4 is a multiple of 2, so the lowest common denominator is 4, and 1/2 = 2/4. Then 2/4 + 3/4 = 5/4 tablets. Reduced and converted, 5 ÷ 4 = 1 with remainder 1, so 5/4 = 1 1/4 tablets. The total is more than one tablet: one whole tablet and one quarter. The decimal check is 0.5 + 0.75 = 1.25 = 1 1/4.
An order reads 2/3 of a 900 mg daily dose. Calculate the dose in milligrams using a fraction multiplication, and show the reduction.
Hint
The word “of” means multiply. Write 900 over 1, multiply across, then reduce before you divide.
Show the worked answer
2/3 of 900 mg is 2/3 × 900/1 = (2 × 900)/(3 × 1) = 1,800/3 mg. The reduction is 1,800/3 = 600 mg, since 1,800 ÷ 3 = 600. Equivalently, cancel before multiplying: 900 ÷ 3 = 300, then 2 × 300 = 600 mg. No common denominator was needed because the operation is multiplication.
A stock bottle holds 2 1/4 tablets, and each dose is 1/8 tablet. How many full 1/8-tablet doses can be prepared, and is anything left over?
Hint
Change 2 1/4 to an improper fraction, invert the divisor 1/8, multiply, and reduce the quotient.
Show the worked answer
2 1/4 = (2 × 4 + 1)/4 = 9/4 tablets. Then 9/4 ÷ 1/8 = 9/4 × 8/1 = 72/4 = 18. Eighteen full doses of 1/8 tablet can be prepared, and nothing is left over, because 18 eighths is exactly 9/4 written over a denominator of 8. Check: 18 × 1/8 = 18/8 = 9/4.
A prescription is written 26/8 tablets. Reduce it to lowest terms, then write it as a mixed number and say which form you would read aloud to a colleague.
Hint
Find the largest number that divides both 26 and 8, reduce, then divide the numerator by the denominator for the whole part.
Show the worked answer
Both 26 and 8 divide by 2, so 26/8 = 13/4; 13 and 4 share no further common factor, so 13/4 is in lowest terms. As a mixed number, 13 ÷ 4 = 3 with remainder 1, so 13/4 = 3 1/4 tablets. Read 3 1/4 aloud, because a whole number and a fraction are harder to mishear than 26/8, and confirm the decimal 3.25 if any doubt remains.

Common misconception

The most common beginner error is to add fractions “straight across”, writing 1/2 + 1/3 = 2/5 or 3/4 + 1/2 = 4/6. It feels symmetrical and it is wrong, because denominators are names, not quantities: a half and a third are parts of different sizes, and only equal parts can be counted together. Over sixths, 1/2 + 1/3 = 3/6 + 2/6 = 5/6, which is more than either fraction and less than one whole. A second trap is carrying the same habit into multiplication, where learners invent a common denominator that is not needed: 2/3 × 3/5 = 6/15 = 2/5 with no rewriting at all. The rule divides the four operations cleanly — match to add or subtract, never match to multiply or divide.

Check your reasoning

Explain in your own words why 1/2 + 1/3 must be rewritten as 3/6 + 2/6 before it can be added, while 1/2 × 1/3 can simply be multiplied to 1/6 with no rewriting at all.

Reveal an answer

Addition counts how many parts you have, so every part must be the same size: a half and a third are different sizes, and sixths make them comparable, since 1/2 = 3/6 and 1/3 = 2/6, giving 5/6. Multiplication does not count parts; it takes a part of a part. Half of a third is one sixth whatever the original sizes were, so the numerators and denominators are simply multiplied. The distinction is the section’s one rule: denominators are matched for addition and subtraction, and left alone for multiplication and division.

Where this comes from

BMS 1110 Applied Mathematics and Medical Biostatistics lecture notes (supplied) · Unit Two “Basic Maths Review”, Lesson 1 “Fractions”, lines 84–176: fractions as parts of a whole; numerator as dividend and denominator as divisor (lines 92–97); the four reminders (98–102); proper, improper, complex and mixed fractions (103–107); improper-to-mixed and mixed-to-improper conversions (108–118); lowest common denominator and equivalent fractions (119–128); addition, subtraction, multiplication and division of fractions and mixed numbers (129–176).
Beyond the supplied notes: why inverting the divisor works · The notes state “invert (turn upside down) the divisor” at lines 170–171 without giving a reason. The common-denominator argument, the missing-factor argument and the tablet check used here are standard arithmetic that the notes assume rather than teach.
Beyond the supplied notes: complex fractions and reduction · Line 91 lists “reducing a complex fraction” as an objective and lines 103–107 define a complex fraction, but the notes give no method. The two reduction methods for complex fractions, the greatest-common-factor rule, prime factorisation and cancelling before multiplying are standard arithmetic supplied here.
A defect in the supplied equivalence examples · Lines 126–128 ask the learner to complete 2/3 = ?/12, 2/5 = ?/8 and 2/7 = ?/14. The middle item has no whole-number solution, because 8 is not a multiple of 5 and so is not a valid denominator for 2/5. That item is not reproduced; this section uses 2/3 = 8/12 and 2/7 = 4/14 instead, and states the rule that a new denominator must be a multiple of the old one.
Note on the course code · The supplied cover page reads “Course Number: BMS 110”. The curriculum catalogue in this project records “BMS 1110”; the catalogue is treated as authoritative and the notes’ cover value is retained here for provenance.

Lesson progress is saved on this device for the current learner. Practice answers stay separate and are not stored.

About this course and its sources

Bilingual lessons organised from the supplied BMS 1110 Applied Mathematics and Medical Biostatistics lecture notes (Mr. Ebangan Prosper Achuanoh). Diagrams are original educational schematics; sections that go beyond the supplied notes say so in their sources. Mathematics, clinical and French-language review are pending.