BMS 1110 · Common Core (Tronc commun)Section 1.1 · Number foundations: algebra and fractions

Section 1 of 30

Algebra: the language of unknowns

Name the variable, coefficient and constant in a simple expression, and add, subtract or multiply signed numbers correctly.

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

Every dose you will ever calculate is a small equation in disguise. “Give 4 mg of a drug that comes as 2 mg per mL” is really the question “what number times 2 gives 4?” Before you can answer it quickly and safely, you need the vocabulary that names the parts and the sign rules that keep the arithmetic honest. Take one step at a time: name the unknown, write the operation and check the answer by putting it back into the equation.

What you need to know first

You need only arithmetic you already have: adding, subtracting, multiplying and dividing whole numbers. Negative numbers may feel rusty, and that is expected — the first idea below rebuilds them from a picture you already understand, a number line.

Words to know (6)

Variable: A quantity that can change, usually written as a letter, standing in for a number you do not know yet or one that differs from patient to patient. In 4 × W = dose, the letter W is the variable: it is the patient’s weight, and it changes from one patient to the next.
Coefficient: The number written immediately in front of a variable, so that “3x” means “three times x”. A variable written alone has an invisible coefficient of 1. In 2.5 mL per kg, the 2.5 is the coefficient attached to the weight in kilograms.
Constant: A quantity whose value does not change: a plain number with no letter attached to it. In x + 4 = 28, both 4 and 28 are constants. They stay the same whatever x turns out to be.
Signed number: A number that carries a + or − sign, showing both a size and a direction. Negative numbers lie to the left of zero, positive numbers to the right. A fluid balance of −200 mL means the patient lost 200 mL more than they took in.
Expression: A mathematical phrase built from numbers, variables and operators. It states a relationship but makes no claim that two things are equal. 3x + 4 is an expression. Add an equals sign and something on the other side and it becomes an equation.
Operator: A symbol that tells you what to do with two quantities: +, −, × and ÷ are the operators used in arithmetic. In 3x + 4, the + is the operator that joins the term 3x to the constant 4.

One idea at a timeIdea 1 / 5

In one sentence Algebra replaces a number you do not know yet with a letter, so that one rule works for every patient.

Letters stand for numbers you do not know yet

Algebra is the branch of mathematics in which letters and other general symbols stand for numbers and quantities. Instead of writing out a separate calculation for every patient, you write one expression containing a letter and use it again and again. In the equation 3x + 4 = 28, the letter x is the variable — the quantity that can change. The number 3 attached to it is the coefficient, telling you to multiply. The numbers 4 and 28 are constants: they do not change. Expressions, equations, inequalities and functions are all built from these same parts. Recognising which part is which is not a formality; it is what tells you which quantity in a medication order you are actually solving for.

For example

A paediatric syrup gives 2 mg of drug for every kilogram of body weight. Written as algebra, dose = 2 × W, where W is the child’s weight. One rule, any child.

Pause and check yourself

In the medication rule dose = 15 × W, which symbol is the variable and which is the coefficient?

Show the answer

W is the variable, because it stands for the patient’s weight and changes from patient to patient. 15 is the coefficient: it is the fixed number the variable is multiplied by, here 15 mg per kilogram.

Read detailed explanations

Why the sign rules are forced, not chosen: It is worth proving to yourself that the sign rules are the only ones available, because a rule you have derived is a rule you will not forget. Start with the two facts everyone accepts: multiplying any number by zero gives zero, and the distributive law works, meaning a(b + c) = ab + ac. Now take a negative a, say −2, and multiply it by the sum 3 + (−3), which is zero. Distributing gives (−2)(3) + (−2)(−3) = 0. The first product, (−2)(3), is −6 by the familiar rule. So −6 + (−2)(−3) = 0, which forces (−2)(−3) to be +6. There is no freedom here: if you break the rule that a negative times a negative is positive, you also break the distributive law, and then ordinary algebra — opening brackets, collecting terms, solving equations — stops working. That matters to a nurse far beyond the classroom, because every dosage formula is an equation and every conversion factor is a fraction, and both rely on the distributive law. When a colleague says the rules are arbitrary, the counter is concrete: choose the other rule and (dose ÷ stock) × volume no longer means the same thing to two people. The same argument extends to division. Dividing by −3 is multiplying by the reciprocal −1/3, and the reciprocal carries a negative sign, so two negatives in a division again give a positive. The rule is one rule, not two.
The real number line, intervals and why −2 beats −7: The real numbers are the union of the rational numbers — those you can write as a fraction of two integers — and the irrational numbers, which cannot be written that way, such as the square root of 2 or pi. Between any two real numbers there is always another one, so the line has no gaps, and this is what allows you to treat every physical measurement as a point on it. The ordering of the line is the source of an everyday clinical trap. On the line, larger means further right, so −2 is greater than −7 even though the digit 7 is larger than the digit 2. A patient whose recorded fluid balance is −2 units is in a better position than one at −7 units, and a temperature change of −2 °C is a larger increase in value than −7 °C. Absolute value is the second idea the line teaches: it discards direction and keeps size, written with two vertical bars. |−7| = 7 and |−2| = 2, so the patient at −7 has the bigger deficit while the value −2 is the larger number. Beginners who confuse “bigger number” with “bigger magnitude” misread exactly these charts. Finally, the line gives us intervals, which are how reference ranges are written. The notation “normal range 3.5–5.0” is an interval, and a measurement is either inside it or outside it; an inequality such as 3.5 ≤ x ≤ 5.0 means x is at least 3.5 and at most 5.0. Reading an inequality as a statement about a stretch of the number line, rather than as a mysterious symbol, makes the next section’s inequality work almost automatic.
Carrying units through every calculation: Unit analysis is the cheapest safety check in clinical arithmetic and it costs nothing but discipline. Treat a unit like an algebraic symbol: it can be multiplied, divided and cancelled exactly as a variable can. If a coefficient is written in milligrams per kilogram and you multiply by a weight in kilograms, the kilogram symbol appears once above and once below the line and cancels, leaving milligrams. If the unit that survives is not the unit your answer requires, you have made an error — either you used the wrong conversion or you inverted a fraction — and you know it before the answer reaches a patient. This is why the notes insist on expressing weight in kilograms whenever an order is written per kilogram: a coefficient in mg/kg multiplied by a weight in pounds produces a meaningless unit, and a meaningless unit means a wrong dose. The method also catches the classic inversion, in which a learner divides where they should multiply. Multiplying by the wrong form of a conversion factor changes the unit into its inverse, and the mismatch is visible immediately. Two habits make this work in practice. First, write the unit on every line, not only on the answer — the intermediate lines are where the cancellation happens. Second, when you convert between units, write the conversion as a fraction equal to one, for example 1,000 mg over 1 g, and arrange it so the unwanted unit cancels. There is no need to remember whether to multiply or divide; the units decide, and they never make an arithmetic slip.

Explore the diagram

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

Signed numbers: direction on the line

Three panels. The first shows addition and subtraction as movement, with both example journeys starting at 3 and ending on the tick they claim. The second shows ordering by position, so −2 is greater than −7. The third names the parts of an expression and states the order of operations.

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 / 4Adding and subtracting as movement

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

Adding and subtracting as movement

Both arrows start on the tick for 3 because 3 is the number being changed. Adding 2 moves two ticks right and lands exactly on 5; subtracting 5 moves five ticks left, crosses zero, and lands exactly on −2. Reading arithmetic as movement makes the sign of an answer predictable instead of guessed, which is why a signed fluid balance is easier to check than to remember.

Schematic number lines at a regular scale; ticks mark whole numbers only, and each label sits on the tick it names. The two journey arrows in the first panel are drawn at different heights so they stay distinguishable where they cross.

Key points to remember

A variable is a letter standing for a quantity that changes; a coefficient is the number multiplied by it; constants never change.
On the number line, adding moves right and subtracting moves left; subtracting a negative moves right.
Two numbers with the same sign multiply or divide to a positive; opposite signs give a negative.
BODMAS: brackets, then orders, then division and multiplication left to right, then addition and subtraction left to right.
Write units on every line. A coefficient carries the unit “per” the variable, and the units should cancel to the unit of the answer.

Go deeper

Worked example

Reading a rule as algebra before using it. An intensive-care unit uses a rule of thumb written as Rate(mL/hr) = 3 × W, where W is the patient’s weight in kilograms. A patient weighs 70 kg. Before touching the infusion, the nurse must say out loud which part is the variable, which is the coefficient, and what the rule predicts.

Name W. It stands for the patient’s weight, which is different for every patient, so W is the variable.
Name 3. It is the number fixed by the rule and multiplied by the weight, so 3 is the coefficient. Its unit is mL/hr per kg.
Substitute the patient’s value for the variable: Rate = 3 × 70.
Multiply and carry the unit: 3 × 70 = 210, so the rule predicts 210 mL/hr.
Check the unit result for sense. The coefficient measured mL/hr per kg and the variable measured kg, so kilograms cancel and the answer is left in mL/hr — the unit the pump needs.

Naming the parts before substituting is what makes the arithmetic safe. The same five moves work for every proportional rule you will meet: identify the variable, identify the coefficient, substitute, calculate and check the unit.

Tutorial 1 — working with a signed fluid balance

A patient’s chart shows a deficit of 800 mL on the morning shift, an intake of 1,150 mL on the afternoon shift, and a further deficit of 250 mL overnight. Find the running balance, using signed numbers rather than words.

Rule used: Write a deficit as a negative number and an intake as a positive number, then add them in order from left to right. On the number line, each addition is a movement: positive moves right, negative moves left.

Translate each sentence into a signed number. Write the three numbers in the order the shifts occurred.
Reveal this step
A deficit of 800 mL is −800. An intake of 1,150 mL is +1,150. A deficit of 250 mL is −250. The sequence is −800, +1,150, −250.
In which direction will the balance move during the afternoon shift, and why?
Reveal the answer
To the right, because the afternoon entry is positive. A positive number always moves you to the right on the number line, which means the balance becomes less negative — the patient is catching up.
Add the first two numbers. Do not jump to the third yet — keeping the moves separate is what prevents sign errors.
Reveal this step
−800 + 1,150. Start at −800 and move 1,150 to the right. You pass zero after 800 of the movement and have 350 left, so the running balance is +350 mL.
Add the overnight entry to the running balance.
Reveal this step
+350 + (−250) = +100. Starting at +350 and moving 250 to the left lands on +100 mL.
The patient asks: “Am I in deficit?” What is the correct answer, and what would you say if the total had been −100 instead?
Reveal the answer
The total is +100 mL, a small positive balance, so the patient is not in deficit. If the total had been −100 mL, the answer would be yes: a negative sign means more fluid has gone out than come in, so the patient is still 100 mL short.

Answer: The running balance is +100 mL, meaning the patient finished the 24 hours approximately 100 mL ahead.

Keep this habit: Write every deficit as a negative number before you add anything. Adding signed numbers left to right, one move at a time, is faster than trying to hold several directions in your head — and it is much easier to check.

Tutorial 2 — evaluating a dosage expression in the right order

A stock ampoule contains 250 mg of a drug in 5 mL. A doctor has crossed out and rewritten the order twice, and the ward record now reads: 2 × (500 mg ÷ 250 mg) × 5 mL + 10 mL. Before you decide whether that is a sensible order at all, evaluate the expression correctly.

Rule used: BODMAS: Brackets first, then Orders, then Division and Multiplication from left to right, then Addition and Subtraction from left to right. Brackets override everything, and multiplication and division rank equally.

Which part of 2 × (500 ÷ 250) × 5 + 10 do you evaluate first, and why?
Reveal the answer
The bracket, (500 ÷ 250), because Brackets come first in BODMAS and they override the normal ranking. Nothing outside the bracket can be done until the bracket has a single value.
Evaluate the bracket and write the expression again.
Reveal this step
500 ÷ 250 = 2, so the expression becomes 2 × 2 × 5 + 10.
Work through the multiplications from left to right.
Reveal this step
2 × 2 = 4, then 4 × 5 = 20. The expression is now 20 + 10.
The addition is still waiting. Would the answer change if it were 2 × 2 × (5 + 10) instead? Show both.
Reveal the answer
Yes, and by a large margin. 2 × 2 × 5 + 10 = 20 + 10 = 30. But 2 × 2 × (5 + 10) = 2 × 2 × 15 = 60. Moving the bracket doubles the answer — which is exactly why nurses never move a bracket that a colleague wrote.
Finish the addition and state what the answer means in millilitres.
Reveal this step
20 + 10 = 30. Read with the units of the ampoule, the expression gives 30 mL — six times the 5 mL that the ampoule contains, so the order is not administrable and must be queried.

Answer: 2 × (500 ÷ 250) × 5 + 10 = 30, read as 30 mL — an impossible volume for a 5 mL ampoule.

Keep this habit: Brackets first, then division and multiplication from left to right, then addition and subtraction. Finish by attaching the unit and asking whether the size of the answer makes sense for the container you have.

Now you try

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

A child’s weight is recorded as 18 kg and the syrup supplies 2.5 mg per kg. Write the rule as an expression, name the variable and the coefficient, then calculate the dose.
Hint
The coefficient carries the unit “mg per kg”. Substitute the weight only after you have written the expression with the letter in it.
Show the worked answer
Dose = 2.5 × W, where W is the variable (weight in kg) and 2.5 is the coefficient (mg per kg). Substituting W = 18 gives 2.5 × 18 = 45 mg. The kilograms cancel, leaving milligrams.
Evaluate −6 − (−9) and explain the sign of the result using the number line.
Hint
Subtracting a negative is adding. Rewrite the expression with a plus sign before you calculate anything.
Show the worked answer
−6 − (−9) = −6 + 9 = +3. Starting at −6, subtracting a negative reverses the direction and moves you right by 9; you pass zero and land 3 places to the right of it, so the answer is positive.
A colleague calculates 40 ÷ 2 + 3 × 6 and writes 48. Evaluate the expression correctly and explain where the colleague went wrong.
Hint
Division and multiplication both outrank addition, and they are done from left to right as they appear.
Show the worked answer
40 ÷ 2 = 20 and 3 × 6 = 18, so the correct value is 20 + 18 = 38. The colleague’s 48 comes from reading the sum 2 + 3 as though it were bracketed: 40 ÷ (2 + 3) × 6 = 8 × 6 = 48, which inserts a bracket that was never written. The other plausible misreading, 40 ÷ (2 + 3 × 6) = 40 ÷ 20 = 2, is equally unsupported. When a colleague’s answer cannot be reproduced from the expression as written, the expression itself is what has to be checked.
A patient’s fluid chart reads −1,500 mL, then −400 mL, then +2,000 mL, then −50 mL. Find the final balance and state whether the patient is ahead or behind.
Hint
Add the two deficits first to make one large negative number, then add the intake, then the last small deficit.
Show the worked answer
−1,500 + (−400) = −1,900. Then −1,900 + 2,000 = +100. Then +100 + (−50) = +50 mL. The patient finishes 50 mL ahead, so there is no deficit to report.

Common misconception

Beginners often believe that two negatives always make a negative, because “minus is bad”. That is true for addition — −5 + (−3) = −8 — but false for multiplication and division, where −5 × (−3) = +15. The confusion comes from treating the minus sign as a mood rather than an instruction. In signed arithmetic a minus sign means “in the opposite direction”, and a product of two negative numbers reverses twice, arriving back at positive. Keeping addition and multiplication rules apart is the whole skill.

Check your reasoning

Explain in your own words why writing (dose ÷ stock) × volume with brackets is safer than writing dose ÷ stock × volume without them, even though the arithmetic is identical.

Reveal an answer

The arithmetic is not identical in meaning to a reader who is tired or distracted. Without brackets, a colleague must know the left-to-right rule for division and multiplication and apply it correctly under pressure. With brackets, the intended first move is visible and cannot be misread. In a setting where two nurses check a calculation independently, brackets make the intended order explicit for both, which is a safety gain even though the numbers come out the same.

Where this comes from

BMS 1110 Applied Mathematics and Medical Biostatistics lecture notes (supplied) · Unit One “Algebra”, lines 20–57: variable, coefficient, operator and constant; real numbers; number-line addition and subtraction; products of signed numbers.
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.
Beyond the supplied notes: the derivation of the sign rules · The distributive-law argument that a negative times a negative is positive, absolute value, intervals and unit analysis are standard algebra that the notes assume rather than teach.

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.