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I am looking for a formula to help me work out the following:

On average, how many calories would a 3 year old (weighing 11kg) have to consume daily to weigh 16kg by the time she is 4 years old?

The question assumes average activity levels and average height.

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The normal weight for a 4-year-old boy or girl is from 27 to 50 lbs (or 12-22 kgs).

Unless the child in question is deprived of average nutrition, normal growth and normal activity will get the girl (or boy) you describe to the vicinity of 16kg without any particular formula or intervention in the normally recommended diet/calories for a child that age.

The normally recommended calories that an average 4-year old needs to grow as expected (i.e., to reach the 16 kg range you are looking toward) is in the area of 1500-1700 kcals per day.

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