Page:Elementary algebra (1896).djvu/185

167 Rh EXAMPLES XIX.

Make such necessary changes in the statements of the following problems as will render them possible arithmetically.

1. A is 27 years old and B 15 ; in how many years will A be twice as old as B ?

2. What are the two numbers whose difference is 50, and sum 40 ?

3. If to the sum of twice a certain number and {1}{5} of the same number 10 be added, the result is equal to twice the number.

4. A man loses $400, and then finds that 6 times what he had at first is equal to 5 times what he has left.

5. What fraction is that which becomes {4}{7} when 1 is subtracted from its numerator, and {1}{2} when 1 is subtracted from its denominator?

6. A is to-day 25 years old, and B's age is {4}{5} of A's : find the date when A's age is twice that of B.

MEANING OF {a}{\infty }, {a}{\infty}, {0}{0}, {\infty}{\infty}.

181. Meaning of {a}{0}. Consider the fraction {a}{x} in which the numerator a has a certain fixed value, and the denominator x is a quantity subject to change; then it is clear that the smaller x becomes, the larger does the value of the fraction {a}{x} become. For instance, {a}= 10a, {a} = 1000 a, {a} = 100000 a.

By making the denominator x sufficiently small, the value of the fraction {a}{b} can be made as large as we please ; that is, as the denominator x approaches to the value 0, the fraction becomes infinitely great. The symbol \infty is used to express a quantity infinitely great, or more shortly infinity. The full verbal statement, given above, is sometimes written {a}{0} = \infty.

182. Meaning of {a}{\infty}. If, in the fraction {a}{x}, the denominator x gradually increases and finally becomes infinitely large,