BMS 1110 · Common Core (Tronc commun)Section 2.4 · Measurement, ratios and conversions

Section 9 of 30

Concentration, ratios and dilutions

Read a stock label such as 500 mg/5 mL, 1:1000 or 0.9% as an amount of drug in a stated volume, work out the volume that contains a prescribed dose, and dilute a stock solution while keeping the amount of drug constant.

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Why this matters

Almost no medicine reaches a patient exactly as it left the manufacturer. A stock solution is measured out, diluted, drawn into a syringe or added to a bag, and every one of those moves changes the volume the drug sits in. The strength on the label therefore describes only the starting point. Nurses who understand that concentration is an amount divided by a volume, and that dilution changes the volume without changing the amount, can follow a drug through every container it enters. Nurses who do not are reduced to trusting a remembered factor, and a remembered factor is exactly what fails at the end of a long shift.

What you need to know first

You need division and the metric conversions of the previous sections: 1 g = 1,000 mg and 1 L = 1,000 mL. Be comfortable changing grams into milligrams before dividing, because every strength on this page eventually has to be read in mg/mL. No algebra beyond a simple multiplication — and one rearrangement, written out where it is used — is required.

Words to know (5)

Concentration: The amount of drug contained in a stated volume of liquid, always written as an amount per volume, for example 100 mg/mL or 500 mg/5 mL. A vial labelled 500 mg/5 mL has a concentration of 100 mg/mL, because 500 ÷ 5 = 100.
Stock solution: The concentrated solution you start from, the one named on the vial or ampoule label, before any diluent is added. Lidocaine 2% is the stock solution; the 0.5% the patient receives is made from it.
Diluent: The fluid added to a stock solution to increase its volume — usually sterile water, saline or glucose — which contains no drug of its own. Adding 8 mL of saline to 2 mL of solution makes the diluent and raises the volume to 10 mL.
Ratio strength: A strength written with a colon, such as 1:1000, meaning 1 g of drug made up to 1,000 mL of solution. The left number is a mass, the right number a volume. Adrenaline 1:1000 is 1,000 mg in 1,000 mL, that is 1 mg/mL.
Percent solution: A strength written as a percentage, meaning grams of drug per 100 mL of solution for a dissolved solid. A 0.9% sodium chloride solution holds 0.9 g, that is 900 mg, in every 100 mL.

One idea at a timeIdea 1 / 5

In one sentence A concentration states how much drug is dissolved in a given volume, written as an amount per volume such as 500 mg/5 mL.

Concentration is an amount of drug inside a stated volume

A stock label does not tell you how much drug the bottle holds in isolation; it tells you how much drug sits in the volume printed beside it. The form is always amount per volume: 500 mg/5 mL, 250 mg/1 mL, 1 g/10 mL. The first number is a mass of drug and the second is the volume of liquid that contains it. Because both numbers matter, the same bottle can be described in two ways: 500 mg/5 mL and 100 mg/mL are the same strength, since 500 ÷ 5 = 100. Reading the label as a rate — this many milligrams in every millilitre — is what lets you scale it to any volume you need. Before any calculation, write the strength as a fraction, with the drug above the line and the volume below it, so that the units can cancel later.

For example

A vial is labelled 500 mg/5 mL. Dividing both numbers by 5 gives 100 mg/mL, so every millilitre contains 100 mg; two millilitres therefore contain 200 mg.

Pause and check yourself

A vial is labelled 1 g/10 mL. What is its concentration in mg/mL, and how much drug is in 4 mL?

Show the answer

1 g = 1,000 mg, so the concentration is 1,000 ÷ 10 = 100 mg/mL. Four millilitres therefore contain 4 × 100 = 400 mg of drug.

Read detailed explanations

What a strength statement really says: weight/volume, volume/volume and the ratio convention: A strength statement is a small specification, and the sign that introduces it decides which quantities are being compared. A percentage of a solid dissolved in a liquid is a weight/volume strength: the drug is weighed in grams and the liquid is measured up to 100 mL. That is why 5% glucose holds 5 g of glucose in every 100 mL of solution, and not 5 g of glucose in 100 g of fluid. When both components are liquids, the same sign usually means volume/volume instead: 70% alcohol is 70 mL of alcohol made up with water to a final volume of 100 mL. Creams and ointments are different again, because they are weighed rather than poured, so 1% hydrocortisone cream means 1 g of drug in 100 g of cream. There the density of the base is needed before any mg/mL figure can be produced, and the arithmetic of a dilution no longer applies unchanged. Ratio strengths follow the weight/volume convention whenever the drug is a solid: 1:1000 means 1 g made up to 1,000 mL of solution, which is 1 mg/mL, and 1:10,000 means 0.1 mg/mL. The notation is a survival from apothecary practice, when a grain of a substance was dissolved in an ounce of liquid, and it still appears on adrenaline, lidocaine and potassium permanganate labels today. Its danger is ambiguity: 1:1000 can be read as one part of drug in one thousand parts of the finished product, which for a dilute solid gives the same answer because the dissolved solid adds almost no volume, but for two concentrated liquids 1:1 means equal volumes rather than one part in one part. Convert every ratio strength into milligrams per millilitre before you calculate, and say the units out loud; reading the 1 of 1:1000 as one milligram instead of one gram understates the dose a thousandfold.
Making up a dilution safely: dilute to a final volume, then let the arithmetic follow: Every dilution question hides a wording decision, and the decision is whether the figure in the prescription is the volume you end up with or the volume you pour in. “Dilute 10 mL to 100 mL” fixes the final volume at 100 mL, so the denominator of the new concentration is known. “Add 100 mL of saline to 10 mL” fixes only the volume added, so the final volume is 110 mL and the concentration is lower than the prescriber may have expected. Read the verb: make up to, dilute to and bring to volume all fix the final figure, while add and dilute with fix only the diluent. The arithmetic itself starts from conservation of mass. The amount of drug in the stock is C1 × V1 and the amount in the finished solution is C2 × V2; because no drug is created or lost, the two expressions are equal. Rearranged, V1 = (C2 × V2) ÷ C1 gives the volume of stock to withdraw, and V2 = (C1 × V1) ÷ C2 gives the final volume that produces a wanted strength. A practical difficulty appears when V1 is larger than the container of stock: 100 mL of 5 mg/mL needs 500 mg, which is two 5 mL ampoules of 50 mg/mL, so either more ampoules are opened or a more concentrated presentation is requested. The opposite difficulty appears when V1 is too small to measure accurately, for instance 0.02 mL. Then a serial dilution is used: a first dilution of 1 in 100 is prepared, and a measured portion of that is diluted again by 1 in 10, so the overall dilution is 1 in 1,000 and every step stays within the range of a syringe. Accuracy decides the equipment, because 0.1 mL read in a 10 mL syringe is a poor measurement; choose the smallest syringe that comfortably holds the volume. Two further cautions belong with the method. Volumes are not perfectly additive — ethanol and water contract slightly when mixed — which is why a pharmacy specifies a final volume instead of assuming that millilitres simply add. And the diluent is not interchangeable: some drugs precipitate or are inactivated in the wrong fluid, so the name of the diluent is part of the order. Finish by labelling the container with the drug, the total amount, the final volume, the resulting strength, the time and your initials; a diluted product is less stable than the concentrate and should not be kept beyond its stated expiry.
Concentration inside the patient: why serum levels are measured: Concentration is not only a property of bottles; it is the quantity clinicians measure inside the patient. The plasma concentration of a drug is the amount circulating in the blood divided by the volume it appears to occupy, and that apparent volume is called the volume of distribution. It is a mathematical volume rather than an anatomical one: a drug that leaves the blood and dissolves in fat has a large apparent volume and therefore a low plasma concentration, even though the body has not become larger. The rule concentration = amount ÷ volume then does real work at the bedside. To reach a target concentration quickly, a loading dose is given, calculated from the target concentration multiplied by the volume of distribution and corrected for how much of the dose survives absorption. It is larger than the maintenance dose because it must fill the whole apparent volume, not merely replace what the body has already removed. Elimination normally removes a constant fraction of the drug per unit time — a first-order process — and the time needed to remove half of the amount present is the half-life. Because of this, a continuous infusion climbs until the rate of drug entering the body equals the rate being removed; that plateau is steady state, and it is reached after roughly four to five half-lives. Doubling the infusion rate doubles the steady-state concentration, which is the same proportionality that governs a dilution. Laboratories exploit all of this by reporting a serum concentration against a therapeutic range, the interval between the level below which the drug does not work and the level above which it becomes toxic. Nursing actions keep those numbers meaningful: give the infusion at the prescribed rate and check the pump setting, and draw a trough level just before the next dose and a peak level at the interval the protocol names, because a sample taken at the wrong moment produces a concentration that looks dangerous or useless when neither is true. Never adjust a rate on the basis of a level drawn at the wrong time; report the value and the timing to the prescriber.

Explore the diagram

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Concentration, ratios and dilutions

Three linked panels. The first reduces a stock label — 500 mg/5 mL, 1:1000, 0.9% or 5% — to a single-unit strength in mg/mL; the second turns an ordered dose into a volume with volume = dose ÷ concentration; the third dilutes 2 mL of a 1 mg/mL solution to 10 mL and shows that the volume grows while the 2 mg of drug stays the same.

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01 / 4Reading a label as a rate

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

Reading a label as a rate

A strength statement is always an amount of drug inside a stated volume, so the first move is to reduce it to one millilitre. 500 mg in 5 mL is 500 ÷ 5 = 100 mg in each 1 mL. Once the strength is in mg/mL, any dose can be turned into a volume by a single division, which is why this step comes before a syringe is touched.

Schematic teaching plate: the beakers are drawn to different widths to show the change in volume, not to a measured scale. All concentrations shown are worked from the labels printed beside them.

Key points to remember

Concentration is an amount of drug per stated volume; reduce any label to a single unit such as mg/mL before calculating.
Ratio strength 1:1000 means 1 g of drug in 1,000 mL of solution, which is 1 mg/mL; the left side is a mass and the right side a volume.
Percent strength means grams per 100 mL, so multiplying the percentage of a dissolved solid by 10 gives its strength in mg/mL.
Volume to give = prescribed dose ÷ concentration, with the dose and the concentration written in the same unit.
Adding diluent changes the volume and the concentration but never the amount of drug: C1 × V1 = C2 × V2.

Go deeper

Worked example

From a percent label to a volume to withdraw. A ward stocks lidocaine 2% for a minor procedure. The order asks for 60 mg. Turn the label into a concentration, turn the concentration into a volume, and check the answer against the 10 mL vial before drawing anything up.

Read the percent as grams per 100 mL: 2% means 2 g of lidocaine in every 100 mL of solution.
Convert the mass into the unit of the order: 2 g × 1,000 = 2,000 mg per 100 mL.
Divide by 100 mL to reach a single-unit concentration: 2,000 ÷ 100 = 20 mg/mL.
Divide the ordered dose by the concentration: volume = 60 ÷ 20 = 3 mL.
Check the container and the syringe. Three millilitres fits comfortably in a 10 mL vial and in a 5 mL syringe, and it is far less than the vial’s contents, so the order is administrable as written.

Five moves turn any strength statement into an actionable volume: read the label, convert to milligrams, reduce to one millilitre, divide the dose by the concentration, and check the result against the container you actually have.

Tutorial 1 — diluting a stock solution and recalculating its strength

A ward has a stock solution containing 250 mg of a drug in 5 mL. The prescription calls for 100 mL of a solution whose strength is 5 mg/mL. Work out how much stock to withdraw and how much diluent to add, then state the final strength as a check.

Rule used: Dilution conserves the amount of drug: C1 × V1 = C2 × V2, where C1 and V1 are the concentration and volume of the stock, C2 is the wanted concentration and V2 is the final volume. Rearranged for the volume of stock, V1 = (C2 × V2) ÷ C1. Amount of drug (mg) = concentration (mg/mL) × volume (mL). Add diluent until the total volume reaches the figure the prescription states.

Reduce the stock label to a single-unit concentration. Write the amount over the volume and divide.
Reveal this step
The stock is 250 mg/5 mL, so 250 ÷ 5 = 50 mg/mL. That is C1, the concentration you are starting from.
The prescription asks for 100 mL at 5 mg/mL. How many milligrams must the final bag contain, and where will those milligrams come from?
Reveal the answer
C2 × V2 = 5 mg/mL × 100 mL = 500 mg. All 500 mg must come from the stock solution, because saline contains no drug. Dilution spreads those 500 mg through a larger volume; it does not remove any of them.
Using C1 × V1 = C2 × V2, what volume of stock solution supplies those 500 mg?
Reveal the answer
V1 = (C2 × V2) ÷ C1 = 500 mg ÷ 50 mg/mL = 10 mL. Ten millilitres of stock contains exactly 500 mg, because 10 × 50 = 500.
Work out how much diluent to add, and say which wording on a prescription tells you to do this rather than something else.
Reveal this step
100 mL final − 10 mL of stock = 90 mL of diluent. The prescription says dilute to 100 mL, which fixes the total volume. If it had said add 100 mL, the final volume would be 110 mL and the strength 500 ÷ 110 ≈ 4.5 mg/mL, not 5 mg/mL.
Check the final strength directly from the amount of drug and the final volume.
Reveal this step
C2 = 500 mg ÷ 100 mL = 5 mg/mL, which matches the prescription. The stock became ten times weaker because the volume went from 10 mL to 100 mL, a factor of ten.
The stock comes in 5 mL ampoules but the calculation needs 10 mL. What do you do, and what must appear on the label of the finished bag?
Reveal the answer
Use two 5 mL ampoules, because one is not enough. Label the bag with the drug name, the total amount (500 mg), the final volume (100 mL), the resulting strength (5 mg/mL), the time of preparation and your initials, so the next nurse never has to guess the concentration.

Answer: Withdraw 10 mL of the 250 mg/5 mL stock, add 90 mL of diluent to make a final volume of 100 mL, and the finished solution contains 500 mg at 5 mg/mL.

Keep this habit: Never dilute by a remembered factor. Calculate the milligrams the final container needs, divide by the stock concentration to obtain the volume to withdraw, then make the total volume up to the figure the prescription names — and write the resulting strength on the container.

Tutorial 2 — how much 1:1000 solution holds a prescribed amount

A doctor prescribes 0.4 mg of adrenaline subcutaneously. The ampoule is labelled adrenaline 1:1000 and contains 1 mL. Work out how many millilitres to draw up, and check whether one ampoule is enough.

Rule used: A ratio strength 1:1000 means 1 g of drug in 1,000 mL of solution. Convert the grams to milligrams, then divide by the volume in millilitres to obtain the concentration in mg/mL. Volume to give (mL) = prescribed dose (mg) ÷ concentration (mg/mL).

Write down what each side of 1:1000 means, in words, before converting anything.
Reveal this step
The left number, 1, is the mass of drug: 1 g. The right number, 1000, is the volume of solution: 1,000 mL. The ampoule therefore has a strength of 1 g per 1,000 mL.
Convert 1:1000 into a concentration in mg/mL. Which conversion must happen first, and why?
Reveal the answer
Convert the mass first, because the ratio mixes grams with millilitres while the prescription is written in milligrams. 1 g = 1,000 mg, so the strength is 1,000 mg per 1,000 mL; dividing gives 1 mg/mL.
The order is 0.4 mg. Without calculating yet, is the volume you need larger or smaller than 1 mL? Explain your reasoning.
Reveal the answer
Smaller than 1 mL, because each millilitre contains 1 mg and 0.4 mg is less than 1 mg. Estimating the size before calculating is how a misplaced decimal point is caught.
Calculate the volume with volume = dose ÷ concentration, keeping the unit on every line.
Reveal this step
Volume = 0.4 mg ÷ 1 mg/mL = 0.4 mL. The milligrams cancel and millilitres remain, so 0.4 mL is the volume to draw up.
Check the answer from the amount, and state whether one 1 mL ampoule is enough.
Reveal this step
0.4 mL × 1 mg/mL = 0.4 mg, which matches the order. One ampoule holds 1 mg, so it contains enough; the remaining 0.6 mL is discarded and never kept for another patient.
The next prescription asks for 1.5 mg from the same 1:1000 stock. What volume is needed, and what is the problem?
Reveal the answer
Volume = 1.5 ÷ 1 = 1.5 mL, which is more than one ampoule. Either two ampoules of the same strength are used, with the surplus discarded, or the prescriber is asked whether a stronger preparation was intended. Measuring 1.5 mL from a container holding only 1 mL would be a serious error.

Answer: 0.4 mg of adrenaline 1:1000 is contained in 0.4 mL, which a single 1 mL ampoule supplies. A 1.5 mg order would need 1.5 mL and therefore more than one ampoule.

Keep this habit: Read the ratio as grams per thousand millilitres, convert it to mg/mL, then divide the prescribed dose by that concentration. Finish by comparing the volume with the ampoule and with the syringe, so a decimal error is caught before it reaches the patient.

Now you try

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

A bag contains 0.9% sodium chloride. What mass of sodium chloride, in grams, is dissolved in 250 mL of this solution? Show the concentration in mg/mL on the way.
Hint
Percent means per hundred: start from 0.9 g in 100 mL, find the mass in 1 mL, then scale to 250 mL.
Show the worked answer
0.9% means 0.9 g per 100 mL. In milligrams that is 900 mg per 100 mL, so the concentration is 900 ÷ 100 = 9 mg/mL. In 250 mL the mass is 9 × 250 = 2,250 mg, which is 2.25 g of sodium chloride.
A vial is labelled 1 g/10 mL. A prescription asks for 350 mg. What volume must be withdrawn, and is it available from a single 10 mL vial?
Hint
Turn the label into mg/mL first, then divide the ordered dose by that concentration, and compare the result with the stated vial volume.
Show the worked answer
1 g = 1,000 mg, so the concentration is 1,000 ÷ 10 = 100 mg/mL. Volume = 350 ÷ 100 = 3.5 mL. The vial holds 10 mL, so 3.5 mL is available from one vial, and the answer is smaller than the container as it should be.
You have 4 mL of a solution containing 25 mg/mL and you add 6 mL of sterile water. What is the new concentration, and what mass of drug does the diluted solution contain?
Hint
Find the mass of drug first, because the water does not change it. Then divide that mass by the total volume after the water is added.
Show the worked answer
Mass = 4 mL × 25 mg/mL = 100 mg, and sterile water adds no drug, so the diluted solution still contains 100 mg. The total volume is 4 + 6 = 10 mL, so the new concentration is 100 ÷ 10 = 10 mg/mL.
A 1:1000 solution is available and a patient is prescribed 0.25 mg. What volume contains that dose, and how much drug does one 2 mL ampoule hold?
Hint
Convert the ratio to mg/mL, divide the dose by that concentration, then work out the mass held in 2 mL.
Show the worked answer
1:1000 is 1,000 mg in 1,000 mL, which is 1 mg/mL. Volume = 0.25 ÷ 1 = 0.25 mL. A 2 mL ampoule holds 2 × 1 = 2 mg, so it contains eight such doses by volume, but one ampoule must never be used for more than one patient and the surplus is discarded.

Common misconception

The commonest wrong model is that dilution “weakens” the drug by destroying or removing some of it, so a diluted solution is thought to contain less drug than the same solution held before dilution. Nothing is destroyed: the same milligrams are simply spread through more millilitres, and you can prove it by multiplying the new concentration by the new volume and getting the original amount back. A related error is to treat “dilute to 100 mL” and “add 100 mL of diluent” as the same instruction. They are not, because only the first fixes the final volume, and only the final volume gives the new concentration.

Check your reasoning

Explain, without using the formula, why doubling the volume of a solution by adding saline halves its concentration but leaves the mass of drug unchanged. Then say why that means you must record both the old and the new volume.

Reveal an answer

The drug is already dissolved and nothing removes it, so the number of milligrams stays the same; only the space it occupies grows. Suppose 20 mg sits in 10 mL, giving 2 mg/mL. Doubling the volume to 20 mL keeps the 20 mg but shares it over twice as many millilitres, so each millilitre now holds half as much: 1 mg/mL. The numerator of a concentration is fixed while the denominator changes. Recording both volumes matters because a concentration is only meaningful with the volume it refers to: a bag labelled 500 mg tells the next nurse the amount, but without the final volume nobody can tell whether the strength is 5 mg/mL or 2.5 mg/mL.

Where this comes from

BMS 1110 Applied Mathematics and Medical Biostatistics lecture notes (supplied) · Chapter Two, section 2.5 “Ratio and Proportion”, lines 411–412: the same dosage problem written as two equal ratios, 250 mg : 5 mL = 500 mg : X, with cross-multiplication giving 250X = 2,500 and X = 10 mL. The formula sheet at line 508 states Concentration = Amount of drug ÷ Volume; beyond that formula the notes mention concentration only in passing.
Beyond the supplied notes: the ratio-strength notation · The ratio-strength form 1:1000 — 1 g of drug in 1,000 mL of solution, hence 1 mg/mL — and its use on adrenaline and lidocaine labels are standard prescribing conventions that the notes do not teach. Listed as a required addition in section-map.md, “What the course must correct or add”.
Beyond the supplied notes: percent solutions and the dilution rule · Percent strength as grams of drug per 100 mL, and the conservation rule C1 × V1 = C2 × V2 (adding diluent changes the volume and therefore the concentration but never the amount of drug) are standard pharmaceutical arithmetic absent from the supplied notes; the notes give only ratio and proportion.
Beyond the supplied notes: concentration inside the patient · Volume of distribution, loading dose, first-order elimination, half-life, steady state and therapeutic drug monitoring are standard pharmacology used here to show where the same concentration = amount ÷ volume rule reappears at the bedside; the supplied notes do not cover them.

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.