9⅕ - 1⅔ × 3⅜ and 2¾ × 1⅙ - ⅘. 3. Find the square root of 4⅑ to four places of decimals. 4. If, by selling an article for $2, a man gains ⅐ of the cost price, at what price must he sell it so as to gain 8 per cent.? 5. The area of one side of a cubical cistern is 14·0625 square feet; find to the nearest gallon the amount of water which it will hold when full, assuming that one cubic foot weighs 1,000 ounces and that one gallon of water weighs 10 lb. 6. Find the cost of a carpet to cover a floor 22 ft. 6 in. long and 18 ft. 9 in. wide at 5_s._ 4_d._ per square yard. 7. Divide £37. 10_s._ 4½_d._ by 4⅐ and express £3. 14_s._ 7½_d._ as the decimal of £10. 8. A sum of $2,500 is lent at compound interest at 3½ per cent. per annum. What is due to the lender at the end of three years? 9. _A_ can do a piece of work in 24 days which _B_ can do in 36 days. What fraction will remain to be done if both are engaged upon the work for 6 days? II. ALGEBRA OR GEOMETRY. (_a_) ALGEBRA. The full working must be shown in all cases. 1. If 2_p_ - 3_q_ = 8, ½_q_ = _p_ - 6, find the value of (_p_ - _q_)² - 7(_p_² - _q_²) + 12(_p_ + _q_)². 2. Divide _x_⁶ + _ax_⁵ - 12_a_²_x_⁴ + 19_a_³_x_³ + 15_a_⁴_x_² - 14_a_⁵_x_ + 2_a_⁶ by _x_² + 4_ax_ - 2_a_². 3. Find the highest common factor of 54_p_⁵ - 11_p_²_q_³ - _q_⁵ and 12_p_⁵ + 11_p_⁴_q_ + _q_⁵, and the least common multiple of _a_²_c_(_ab_ - _b_²), 4(_a_² - _b_²)_c_³, 6_b_²_c_. 3(_ab_² + _a_²_b_). 4. Simplify: (1) ((_p_ + 3)/(_p_² + 2)) + (1/(2_p_ + 2)) - (1/(_p_ - 1)); (2) {_p_(_p_ + _q_) - _q_(_p_ - _q_)} {_p_(_p_ - _q_) - _q_(_q_ - _p_)} ÷ (_p_³ - _q_³). 5. Solve the equations: (1) (_x_ + 3)/6 - (11 - _x_)/7 = (⅖)(_x_ - 4) - (⅟₂₁)(_x_ - 3); (2) 1/(_x_ + 1) + 2/(_x_ + 2) = 3/(_x_ + 3); (3) 17_x_/a + 3_y_/b = 9, 3_x_/a - 2_y_/b = 37. 6. Find the remainder (free from _x_) when _ax_² + _bx_ + _c_ is divided by _x_ - _p_. What inference is suggested by the result? 7. By the investment of £400, partly in a 2½ per cent. stock at 75 and partly in a 4 per cent. stock at 96, a total income of £15. 8_s._ 4_d._ is obtained. How much money is spent on each stock? 8. The perimeter of a room is _a_ feet, and the height of its walls is _b_ feet: find the cost of papering the walls of the rooms with paper _x_ inches wide at _y_ pence per foot, the area of the windows, door, and fireplace being (⅒)_ab_ square feet in all. (_b_) GEOMETRY. The use of reasonable symbols and abbreviations is permitted. 1. If two angles of a triangle are equal, the sides opposite to them are equal. 2. Find the locus of points which are equidistant from two given points. 3. The sum of any two sides of a triangle is greater than the third side. 4. Make a triangle equal in area to a given triangle and having one of its angles equal to a given angle. 5. Show that the bisector of the exterior angle at the vertex of an isosceles triangle is parallel to the base.