How to use the Pearson square method

Guide · Updated 12 October 2026 · Worked examples are calculated with the pearson square calculator

The Pearson square tells you how to mix two ingredients to reach a target percentage of one nutrient or property: protein in a feed, fat in milk, sugar in a syrup. Write the two concentrations on the left, the target in the middle, subtract along the diagonals, and the results are the parts of each ingredient. Below: the steps, why it works, and worked feed examples, including what to do with a third ingredient.

The method in five steps

1. Draw a square. Write ingredient A's concentration at the top left and ingredient B's at the bottom left.
2. Write the target concentration in the middle. It must lie between A and B, or no mix of the two can reach it.
3. Subtract along each diagonal, ignoring the sign: |target − B| goes at the top right, |A − target| at the bottom right.
4. The top-right number is the parts of A; the bottom-right number is the parts of B. Add them for the total parts.
5. Each ingredient's share is its parts ÷ total parts. Multiply by the batch size for the amounts.

Example: calculate fat in the meat block

If your two ingredients' labels or measured analyses report 10% and 50% fat by mass on the same basis, a 20% target lies between them. For a 5 kg meat-and-fat block, the square gives 30 parts of the 10% ingredient to 10 parts of the 50% ingredient: 3.75 kg and 1.25 kg respectively.

Check: 3.75 × 0.10 + 1.25 × 0.50 = 1.00 kg fat; 1.00 ÷ 5.00 = 20%. These are conditional example values, not assumed fat contents of any particular cut. Use the label or analysis for the ingredients you actually have.

The 5 kg denominator is the meat-and-fat block. Water, salt and other additions change the percentage of the entire recipe. For example, adding 0.5 kg water contributes no fat: the same 1 kg fat is then 18.18% of a 5.5 kg total mix. Do not use a whole-recipe target as a meat-block target without accounting for all additions.

The formula, and why it works

The square is a shortcut for a mass balance. If a fraction x of the mix is A and the rest is B, the nutrient in the mix is x × A + (1 − x) × B, which must equal the target T. Solving gives x = (T − B) ÷ (A − B): exactly the share the square produces. That is also why it only works when the percentages are of the same kind (all by mass, on the same moisture basis) and why the target must sit between the two.

Example 1: a 16% protein mix from maize and soybean meal

Maize at 8.5% crude protein, soybean meal at 44%, target 16%, for 1,000 kg. (Illustrative analyses; use your own feed tests or supplier values.)

• Parts of maize = 44 − 16 = 28; parts of soybean meal = 16 − 8.5 = 7.5; total 35.5 parts.
• Maize = 28 ÷ 35.5 = 78.87% of the mix, so 788.7 kg; soybean meal 21.13%, so 211.3 kg.
• Check: 788.7 kg × 8.5% + 211.3 kg × 44% = 160.0 kg of protein in 1,000 kg, which is 16%.

Example 2: a 30% protein fish feed

Fish meal at 60% protein, rice bran at 12%, target 30%, for a 100 kg batch (illustrative values):

• Parts of fish meal = 30 − 12 = 18; parts of rice bran = 60 − 30 = 30.
• Fish meal = 37.50% (37.5 kg), rice bran = 62.50% (62.5 kg).

Three ingredients

A single square handles two ingredients and one nutrient. With a third ingredient there are many possible answers for one target, so you fix something first. Two common ways:

Fix the amount of one ingredient

Say the mix must contain 3% of a mineral and vitamin premix with no protein. Only 97% of the mix is left for maize and soybean meal, and that part must carry all the protein, so its target rises to 16 ÷ 0.97 = 16.49%. Square maize (8.5%) and soybean meal (44%) to 16.49%, then scale to the 970 kg left in a tonne: 751.5 kg maize, 218.5 kg soybean meal and 30.0 kg premix. Check: 160.0 kg of protein per tonne.

Fix a ratio between two ingredients

To use maize and wheat (11.5%, illustrative) in a 2 : 1 ratio, treat them as one "grain" ingredient with the weighted concentration (2 × 8.5 + 1 × 11.5) ÷ 3 = 9.50%. Square the grain against soybean meal to 16%: 81.16% grain (811.6 kg, split 541.1 kg maize and 270.5 kg wheat) and 18.84% soybean meal (188.4 kg).

When the square is the wrong tool

• More than one nutrient at once (say protein and energy, or fat and solids-not-fat): two targets need simultaneous equations or a least-cost formulation program.
• Mixed bases: dry-matter and as-fed percentages, or mass and volume, must be converted to the same basis first.
• Targets outside the range of the ingredients cannot be reached by any blend of them.

For dairy, the same square standardises milk fat; see milk fat standardisation for cream, skim and separation examples.