gether they can build it in 6 days: find the time it would take each alone. 6. Solve the equations - + - = -,- = 2, —[--==„. 'L x 1 y . z z y x z o x _i_ 2 4 x 7 7. Solve the equation , ~ = .=. u x — 1 2 x 3 8. The length of a rectangular field exceeds the breadth by one yard, and the area is three acres; find the dimensions. 1\» 9. Expand the expression ^2 a + pj 10. What is Elimination? How many methods are you familiar with? Explain them in full XXIII. 1. Simplify (a + b) (6 + e) — (e + d) (d + a) — (a + c) (b — d). 2. Reduce to its simplest form ^ X 2 T" -g X a — 6 a — r, I . 2a; 3 / 4. Find a number such that three times its square diminished by five times the number itself shall amount to 50. Solve completely. 5. What fraction is that which becomes equal to f when 6 is added to its numerator, and equal to J when 2 is subtracted from its denominator? e _ . ., ,. 2x — 3 5 3x — 5 6. Solve the equation ^—g = g - ^-3. 7. A and B find a purse of dollars. A takes out 2 dol 2 X X i j. « —^— ^ 6. Solve the equations y -4- ^ = g 4- 5, —j g— _ z-4-3 _ 2y —5 _ 7 _ z_ — 1Q , * — 3 — 4 12 7. A man hires a certain number of acres of land for % 336. He cultivates 7 acres for himself, and lets the rest for $ 4 an acre more than he pays for it. He receives for the portion that he lets what he pays for the whole, or % 336. Find the number of acres. 8. The value of a fraction, if its numerator is doubled and its denominator increased by 7, is \; while, if its denominator is doubled and its numerator increased by 2, its value is f. What is the fraction? XXV. 1. A certain piece of work can be done by A and B working together in 3| days, by B and C in 4f days, and by C and A in 6 days. Required the time in which either can do it alone, and the time in which all can do it together. 2. Solve the equation ^— 1 . = r. 3. Solve the equation x* — (a — b -f- c) x = (6 — a)c. ax + V. a2— (b — 2x)b+2b> 4. Divide 1 — by r-1—! 2 b, cr -\- ax J a -f- x and reduce the result to its lowest terms. 5. Divide 9 a2n — a3n — 27 a" + 27 by a" — 3. 6. Divide ft- by y/-. 7. What is the reason that aman = am+n? XXVI. 1. Solve the equation % — 3 = 4 x -. 2. What are the three methods of Elimination? Solve the following equations by any two of the three methods: 6 x + J y = 0, 2 (4 x — 1) = 3 {y - 8). 3. M's age is to N's as a is to b; but c years ago ATs age was to N's as a' to V. Required the present ages of both. 4. Divide 1 - I X by ^"f^ - *; and reduce the answer to its lowest terms. Simplify the division by cancelling. 5. Find the fourth term of XXVII. 1. Solve the equation — — .=—-— = -. c ox — a c 2. What are eggs a dozen when two more in a shilling's worth lowers the price one penny per dozen? 3. A merchant adds yearly to his capital one third of it, but takes from it at the end of each year $ 5,000 for his expenses. At the end of the third year, after deducting the last $ 5,000, he finds himself in possession of twice the sum he had at first. How much was his original capital? 4. Divide tfaV6 by 5. Find x from the proportion 6aTM"~2 b : x = 15a365: 40a-(ra-1,. 6. Divide a2 „ a ,„ by a. 7. What is the rule for transposing a term from one side of an equation to the other; and what is the reason of the rule? 8. Solve the equations 4x -f- 3y -f- 2z = 40, 5x — 9y — 7z = 47, 9x — 8y — 3z = 97. 9. Find (a — &)7 by the Binomial Theorem XXVIII. 1. A certain sum of money at simple interest will amount to a dollars in m months, and to b dollars in n months. Find the principal and the rate of interest. Find the answers when a = 1837.50, b = 1890.00, m = 10, 71= 16. 4. Find (x — yf and — by the Binomial Theorem. 5. Divide 13aV — 5a4 — 13ax* + 6a* — 13a3x by 3ax + *2 — 2a?. 6. Find two numbers of which the sum is a and difference 6. State a rule for finding two numbers when their sum and difference are given 7. Find the greatest common divisor and the least common multiple of 12asbci and 27ale'd. |