Those three numbers represent the percentages of nitrogen, phosphorus and potassium. Actually the middle number is not the percent pure phosphorus but rather P2O5, and the third number is not pure potassium but rather the percent K2O. These two are just industry standards for expressing P and K. But for all practical purposes we refer to them as simple the percent phosphorus and the percent potassium.
A 15-5-10 fertilizer for example, thus would contain 15% nitrogen, 5% phosphorus and 10% potassium. Thus the numbers tell us how concentrated a fertilizer is in these 3 nutrients and the ratio of these nutrients to each other.
A lawn needs all three to grow and be healthy. Each nutrient has many functions in a plant. In general, nitrogen promotes growth and vigor. Phosphorus promotes strong roots and is important in fruit and flower production. Potassium is important in fruiting, cold hardiness and other plant functions.
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