Concentrations, Ratios and Dilutions

Introduction

A medicine's strength tells you how much active ingredient is in a stated amount of product. For example, "400 mg per 5 mL" means every 5 mL of the liquid contains 400 mg of the ingredient. Once you can read a strength this way, three common pharmacy calculations follow from the same idea:

• converting a percent or ratio strength into usable units
• finding how much ingredient is in a whole bottle or tube
• predicting what happens when a liquid is diluted

By the end of this lesson, you will be able to:

• Read a percent strength (w/v, w/w or v/v) and a ratio strength as a specific amount of ingredient in a specific amount of product.
• Convert percent and ratio strengths to mg/mL, and compare which of two strengths is stronger.
• Calculate the total amount of ingredient in a final quantity of product.
• Use the fact that dilution does not change the amount of ingredient to solve a simple dilution problem.

Before you start: You need everyday arithmetic and the metric relationships 1 g = 1000 mg and 1 mg = 1000 mcg. The prefix milli- means one-thousandth and micro- means one-millionth. For practice with these conversions, see Metric conversions and dimensional analysis.

All cases in this lesson are arithmetic exercises. They are not instructions for preparing or giving any medicine.

Strength means ingredient per amount of product

A strength is a pair of amounts: how much ingredient in how much product. The ingredient is the active drug. The product is the whole tablet, liquid, cream or other finished form.

Amoxicillin for oral suspension is a real example. One label lists two liquid strengths, 200 mg/5 mL and 400 mg/5 mL. Read 400 mg/5 mL as "every 5 mL of the mixed suspension contains 400 mg of amoxicillin."

This example shows two things:

• The amount of liquid alone tells you nothing. 5 mL of one product contains twice as much amoxicillin as 5 mL of the other.
• The same ingredient name does not mean the same product. Both bottles say amoxicillin, but the strengths differ. You must read the strength every time.

A strength like mg/mL is sometimes called an amount strength. Its two units do not have to match: milligrams on top, milliliters on the bottom is normal.

FDA describes strength in several forms:

• weight per volume (such as mg/mL)
• weight per weight (such as g per g)
• weight per dosage unit (such as mg per tablet)

Some FDA listings also express strength as an equivalent amount of the base drug, marked "EQ," as in "EQ 200 MG BASE." The lesson is the same: the label states what the number measures, and you read that basis rather than assuming it.

Percent strength: the letters tell you the units

Percent means "per hundred." A percent strength becomes usable only once you know what the hundred measures. Pharmacy labels mark this with letters:

• w/v (weight in volume): grams of ingredient per 100 mL of product. Typical of liquids. Plain meaning: a solid dissolved in a liquid, counted in grams per 100 mL.
• w/w (weight in weight): grams of ingredient per 100 g of product. Typical of creams and ointments. Plain meaning: everything is weighed, nothing is measured by volume.
• v/v (volume in volume): mL of ingredient per 100 mL of product. Used, for example, for alcohol. Plain meaning: a liquid mixed into a liquid, both measured in mL.

Percent strengths pair corresponding units: grams go with mL, and milligrams go with microliters. Milligrams do not correspond to milliliters, because they differ by a factor of 1000. Most percent errors come from that factor.

Worked example: 2% w/v

1. Write the meaning: 2% w/v = 2 g in 100 mL.
2. Convert grams to milligrams: 2 g = 2000 mg, so the strength is 2000 mg in 100 mL.
3. Divide to get the amount in 1 mL: 2000 ÷ 100 = 20 mg/mL.

Shortcut and check: 1% w/v = 1 g/100 mL = 1000 mg/100 mL = 10 mg/mL. So a w/v percent times 10 gives mg/mL. For 2%, that gives 20 mg/mL, which matches.

Changed case: 2% w/w cream. Now the basis is weight, so 2% w/w = 2 g of ingredient per 100 g of cream. There is no mL anywhere in that statement. You cannot turn it into mg/mL from the label alone, because that would require knowing how much a mL of cream weighs.

Tempting mistake: reading 2% as "2 mg per mL." That treats the % sign as if it pointed to milligrams and milliliters. The real value, 20 mg/mL, is ten times larger. Always write out "grams per 100" first.

Question 1

A liquid is labeled 3% w/v. How many milligrams of ingredient are in each mL?

Question 2

An ointment is labeled 3% w/w. State its meaning in grams. Can you express its strength in mg/mL from this label alone?

Ratio strength: one part in the total

A ratio strength such as 1:1000 w/v means 1 part of ingredient in 1000 total parts of product. With the w/v basis, the corresponding units are grams and mL, so 1:1000 w/v = 1 g in 1000 mL of finished product.

The second number counts the whole product, not just the liquid you add. This is a part-to-whole comparison.

Worked example: 1:1000 w/v to mg/mL

1. Write the meaning: 1 g in 1000 mL.
2. Convert grams to milligrams: 1000 mg in 1000 mL.
3. Divide: 1000 ÷ 1000 = 1 mg/mL.

Converting to percent: 1 g in 1000 mL is the same as 0.1 g in 100 mL, so 1:1000 w/v = 0.1% w/v.

Real-world context: epinephrine injection was historically labeled "1:1,000." Current FDA labeling guidance and U.S. labeling standards (USP) call for metric units rather than ratio expressions on single-entity injectable drug products, meaning injectables that contain a single active ingredient. That strength is now written as 1 mg/mL. You may still meet ratio strengths in older references and practice problems, so you need to translate them. The current label wording is what you read on the product.

Changed case: 1:10,000 w/v. This means 1 g in 10,000 mL, which is 1000 mg ÷ 10,000 mL = 0.1 mg/mL. The bigger second number makes the product weaker, because the same 1 part is spread through more total product.

Tempting mistake: reading 1:1000 as "1 g plus 1000 mL of liquid." That would give a total volume of more than 1000 mL and a slightly weaker strength than the label means. In a ratio strength, the 1000 is the total.

When you write results, use "mcg" rather than "μg," and put a leading zero before a decimal, as in "0.1 mg." These conventions prevent misreading.

Question 3

Express 1:200 w/v in mg/mL and as a percent. Which is stronger, 1:200 or 1:1000?

Calculating the amount of ingredient in a final quantity

To find the total ingredient, multiply the strength by the final quantity of product:

amount of ingredient = strength × final quantity

Write the units and let them cancel, so the answer comes out in mass, such as mg or g. This works because the strength is a fixed rate: each equal portion of product carries the same amount of ingredient.

Worked example: a bottle of suspension

The amoxicillin label lists a 400 mg/5 mL suspension in a 100 mL bottle size. Assume the mixed bottle contains 100 mL of suspension.

1. Count the 5 mL portions in the bottle: 100 mL ÷ 5 mL = 20 portions.
2. Multiply by the amount per portion: 20 × 400 mg = 8000 mg = 8 g.
3. As one line: 400 mg/5 mL × 100 mL = 8000 mg. The mL units cancel, leaving mg.

Changed case: same bottle size, 200 mg/5 mL. Now it is 20 × 200 mg = 4000 mg. The bottle size is unchanged, the strength is halved, and so the total is halved.

Tempting mistake: using the water instead of the final volume. That label's mixing table says to add 71 mL of water to the 400 mg/5 mL, 100 mL bottle. If you multiply 400 mg/5 mL × 71 mL, you get 5680 mg, which is wrong. The strength describes the finished suspension, not the water. The powder itself takes up part of the final volume, so the water added is less than the final amount.

The water amounts are specific to that product's label. Another product, or even another bottle size of the same product, lists different amounts. Always take mixing amounts from the exact product label in hand.

The same method with percent and ratio strengths

• 300 mL of a 4% w/v solution: 4 g/100 mL × 300 mL = 12 g.
• 60 g of a 1.5% w/w ointment: 1.5 g/100 g × 60 g = 0.9 g = 900 mg. Here, grams of product cancel grams of product.
• 400 mL of a 1:2000 w/v solution: 1 g/2000 mL × 400 mL = 0.2 g = 200 mg.

Magnitude check: compare your answer against the actual percent. 4% w/v means 4 g in every 100 mL, so the grams of ingredient should be 4/100 of the mL of product: 0.04 × 300 = 12 g. Suppose you got 120 g instead. Reverse the calculation: 120 g in 300 mL is 40 g per 100 mL, or 40% w/v. That is ten times the labeled 4%, which signals a misplaced decimal. A broad rule such as "the grams of ingredient must be less than the mL of product" would not catch this error, because 120 is still less than 300. Check against the specific ratio, not just a general bound.

A correct calculation is only one part of filling a prescription. Whether a product, strength and quantity are appropriate for a patient is checked by the pharmacist. For when a technician should stop and ask, see Recognize when the pharmacist must intervene. Who may mix or check a preparation is set by state rules and pharmacy procedures; this lesson teaches only the arithmetic.

Question 4

A bottle of amoxicillin 200 mg/5 mL suspension has a labeled bottle size of 75 mL. Assume it contains 75 mL once mixed. How much amoxicillin does it contain? The label lists 57 mL of water for this bottle. Why is 57 mL not used in the calculation?

Dilution: the ingredient amount stays the same

To dilute means to add a diluent, a liquid containing none of the ingredient, so the product becomes weaker. The key fact:

Adding diluent spreads the same amount of ingredient through more product. The amount of ingredient does not change.

For the classroom version, three assumptions must hold:

1. The diluent contains none of the ingredient.
2. No ingredient is lost.
3. You know the final quantity. If a problem says you "add" diluent, assume the volumes simply add together, unless told otherwise.

Because the ingredient amount stays the same:

ingredient before = ingredient after C1 × Q1 = C2 × Q2

• C1 = starting (stock) strength
• Q1 = quantity of stock used
• C2 = final strength
• Q2 = final quantity

Each side is just "strength × quantity = amount of ingredient," the method you learned in the previous section. C1 and C2 must be in the same units, such as both % or both mg/mL. Q1 and Q2 must also be in the same units.

Worked example 1: find the new strength

Take 20 mL of a 5% w/v stock solution and dilute it to a final volume of 100 mL.

1. Ingredient present: 5 g/100 mL × 20 mL = 1 g.
2. After dilution, the same 1 g is in 100 mL, so the strength is 1% w/v.
3. With the equation: 5% × 20 mL = C2 × 100 mL, so C2 = 1%.

Check: the volume grew five-fold (20 → 100 mL), so the strength should drop five-fold (5% → 1%). It does.

Worked example 2: find how much stock to use

You need 250 mL of a 2% w/v solution, starting from a 10% w/v stock.

1. Set up the equation: 10% × Q1 = 2% × 250 mL.
2. Solve: Q1 = (2 × 250) ÷ 10 = 50 mL of stock.
3. Diluent needed, assuming the volumes add: 250 − 50 = 200 mL of diluent.

Changed case: you need 1% instead. Q1 = (1 × 250) ÷ 10 = 25 mL of stock, with 225 mL of diluent. A weaker target needs less stock in the same final volume.

Tempting mistake: confusing the diluent added with the final volume. In example 2, the final volume is 250 mL, but only 200 mL of diluent is added. Using 250 mL as the diluent amount would give 300 mL total, which is too weak.

Reasonableness check: a dilution always makes the strength weaker. If your answer is stronger than the stock, or calls for more stock than the final volume, an amount was put in the wrong place.

Question 5

10 mL of a 1 mg/mL solution is diluted to a final volume of 50 mL. What is the new strength?

Independent practice

Work each question fully before opening the answer. State the basis (w/v, w/w or ratio) and your units at every step.

Question 6

How many grams of ingredient are in 500 mL of a 0.5% w/v solution?

Question 7

How many milligrams of ingredient are in a 30 g tube of a 2.5% w/w ointment?

Question 8

A solution is labeled 1:2500 w/v. What is its strength in mg/mL, and how many mg are in 250 mL?

Question 9

You need 200 mL of a 0.25% w/v solution. The stock is 5% w/v. Assuming the volumes add, how much stock and how much diluent do you need?

Question 10

A coworker adds 60 mL of diluent to 15 mL of a 20 mg/mL solution and labels the result "5 mg/mL." Assuming the volumes add, is the label correct? Explain the error, if any.

Question 11

Your stock solution is 2% w/v. How much stock is needed to make 100 mL of a 4 mg/mL solution? Then: could you make 100 mL of a 40 mg/mL solution from this stock by dilution?

Question 12

Two amoxicillin suspensions are on the shelf: 200 mg/5 mL and 400 mg/5 mL. How much amoxicillin is in 10 mL of each? What does this show about choosing between bottles by ingredient name alone?

What to remember

• Strength = ingredient per amount of product. Read both parts of the label every time.
• w/v = g per 100 mL; w/w = g per 100 g; v/v = mL per 100 mL. For w/v, 1% = 10 mg/mL.
• Ratio strength (such as 1:1000 w/v) = 1 g in 1000 mL of total product = 1 mg/mL. A larger second number means a weaker product.
• Amount of ingredient = strength × final quantity. Use the finished quantity, not the water or diluent added.
• Dilution keeps the ingredient amount constant: C1 × Q1 = C2 × Q2, with matching units on both sides. The result must be weaker than the stock.
• Product-specific details, such as mixing water amounts, come from that exact product's label. A correct calculation does not by itself approve a product for dispensing.