Formulating animal feed rations with a Pearson Square

When developing a ration for livestock, especially when using only a few ingredients, a Pearson Square may be easiest to use. A Pearson Square is a simple and easy way to develop a total mixed ration (TMR) for livestock. The main consideration is knowing the final requirement or percentage of the nutrient needed in the ration. This article will explain how to use a Pearson Square effectively to formulate a ration.

PEARSON SQUARE
The Pearson Square is an easy and simple way to calculate livestock rations, especially when using only a few ingredients. Figure 1 is the main model for the Pearson Square. In Figure 1, we are formulating a ration to meet crude protein (CP) requirements using chopped hay and corn silage. In this example a final diet formulation of 11% CP is required.

HOW TO USE A PEARSON SQUARE
1. The value in the middle MUST be intermediate between the two values used in the left side of the square. In this example, 11% is the CP requirement for the animals. Chopped hay has 12.25% CP and corn silage has 10.8% CP.

2. Disregard any negative numbers during subtraction.

3. Subtract the nutrient value from the nutrient requirement on the diagonal.
a. 10.8 – 11.0 = 0.2 parts chopped hay
b. 12.25 – 11.0 = 1.25 parts corn silage

4. Add the parts of each ingredient and divide by the total to calculate the percent of the ration that each ingredient will represent.
a. 0.2 + 1.25 = 1.45 total parts
b. Chopped Hay: (0.2 ÷ 1.45) × 100 = 13.8%
c. Corn Silage: (1.25 ÷ 1.45) × 100 = 86.2%

5. Mixing a 1,000 lb batch on a dry matter basis
a. Chopped hay: 1,000 lb × (13.8 ÷ 100) = 138 lb
b. Corn Silage: 1,000 lb × (86.2 ÷ 100) = 862 lb

The Pearson Square works the best for rations using only two ingredients. When using the Pearson Square, nutrient concentrations of the feeds used and the requirement need to be on the same basis, as-fed or dry matter. Most feed analysis is provided on a dry matter basis. Therefore, once the formulation is complete, the numbers should be converted back to as-fed for easy use when feeding. It is always important to check your math after completing a Pearson Square.

To check the math using the Figure 1 example:
• 138 lb chopped hay × 12.25% CP = 16.9 lb CP
• 862 lb corn silage × 10.8% CP = 93.1 lb CP
• Cow’s requirement = 1,000 lb × 11% CP = 110 lb CP
• Check: 16.9 lb CP + 93.1 lb CP = 110 lb CP
Since using the dry matter basis for our formulations, we need to convert to as-fed for ease of use when mixing the diet. The chopped hay has a dry matter of 88% and the corn silage has a dry matter of 35%. To convert dry matter to as-fed, we will take the weights of the 1,000 lb batch and divide by the dry matter percent.

• Chopped hay: 138 lb ÷ (88 ÷ 100) = 157 lb as-fed
• Corn silage: 862 lb ÷ (35 ÷ 100) = 2,463 lb as-fed

USING A PEARSON SQUARE WITH MORE THAN TWO INGREDIENTS
Although a Pearson Square is easiest and only includes options for two ingredients, formulations can be made with more. Figure 2 is an example of using more than two ingredients. In this example, a final diet formulation of 15% CP is needed.

In this example, there is a supplement containing 50% soybean meal (SBM) and 50% corn gluten feed (CGF) and a grain mixture containing 60% corn grain and 40% soybean hulls (SBH).

Supplement mixture
• 50% Soybean meal with 54% CP
• 50% Corn gluten feed with 24% CP

Grain mixture
• 60% Corn grain with 10% CP
• 40% Soybean hulls with 12% CP

Step 1: Calculate the CP in the supplement and grain mixtures.
• Supplement mixture
    o 50% SBM × (54 ÷ 100) = 27% CP
    o 50% CGF × (24 ÷ 100) = 12% CP
    o 27% + 12% = 39% CP

• Grain mixture
    o 60% Corn × (10 ÷ 100) = 6% CP
    o 40% SBH × (12 ÷ 100) = 4.8% CP
    o 6% + 4.8% = 10.8% CP

Step 2: Place the CP for each mixture on the Pearson Square and subtract diagonally.
• 39.0 – 15.0 = 24 parts grain mixture
• 10.8 – 15.0 = 4.2 parts supplement mixture
• 24 + 4.2 = 28.2 total parts
• Grain mixture: (24 ÷ 28.2) × 100 = 85.1%
• Supplement mixture: (4.2 ÷ 28.2) × 100 = 14.9%

Step 3: Determine how much of each ingredient are needed.
• Supplement mixture
    o 4.2 parts × 50% = 2.1 parts SBM and CGF each
    o (2.1 parts ÷ 28.2 parts) × 100 = 7.45% SBM and CGF each

• Grain mixture
    o 24 parts × 60% = 14.4 parts Corn
    – (14.4 ÷ 28.2) × 100 = 51.06% Corn
    o 24 parts × 40% = 9.6 parts SBH
    – (9.6 ÷ 28.2) × 100 = 34.04% SBH

Step 4: Check the math.
• 7.45 lb × 54% CP = 4.02
• 7.45 lb × 24% CP = 1.79
• 51.06 lb × 10% CP = 5.11
• 34.04 lb × 12% CP = 4.08
• 4.02 + 1.79 + 5.11 + 4.08 = 15 lb CP

Step 5: Convert to as-fed basis.

CONCLUSION
The Pearson Square is an excellent tool to use when formulating livestock rations. It is most efficient when only two ingredients are being fed. However, more ingredients can be used with a bit more math involved. The Pearson Square can be effective if access to diet formulation software is limited. A Pearson Square can often be used when mixing high- and low-quality forages together. Remember when using a Pearson Square, it’s important to work within the same units, such as dry matter or as-fed or pounds and kilograms. Any nutrient can be used to balance a diet with a Pearson Square as long as the nutrient requirement falls between the nutrient concentrations in the feeds used. Remember to always check your math when finished with a Pearson Square.

About Dr. Megan Van Emon
Dr. Megan Van Emon is an Extension Beef Cattle Specialist at Montana State University. Her research focuses on supplementation strategies, cow-calf production, water quality, and bull development. Dr. Van Emon’s research and extension programs focus on ruminant nutrition and improving beef cattle production in Montana and throughout the United States. She is located at USDA-ARS Fort Keogh in Miles City, Montana.