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Alligation Practice Problems

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Alligation mixes two preparations of different strengths to get an intermediate strength. Alligation alternate starts from two known strengths and finds the quantity of each to mix; alligation medial starts from known quantities and finds the resulting strength. The alternate method uses a grid where you take diagonal differences to get the parts of each component.

Formulaparts of high = target − low  •  parts of low = high − target  •  volume of each = (its parts ÷ total parts) × final volume
Worked example

Make 250 mL of 25% dextrose from 70% and 10% stock. Parts of 70% = 25 − 10 = 15; parts of 10% = 70 − 25 = 45; total = 60. Volume of 70% = (15 ÷ 60) × 250 = 62.5 mL (and 187.5 mL of the 10%).

Don't confuse the two: medial = known quantities → find strength; alternate = known strengths → find quantities.

The alligation grid Two diagrams. The first is the general form: the higher strength sits top left, the lower strength bottom left, and the desired strength in the middle. Subtracting diagonally gives the parts of each. Parts of the higher strength equal desired minus lower; parts of the lower strength equal higher minus desired. Total parts is the sum. The second diagram works the same grid for 95 percent and 70 percent isopropyl alcohol mixed to 90 percent, giving 20 parts of 95 percent and 5 parts of 70 percent, 25 parts in total. THE GENERAL FORM Higher strength Lower strength Desired strength Desired − Lower = parts of the HIGHER Higher − Desired = parts of the LOWER Total parts = higher parts + lower parts WORKED: 95% AND 70% IPA TO MAKE 90% 95% IPA 70% IPA 90% desired 90 − 70 = 20 parts of the 95% IPA 95 − 90 = 5 parts of the 70% IPA Total = 25 parts, so 20/25 is 95% stock and 5/25 is 70% stock
Subtract along the diagonals. The number that lands on the right is the parts of the strength diagonally opposite it. That crossover is the step people get backwards.
Streak 0
Correct 0/0
Alligation Medial
You mix 150 mL of 50% dextrose with 1000 mL of 10% dextrose. What is the percentage strength of the final mixture? Round your answer to one decimal place.
%

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How to set up an alligation problem

Alligation answers one question: if I mix two preparations of different strengths, what do I get, or how much of each do I need? Every strength you mix from is either stronger or weaker than the target, and the target sits somewhere between them. The method is a weighted average written in a shape that keeps the bookkeeping straight. There are two directions, and the first thing to decide is which one the question is asking.

Alligation alternate: the grid

Write the higher strength top-left, the lower strength bottom-left, and the target in the middle. Then subtract along the diagonals and write each difference on the right, ignoring the sign. The number that lands next to a strength is the number of parts of that strength.

parts of high = target − low   •   parts of low = high − target   •   amount of each = (its parts ÷ total parts) × final quantity

Two things are easy to get backwards. The parts of the high strength come from the difference between the target and the low strength, not from the high strength's own number. And the diluent counts as a strength: water, petrolatum or normal saline is a 0% preparation and goes in the low position like anything else.

Worked example 1: two stocks

Prepare 500 mL of 20% dextrose from 50% and 10% stock.

parts of the stronger stock = 20 − 10 = 10

parts of the weaker stock = 50 − 20 = 30

total parts = 10 + 30 = 40

volume of the stronger stock = 10 ÷ 40 × 500 = 125 mL

volume of the weaker stock = 500 − 125 = 375 mL

Check it the way alligation medial would: 125 mL × 50% is 62.5 g and 375 mL × 10% is 37.5 g, 100 g in 500 mL, which is 20%. The check takes ten seconds and catches a swapped diagonal every time.

Worked example 2: the parts don't divide evenly

Prepare 240 mL of 70% alcohol from 95% and 50% alcohol.

parts of the stronger stock = 70 − 50 = 20

parts of the weaker stock = 95 − 70 = 25

volume of the stronger stock = 20 ÷ 45 × 240 = 106.67 → 106.7 mL

volume of the weaker stock = 240 − 106.67 = 133.33 → 133.3 mL

Divide the parts by the total parts first, then multiply by the volume. Doing it in the other order works too, but rounding the ratio 20/45 to 0.44 before multiplying gives 105.6 mL, a volume the grader will not accept. Carry the fraction, or at least four decimals, until the last step.

Alligation medial: a weighted average

Here you already know what went into the mixture. Work out the drug in each component, add, and divide by the total quantity. There is no grid.

resulting % = (q₁ × c₁ + q₂ × c₂ + …) ÷ (q₁ + q₂ + …)

Worked example 3: alligation medial

You mix 200 mL of 50% dextrose with 800 mL of 10% dextrose. What is the strength of the mixture?

drug from the first = 200 × 50% = 100 g

drug from the second = 800 × 10% = 80 g

resulting strength = (100 + 80) ÷ (200 + 800) × 100 = 18%

The result has to land between the two starting strengths, and closer to the one you used more of. 18% is between 10% and 50%, and closer to 10%, which is the one that made up 80% of the volume. If your answer is outside that range, an amount and a strength were swapped somewhere.

Where students lose the points

When it's a proportion, not an alligation

If one component is pure drug (100%) or pure diluent (0%) and the question only asks how much diluent to add, C₁V₁ = C₂V₂ from the dilution topic gets there in one line. Alligation is the tool for two stocks that are both partly drug, and it still works in the simple cases; it's just longer than it needs to be.

10 Alligation practice problems with worked solutions

Every problem below is worked start to finish. Try it yourself first, then check the steps. For a fresh problem every time, use the practice box above. It generates unlimited variations of these same question types.

Problem 1: Alligation Medial

You mix 250 mL of 70% dextrose with 750 mL of 5% dextrose. What is the percentage strength of the final mixture? Round your answer to one decimal place.

Answer 21.3 %

Solution
  1. Grams from each: 250 mL × 70% = 175 g; 750 mL × 5% = 37.5 g.
  2. Total drug = 212.5 g in total volume 1000 mL.
  3. Strength = 212.5 g ÷ 1000 mL × 100 = 21.25%.
  4. Rounded as the question asks: 21.3 %.

Alligation MEDIAL = known quantities → find resulting strength. Alligation ALTERNATE = known strengths → find quantities.

Problem 2: Alligation Alternate

You need 120 g of 2% hydrocortisone ointment, prepared from 2.5% and 0.5% stock. How many g of the 2.5% stock do you need? Round your answer to one decimal place.

Answer 90.0 g

Solution
  1. Alligation grid: high (2.5%) top-left, low (0.5%) bottom-left, target (2%) center.
  2. Parts of 2.5% = 2 − 0.5 = 1.5 parts (diagonal difference).
  3. Parts of 0.5% = 2.5 − 2 = 0.5 parts.
  4. Total parts = 2.
  5. Amount of 2.5% = (1.5 parts ÷ 2 total parts) × 120 g = 90 g (and 30 g of 0.5%).
  6. Rounded as the question asks: 90.0 g.

Sanity check: 90×2.5% + 30×0.5% = 2.4 g = 120 g × 2% ✓

Problem 3: Alligation Medial

You mix 150 mL of 50% dextrose with 200 mL of 10% dextrose. What is the percentage strength of the final mixture? Round your answer to one decimal place.

Answer 27.1 %

Solution
  1. Grams from each: 150 mL × 50% = 75 g; 200 mL × 10% = 20 g.
  2. Total drug = 95 g in total volume 350 mL.
  3. Strength = 95 g ÷ 350 mL × 100 ≈ 27.143%.
  4. Rounded as the question asks: 27.1 %.

Alligation MEDIAL = known quantities → find resulting strength. Alligation ALTERNATE = known strengths → find quantities.

Problem 4: Alligation Alternate

You need 240 mL of 80% alcohol, prepared from 95% and 70% stock. How many mL of the 95% stock do you need? Round your answer to one decimal place.

Answer 96.0 mL

Solution
  1. Alligation grid: high (95%) top-left, low (70%) bottom-left, target (80%) center.
  2. Parts of 95% = 80 − 70 = 10 parts (diagonal difference).
  3. Parts of 70% = 95 − 80 = 15 parts.
  4. Total parts = 25.
  5. Amount of 95% = (10 parts ÷ 25 total parts) × 240 mL = 96 mL (and 144 mL of 70%).
  6. Rounded as the question asks: 96.0 mL.

Sanity check: 96×95% + 144×70% = 192 g = 240 mL × 80% ✓

Problem 5: Alligation Medial

You mix 250 mL of 50% dextrose with 750 mL of 10% dextrose. What is the percentage strength of the final mixture? Round your answer to one decimal place.

Answer 20.0 %

Solution
  1. Grams from each: 250 mL × 50% = 125 g; 750 mL × 10% = 75 g.
  2. Total drug = 200 g in total volume 1000 mL.
  3. Strength = 200 g ÷ 1000 mL × 100 = 20%.
  4. Rounded as the question asks: 20.0 %.

Alligation MEDIAL = known quantities → find resulting strength. Alligation ALTERNATE = known strengths → find quantities.

Problem 6: Alligation Alternate

You need 1000 mL of 85% alcohol, prepared from 95% and 70% stock. How many mL of the 95% stock do you need? Round your answer to one decimal place.

Answer 600.0 mL

Solution
  1. Alligation grid: high (95%) top-left, low (70%) bottom-left, target (85%) center.
  2. Parts of 95% = 85 − 70 = 15 parts (diagonal difference).
  3. Parts of 70% = 95 − 85 = 10 parts.
  4. Total parts = 25.
  5. Amount of 95% = (15 parts ÷ 25 total parts) × 1000 mL = 600 mL (and 400 mL of 70%).
  6. Rounded as the question asks: 600.0 mL.

Sanity check: 600×95% + 400×70% = 850 g = 1000 mL × 85% ✓

Problem 7: Alligation Medial

You mix 300 mL of 50% dextrose with 250 mL of 5% dextrose. What is the percentage strength of the final mixture? Round your answer to one decimal place.

Answer 29.5 %

Solution
  1. Grams from each: 300 mL × 50% = 150 g; 250 mL × 5% = 12.5 g.
  2. Total drug = 162.5 g in total volume 550 mL.
  3. Strength = 162.5 g ÷ 550 mL × 100 ≈ 29.545%.
  4. Rounded as the question asks: 29.5 %.

Alligation MEDIAL = known quantities → find resulting strength. Alligation ALTERNATE = known strengths → find quantities.

Problem 8: Alligation Alternate

You need 250 mL of 20% dextrose, prepared from 70% and 10% stock. How many mL of the 70% stock do you need? Round your answer to one decimal place.

Answer 41.7 mL

Solution
  1. Alligation grid: high (70%) top-left, low (10%) bottom-left, target (20%) center.
  2. Parts of 70% = 20 − 10 = 10 parts (diagonal difference).
  3. Parts of 10% = 70 − 20 = 50 parts.
  4. Total parts = 60.
  5. Amount of 70% = (10 parts ÷ 60 total parts) × 250 mL ≈ 41.667 mL (and 208.333 mL of 10%).
  6. Rounded as the question asks: 41.7 mL.

Sanity check: 41.667×70% + 208.333×10% = 50 g = 250 mL × 20% ✓

Problem 9: Alligation Medial

You mix 100 mL of 50% dextrose with 750 mL of 10% dextrose. What is the percentage strength of the final mixture? Round your answer to one decimal place.

Answer 14.7 %

Solution
  1. Grams from each: 100 mL × 50% = 50 g; 750 mL × 10% = 75 g.
  2. Total drug = 125 g in total volume 850 mL.
  3. Strength = 125 g ÷ 850 mL × 100 ≈ 14.706%.
  4. Rounded as the question asks: 14.7 %.

Alligation MEDIAL = known quantities → find resulting strength. Alligation ALTERNATE = known strengths → find quantities.

Problem 10: Alligation Alternate

You need 240 mL of 85% alcohol, prepared from 95% and 50% stock. How many mL of the 95% stock do you need? Round your answer to one decimal place.

Answer 186.7 mL

Solution
  1. Alligation grid: high (95%) top-left, low (50%) bottom-left, target (85%) center.
  2. Parts of 95% = 85 − 50 = 35 parts (diagonal difference).
  3. Parts of 50% = 95 − 85 = 10 parts.
  4. Total parts = 45.
  5. Amount of 95% = (35 parts ÷ 45 total parts) × 240 mL ≈ 186.667 mL (and 53.333 mL of 50%).
  6. Rounded as the question asks: 186.7 mL.

Sanity check: 186.667×95% + 53.333×50% = 204 g = 240 mL × 85% ✓

Sources and how far they go

Where a convention on this page comes from a published standard, the standard is named. Where it comes from a textbook or from ordinary practice instead, the entry says so. That difference matters when a preceptor or an exam disagrees with you.

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