### Brain Teasers

# Population Explosion

A group of 100 people start a colony on the first day of the year. Each year, the birth rate is 4% of the population at the beginning of the year, and the death rate is 3% of the population at the beginning of the year. The resulting births and deaths are truncated to a whole number. These rates are constant. For example, consider a population of 99 people at the start of a year. A 4% birth rate would be 3.96, truncated to 3 births during the year. A 3% death rate would be 2.97, truncated to 2 deaths during the year. The population at the end of that year would be 99 + 3 - 2 = 100. Based on the stated birth and death rates, what is the LAST year that the population increases by only 1?

### Answer

The last year the population would increase by only one would be the 61st year.1) Beginning with the 1st year, the number of births will be 4 (4% of 100 = 4) and the number of deaths will be 3 (3% of 100 = 3), which results in a population increase of 1. The population will increase at the rate of 1 per year for 25 years, reaching 125 at the end of the 25th year.

2) Beginning with the 26th year, the number of births per year will increase to 5 (4% of 125 = 5), while the deaths will still be 3. The population will increase at the rate of 2 per year, reaching 135 at the end of the 30th year.

3) Beginning with the 31st year, the number of deaths per year will increase to 4 (3% of 135 = 4.05), while the births will still be 5. The population will increase at the rate of 1 per year, reaching 150 at the end of the 45th year.

4) Beginning with the 46th year, the number of births per year will increase to 6 (4% of 150 = 6), while the deaths will still be 4. The population will increase at the rate of 2 per year, reaching 168 at the end of the 54th year.

5) Beginning with the 55th year, the number of deaths per year will increase to 5 (3% of 168 = 5.04), while the births will still be 6. The population will increase at the rate of 1 per year, reaching 175 at the end of the 61st year.

6) Beginning with the 62nd year, the number of births per year will increase to 7 (4% of 175 = 7), while the deaths will still be 5. The population will increase at the rate of 2 per year, reaching 201 at the end of the 75th year.

7) Beginning with the 76th year, the number of births per year will increase to 8 (4% of 201 = 8.04) and the number of deaths will increase to 6 (3% of 201 = 6.03). The population will continue to increase at the rate of 2 per year. The population will never again increase by only 1 per year.

Although the 26th year is the first year that the population increases by more than 1, the 25th year is not the last year that it increases by 1. The last year that the population increases by 1 is the 61st year, when the population reaches 175.

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