What the Pearson square does
The Pearson square finds the proportions of two feeds that produce a mix at an exact nutrient level. Draw a square, put the target in the middle, put the two ingredient analyses at the left corners, subtract diagonally ignoring the sign, and read the parts off the right corners. It is the fastest hand method in animal nutrition and it has been taught in feed-mixing classes for generations because it needs no algebra on paper and it is very hard to get wrong once you have drawn it twice.
Underneath the drawing it is one linear equation. If x is the fraction of feed A, then the blended level is x·A + (1 − x)·B, and setting that equal to the target and solving gives x = (B − T) / (B − A). The square is a diagram of exactly that fraction, which is why the two diagonal differences are the numerator and the denominator.
Its one hard limit follows from the same algebra: because the mix is a weighted average of two numbers, it can only land between them. Corn at 8.5% crude protein and soybean meal at 47.5% can be blended to any level from 8.5% to 47.5% and to nothing outside it. A request for 50% needs a third, richer ingredient, and a request for 6% needs something poorer than corn.
Reading the square, corner by corner
The target goes in the middle. This is what you want in the finished feed — 16% crude protein for a growing ration, 12% for a maintenance mix, or a TDN or calcium figure if that is the constraint you are balancing on.
The ingredients go on the left. Put the lower analysis on one left corner and the higher on the other. Use the analysis on the same moisture basis for both; mixing an as-fed protein for corn with a dry matter protein for silage is the single most common way to get a wrong answer that looks right.
Subtract on the diagonals. Target minus lower analysis gives the parts of the higher feed. Higher analysis minus target gives the parts of the lower feed. The cross is not a mistake: to pull the mix up toward the target you need more of whichever feed is further away on the other side. Take absolute values so it does not matter which corner you started from.
Convert parts to percentages. Divide each parts figure by the sum of the two. When the target is inside the range that sum equals the full spread between the analyses, so the arithmetic is simply how far the target sits along that spread.
Scale to the batch. Multiply each percentage by the finished batch weight. Then verify: multiply each ingredient's pounds by its analysis, add, and divide by the batch weight. You must land back on the target, and if you do not, one of the four numbers you entered is wrong.
Worked example: 16% crude protein from corn and soybean meal
You are mixing a one-ton batch of a 16% crude protein grower from shelled corn analysed at 8.5% CP and 47% soybean meal analysed at 47.5% CP. Corn is $200 a ton delivered and the meal is $420.
- Parts of corn. |47.5 − 16| = 31.5 parts.
- Parts of soybean meal. |16 − 8.5| = 7.5 parts.
- Total parts. 31.5 + 7.5 = 39.0, which is also 47.5 − 8.5, the full spread.
- Percent corn. 31.5 ÷ 39 = 0.807692, so 80.769%.
- Percent soybean meal. 7.5 ÷ 39 = 0.192308, so 19.231%.
- Pounds per ton. 0.807692 × 2,000 = 1,615.4 lb of corn and 0.192308 × 2,000 = 384.6 lb of meal.
- Verify. 1,615.4 × 8.5 = 13,730.8 and 384.6 × 47.5 = 18,269.2. Add them: 32,000.0. Divide by 2,000 lb: 16.00% crude protein. The square is exact.
- Cost. 0.807692 × $200 = $161.54 and 0.192308 × $420 = $80.77, so the blend costs $242.31 a ton and this one-ton batch costs $242.31.
Push the target to 18% and the arithmetic moves in a straight line: parts of corn become 29.5, parts of meal 9.5, so the meal share rises from 19.231% to 24.359% and the blended cost rises to 0.756410 × $200 + 0.243590 × $420 = $151.28 + $102.31 = $253.59 a ton. Two extra points of protein cost $11.28 a ton with these two ingredients.
How to judge the answer before you mix it
First check that the two shares are ones you can physically deliver. A square that calls for 3% of an ingredient in a one-ton batch means 60 lb, which most farm mixers can weigh; the same 3% in a 200 lb batch is six pounds, and that is where inclusion accuracy starts to matter more than the arithmetic does. When an ingredient share falls below about 5%, mix it into a premix or carrier first rather than dropping it straight into the mixer.
Second, look at how close the target sits to one of the analyses. The share of each ingredient is a straight-line function of the target, and its slope is 1 ÷ spread. With corn and soybean meal the spread is 39 points, so one point of target moves the meal share by 1/39 = 2.56 percentage points. With two feeds only three points apart, one point of target moves the shares by 33 percentage points, and a routine laboratory variation on either analysis rewrites the ration. The calculator warns when the spread drops below three points for exactly that reason.
Third, remember what the square does not check. It balances one nutrient. A corn and soybean meal blend at 16% crude protein says nothing about lysine, calcium, phosphorus, salt, vitamins or energy density, and for a monogastric species amino acid balance rather than crude protein is what actually limits performance. Treat the square as the protein step in a longer formulation, not as the formulation.
Corn and soybean meal blends at common protein targets
| Target CP | Corn | Soybean meal | Meal per ton |
|---|---|---|---|
| 10% | 96.154% | 3.846% | 76.9 lb |
| 12% | 91.026% | 8.974% | 179.5 lb |
| 14% | 85.897% | 14.103% | 282.1 lb |
| 16% | 80.769% | 19.231% | 384.6 lb |
| 18% | 75.641% | 24.359% | 487.2 lb |
| 20% | 70.513% | 29.487% | 589.7 lb |
| 22% | 65.385% | 34.615% | 692.3 lb |
Every meal figure is 2,000 × (target − 8.5) ÷ 39. Substitute your own laboratory analyses before mixing; a different corn or a 44% meal changes every row.
Where the square goes wrong
- A target outside the ingredient range. Two feeds average; they never exceed. If you need more protein than your richest ingredient carries, you need a different ingredient.
- Mixed moisture bases. Use as-fed analyses for both feeds or dry matter analyses for both. A dry meal analysis blended against an as-fed silage analysis produces a confident, wrong answer.
- Balancing crude protein for a pig or a bird. Monogastrics need amino acids, and lysine is normally first-limiting. A square on crude protein can pass while lysine falls short.
- Ignoring the rest of the ration. Calcium, phosphorus, salt, trace minerals and vitamins are not touched by a two-ingredient square, and neither is energy density.
- Using book values instead of a lab. Corn between crops and soybean meal between suppliers both vary. The square is exact arithmetic on whatever analyses you give it, and it cannot be more accurate than they are.
- Rounding the shares before scaling. Round the finished pounds, not the percentages, or a large batch drifts off target.
When you need something other than a square
The square handles two ingredients and one nutrient. Three real extensions cover most of what comes next.
Group the ingredients. If you already know the proportions inside a grain mix and inside a protein supplement, compute the weighted analysis of each group, then run the square on those two composite numbers. This is how a square gets used on a five-ingredient ration in practice.
Two nutrients at once. Balancing protein and energy simultaneously with two feeds is two equations in one unknown and generally has no solution; you need a third ingredient and simultaneous equations, or a least-cost formulation program that runs linear programming across dozens of ingredients and constraints.
Cost as the objective. The square hits a nutrient target; it does not find the cheapest way to hit it. When several ingredient pairs can reach the same target, compare their blended cost per ton — this calculator gives you that figure directly, so you can run the square twice with different pairs and pick the cheaper.
Before you mix, size the requirement with the dry matter intake calculator so you know how much finished feed the group will actually consume, and check afterwards what the ration bought you with the feed conversion ratio calculator. If the low-protein side of your square is forage rather than grain, the round bale weight calculator and the hay tonnage per acre calculator tell you how much of it you have.