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That figure seems slightly exaggerated, Wikipedia states he had $9800 saved when he was 20. And that he made $175 a month from his postal route. If he started his paper route at 15 he would have made over $10,000 from just the paper route during those 5 years. And the article lists several more business ventures he was active in.

Few people start working as early as 13-14, but if we extrapolate the $175 a month (which would be almost $2500 in today's dollars), and you had no expenses you could save $150,000 by the time you were 20, today.

Must surprising here is how well paid that paper route was. Average salary was $200 a month in 1944. Today the average salary is $2500 a month, but you can't earn as much from such a simple job today.