Page:Calculus Made Easy.pdf/155

 stocking, or locking it up in his safe. Then, if he goes on for $$10$$ years, by the end of that time he will have received $$10$$ increments of £$$10$$ each, or £$$100$$, making, with the original £$$100$$, a total of £$$200$$ in all. His property will have doubled itself in $$10$$ years. If the rate of interest had been $$5$$ per cent., he would have had to hoard for $$20$$ years to double his property. If it had been only $$2$$ per cent., he would have had to hoard for $$50$$ years. It is easy to see that if the value of the yearly interest is $$\frac{1}{n}$$ of the capital, he must go on hoarding for $$n$$ years in order to double his property.

Or, if $$y$$ be the original capital, and the yearly interest is $$\frac{y}{n}$$, then, at the end of n years, his property will be

(2) At compound interest. As before, let the owner begin with a capital of £$$100$$, earning interest at the rate of $$10$$ per cent. per annum; but, instead of hoarding the interest, let it be added to the capital each year, so that the capital grows year by year. Then, at the end of one year, the capital will have grown to £$$110$$; and in the second year (still at 10%) this will earn £$$11$$ interest. He will start the third year with £$$121$$, and the interest on that will be £$$12$$. $$2$$s.; so that he starts the fourth year with £$$133$$. $$2$$s., and so on. It is easy to work it out, and find that at the end of the ten years the total capital