Elements of Algebra

CHAP. III. OF ALGEBRA. 205 by two pipes running into it, would be filled in $20$ minutes by one alone. Required in what time it would be filled by the other alone. Ans. $30$ minutes. 15. Required two numbers, whose sum may be $s$, and their proportion as $a$ to $b$. Ans. $\frac{as}{a+b}$, and $\frac{bs}{a+b}$. 16. A privateer, running at the rate of $10$ miles an hour, discovers a ship $18$ miles off making way at the rate of $8$ miles an hour: it is demanded how many miles the ship can run before she will be overtaken? Ans. $72$. 17. A gentleman distributing money among some poor people, found that he wanted $10s$. to be able to give $5s$. to each; therefore he gives $4s$. only, and finds that he has $5s$. left: required the number of shillings and of poor people. Ans. $15$ poor, and $65$ shillings. 18. There are two numbers whose sum is the $6$th part of their product, and the greater is to the less as $3$ to $2$. Required those numbers. Ans. $15$ and $10$. N. B. This question may be solved by means of one unknown letter. 19. To find three numbers, so that the first, with half the other two, the second with one-third of the other two, and the third with one-fourth of the other two, may be equal to $34$. Ans. $26$, $22$, and $10$. 20. To find a number consisting of three places, whose digits are in arithmetical progression: if this number be divided by the sum of its digits, the quotients will be $48$; and if from the number $198$ be subtracted, the digits will be inverted. Ans. $432$. 21. To find three numbers, so that $\frac{1}{2}$ the first, $\frac{1}{3}$ of the second, and $\frac{1}{4}$ of the third, shall be equal to $62$: $\frac{1}{3}$ of the first, $\frac{1}{4}$ of the second, and $\frac{1}{5}$ of the third, equal to $47$; and $\frac{1}{4}$ of the first, $\frac{1}{5}$ of the second, and $\frac{1}{6}$ of the third, equal to $38$. Ans. $24$, $60$, $120$. 22. If A and B, together, can perform a piece of work in $8$ days; A and C together in $9$ days; and B and C in $10$ days; how many days will it take each person, alone, to perform the same work? Ans. $14\frac{34}{49}$, $17\frac{23}{41}$, $23\frac{7}{31}$. 23. What is that fraction which will become equal to $\frac{1}{2}$, if an unit be added to the numerator; but on the contrary, if an unit be added to the denominator, it will be equal to $\frac{1}{3}$? Ans. $\frac{2}{5}$. 24. The dimensions of a certain rectangular floor are such, that if it had been $2$ feet broader, and $3$ feet longer, it would have been $64$ square feet larger; but if it had been $3$