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"The US economy in free fall grows at a rate of 4%/yr. China will have to grow at 10% for the next 100 years to equal it in size."
Your calculation is incorrect.
US GDP in 2006 was 5 times of China.
With your assumption that US economy grows 4%/yr, and Chinese 10%/yr.
5 * (1.04)^n = (1.1)^n
n = number of years = 29
Therefore with those grow rates it would take China 29 years to overtake US economy.
In reality the difference in their GDP grow rates is greater, so it should happen much sooner.