Ration formulation and the Pearson square
How livestock rations are formulated: requirements, feed composition, trial and error, the Pearson square, algebra and least-cost methods, with sums.
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Feed is usually the largest single cost of producing milk, meat and eggs, often 60 to 70 per cent of the total. A ration that provides too little protein or energy limits growth and production; one that provides too much wastes money and nitrogen. Ration formulation combines available feeds in proportions that meet an animal's nutrient requirements at the lowest practical cost.
Ration formulation methods, especially the Pearson square, are among the most common calculations in animal nutrition exams.
What a ration must supply
A ration is the feed an animal receives in 24 hours; a balanced ration supplies all nutrients in the right amounts and proportions. Requirements depend on species, body weight, growth rate, pregnancy, milk yield, work and environment, and are published in standards such as the US National Academies (NRC/NASEM) nutrient requirements, the French INRA system and national feeding standards. Requirements cover:
- Dry matter intake (DMI), the amount the animal can eat;
- Energy (as metabolisable or net energy, or total digestible nutrients);
- Protein (crude protein, or metabolisable protein and amino acids such as lysine and methionine for pigs and poultry);
- Minerals (calcium, phosphorus, sodium, trace elements) and vitamins;
- Fibre for ruminants, to maintain rumen function.
Steps in formulating a ration
- Define the animal: species, weight, production level and stage.
- Look up requirements for intake, energy, protein, minerals and vitamins.
- List available feeds with their composition (ideally analysed), price and any limits on inclusion.
- Balance the ration using one of the methods below.
- Check intake, fibre, minerals and palatability; adjust for practical limits.
- Monitor animal performance and adjust.
Methods
Trial and error
Start with a likely mix, calculate the nutrients supplied, and adjust step by step. Simple and widely used by farmers and advisers with spreadsheets.
The Pearson square
A quick method to mix two feeds to reach one nutrient target, such as crude protein, provided the target lies between the two feeds' contents.
- Write the target in the centre and the two feeds' nutrient contents at the left corners.
- Subtract across each diagonal, ignoring signs, and write the results at the right corners.
- Each right-hand number is the parts of the feed on the same horizontal line; divide by the total to get percentages.
When part of the mix is fixed, such as 5 per cent minerals and vitamins with no protein, the target for the remaining feeds is adjusted upward before using the square.
Algebra (simultaneous equations)
Two unknowns can be solved with two equations, one for total quantity and one for the nutrient, giving the same answer as the square and allowing more flexibility.
Least-cost formulation (linear programming)
Commercial feed mills and many farms use linear programming, which finds the cheapest combination of many feeds that meets all nutrient constraints (minimums and maximums) at once. It was first applied to feed mixing in the 1950s and is now built into ration software. Results must still be checked for palatability and practicality.
Mixing a 16 per cent protein concentrate
Pearson square. Maize has 9 per cent CP and soybean meal 44 per cent; the target is 16 per cent.
- Maize: 44 − 16 = 28 parts; soybean meal: 16 − 9 = 7 parts; total 35.
- Maize = 28 ÷ 35 = 80 per cent; soybean meal = 7 ÷ 35 = 20 per cent.
- Check: 0.80 × 9 + 0.20 × 44 = 7.2 + 8.8 = 16.0 per cent.
With a fixed mineral premix. The mix must include 5 per cent of a mineral premix with no protein. The other 95 per cent must then supply all 16 per cent: target for the maize–soybean part = 16 ÷ 0.95 ≈ 16.84 per cent.
- Maize: 44 − 16.84 = 27.16 parts; soybean meal: 16.84 − 9 = 7.84 parts; total 35.
- In the 95 per cent: maize = 27.16 ÷ 35 × 95 ≈ 73.7 per cent, soybean meal = 7.84 ÷ 35 × 95 ≈ 21.3 per cent, premix 5 per cent.
Algebra check. Let M and S be kilograms of maize and soybean meal in 100 kg of the original mix: M + S = 100 and 0.09M + 0.44S = 16. Then 0.09(100 − S) + 0.44S = 16, so 0.35S = 7, S = 20 kg, M = 80 kg.
Cost. At 0.25 dollars per kg of maize and 0.50 per kg of soybean meal, the original mix costs 0.80 × 0.25 + 0.20 × 0.50 = 0.30 dollars per kg.
Practical considerations
- Palatability and physical form: animals must eat the ration; particle size matters for ruminants.
- Maximum inclusion limits for feeds with anti-nutritional factors, such as cottonseed cake (gossypol) or raw soybeans.
- Mixing accuracy, especially of small ingredients such as premixes.
- Total mixed rations (TMR) for dairy cows combine forage and concentrate so each bite is balanced.
- Feed additives where permitted, such as buffers, enzymes and ionophores.
Key points
- A balanced ration meets intake, energy, protein, mineral, vitamin and (for ruminants) fibre needs from published requirements.
- Steps: define the animal, find requirements, list feeds, balance, check and monitor.
- Methods: trial and error, the Pearson square (two feeds, one nutrient), algebra, and least-cost linear programming.
- Pearson square: subtract diagonally; parts ÷ total = proportion.
- Adjust the target when part of the mix is fixed.
What the exams ask
- School-leaving and entrance papers: what a balanced ration is and the nutrients it must supply.
- Degree and research-fellowship entrances: steps of ration formulation, the Pearson square with and without fixed ingredients, algebraic methods, and least-cost formulation.
- Professional examinations: formulating rations for dairy cows, poultry and pigs, feed costing and checking intake and fibre limits.
Described in general terms: the kind of examination, not a past paper. Laisha never reproduces exam questions.
Questions people ask
Can the Pearson square balance more than one nutrient?
Not directly; it balances one nutrient at a time for two feeds or two groups of feeds. For several nutrients, use algebra, spreadsheets or least-cost software.
Why must the target lie between the two feeds?
Mixing two feeds can only give a value between their two contents. A 16 per cent protein feed cannot be made from feeds of 9 and 12 per cent protein.
Does least-cost mean lowest quality?
No. Least-cost formulation meets all the nutrient constraints set by the nutritionist at the lowest cost. Quality depends on setting the right constraints and using accurate feed analyses.
Sources
- McDonald, P. et al. Animal Nutrition, 7th edition. Pearson, 2011.
- National Academies of Sciences, Engineering, and Medicine. Nutrient Requirements of Dairy Cattle, 8th revised edition. National Academies Press, 2021.
- National Research Council. Nutrient Requirements of Swine, 11th revised edition. National Academies Press, 2012.
- Waugh, F. V. "The minimum-cost dairy feed." Journal of Farm Economics 33, 299–310 (1951).
- Pond, W. G., Church, D. C., Pond, K. R. and Schoknecht, P. A. Basic Animal Nutrition and Feeding, 5th edition. Wiley, 2005.
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