PTCB Quiz Prep

Concentrations and Dilutions

Updated

Concentration questions ask how much drug is in a preparation, and dilution questions ask what happens to that strength when you add diluent. Both reduce to a couple of ideas: percentage strength, ratio strength, and one equation. Work the examples here, then test yourself on the PTCB math practice test.

Percentage strength

Percentage strength always means an amount per 100. Which unit depends on the letters:

  • % w/v is grams of solute per 100 mL of solution (a solid in a liquid). A 5% w/v solution holds 5 g per 100 mL.
  • % w/w is grams per 100 g of product (a solid in a solid). A 2% w/w cream holds 2 g per 100 g.
  • % v/v is millilitres per 100 mL of solution (a liquid in a liquid). A 70% v/v solution holds 70 mL per 100 mL.

grams of drug = (percent ÷ 100) × amount of preparation

Percent w/v: dextrose in D5W

How many grams of dextrose are in a 500 mL bag of D5W (5% w/v)?

  1. Step 1. 5% w/v means 5 g per 100 mL.
  2. Step 2. Scale to 500 mL: 5 g × (500 ÷ 100) = 5 × 5 = 25 g.
  3. Step 3. Quick check: the bag is five times 100 mL, so five times 5 g.

Answer: 25 g of dextrose

The trap: Reading 5% as 5 g in the whole bag. The 5 g is per 100 mL, so you must scale up to the actual volume.

The same rule covers ointments and alcohols. A 2% w/w hydrocortisone cream in a 30 g tube holds 0.02 × 30 = 0.6 g of drug. A 70% v/v isopropyl alcohol in 480 mL holds 0.70 × 480 = 336 mL of alcohol.

Ratio strength

Ratio strength writes a concentration as 1 in something. For a w/v ratio, 1:1000 means 1 g in 1000 mL. The fastest move is to convert the ratio into milligrams per millilitre or a percentage before you calculate.

Ratio strength: 1:1000 epinephrine

How many milligrams of drug are in 10 mL of a 1:1000 (w/v) solution?

  1. Step 1. 1:1000 w/v means 1 g per 1000 mL.
  2. Step 2. Convert grams to milligrams: 1 g = 1000 mg, so the concentration is 1000 mg ÷ 1000 mL = 1 mg/mL.
  3. Step 3. Multiply by the volume: 1 mg/mL × 10 mL = 10 mg.

Answer: 10 mg

The trap: Confusing the ratio direction. 1:1000 is 1 part drug to 1000 parts total, not 1000 mg in each millilitre.

The same method handles a 1:200 (w/v) solution: 1 g per 200 mL is 1000 mg ÷ 200 mL = 5 mg/mL, so 15 mL carries 5 × 15 = 75 mg.

The dilution equation

When you add diluent, the amount of drug stays the same while the volume grows, so the strength falls. That is the whole idea behind the dilution equation.

C1 × V1 = C2 × V2

Diluting a stock solution

How many mL of a 10% stock solution are needed to make 250 mL of a 2% solution?

  1. Step 1. Assign the values: C1 = 10%, V1 = unknown, C2 = 2%, V2 = 250 mL.
  2. Step 2. Rearrange: V1 = (C2 × V2) ÷ C1 = (2 × 250) ÷ 10.
  3. Step 3. Solve: 500 ÷ 10 = 50 mL of stock, then add diluent up to 250 mL.

Answer: 50 mL of the 10% stock, q.s. to 250 mL

The trap: Inverting the concentrations. Put the strength you are diluting from as C1; if you flip them you get 1250 mL, which is impossible for a dilution.

The most common mistake

The classic slip is answering with the final volume when the question asked for the diluent to add. If you dilute 120 mL of a 25% solution to 10%, the equation gives V2 = (25 × 120) ÷ 10 = 300 mL, but that is the total. The water you add is 300 - 120 = 180 mL. Always reread the last line: does it want the final volume, or only the diluent?

Try it

  1. How many grams of sodium chloride are in 1 L of 0.9% w/v normal saline?
  2. Express a 1:2500 (w/v) strength as a percentage.
  3. How many mL of a 25% stock solution are needed to make 500 mL of a 10% solution?
Show answers
  1. 0.9% w/v is 0.9 g per 100 mL. In 1000 mL: 0.9 × (1000 ÷ 100) = 0.9 × 10 = 9 g.
  2. 1:2500 is 1 g per 2500 mL, so 1 g ÷ 25 per 100 mL = 0.04 g per 100 mL = 0.04%.
  3. C1 × V1 = C2 × V2: 25 × V1 = 10 × 500, so V1 = 5000 ÷ 25 = 200 mL, then q.s. to 500 mL.

Frequently asked questions

What is the difference between w/v, w/w, and v/v?

They all mean grams or millilitres per 100. Percent w/v is grams of solute per 100 mL of solution, used for a solid dissolved in a liquid. Percent w/w is grams per 100 g, used for ointments and creams. Percent v/v is millilitres per 100 mL, used for a liquid in a liquid such as alcohol.

How do you convert a ratio strength to a percentage?

A ratio strength like 1:1000 (w/v) means 1 g in 1000 mL. Rewrite it as grams per 100 mL by dividing: 1 g in 1000 mL is 0.1 g in 100 mL, which is 0.1%. To go the other way, a 0.9% w/v strength is 0.9 g in 100 mL, or 1 g in about 111 mL, so 1:111.

What is the dilution equation C1V1 = C2V2?

It states that the amount of drug does not change when you add diluent. C1 and V1 are the starting concentration and volume, C2 and V2 are the final concentration and volume. Solve for the one value you do not know, keeping the two concentrations in the same units.

How do you find how much diluent to add?

Use C1 × V1 = C2 × V2 to find the final volume V2, then subtract the starting volume V1. The difference is the diluent you add. A frequent trap is reporting V2, the final volume, when the question asked only for the diluent added.

How many milligrams are in a 1:1000 solution?

A 1:1000 w/v solution is 1 g per 1000 mL, which equals 1000 mg per 1000 mL, or 1 mg/mL. So every millilitre carries 1 mg of drug. Epinephrine 1:1000 is the classic example.

Practise this

When two strengths must be mixed to a target rather than diluted, reach for alligation. Put both skills to work on the pharmacy math practice test, then return to the pharmacy math hub for the rest of the formulas.

PTCB Quiz Prep is an independent study resource, not affiliated with the Pharmacy Technician Certification Board.