EXAMPLE II. Two travellers, A and B, met at an inn. A afked B how far he had travelled. B answered, that he had travelled fo many miles and furlongs. Well, fays A, I travelled only half that distance; and the number of miles I travelled correfponds with your furlongs, and my furlongs with your miles. How far did each travel? Two merchants, A and B, began trade with equal stocks; but A, by frugality and application, gained 6cl. while B, through mifmanagement and bad luck, loft 801. At the year's end A was 8 times richer than B. Required their original ftock. Anf. 1001. QUADRATIC EQUATIONS. When the square and the root of the unknown quantity are joined together, it is called an adfected quadratic equation. RULE. RULE. Tranfpofe the quantities till the unknown quantity stand on one fide of the equation. Divide both fides by the coefficient of the square of the unknown quantity. Add the fquare of one half the coefficient of the fimple power to both fides of the equation. Extract the fquare root, and tranfpofe the half coefficient, which gives the value of the unknown quantity. EXAMPLE I. Required two numbers whose fum is 16, and product 48. x+y=16 xy=48 x=16-y x=48 y y y2-163—48 Per Rule, 1-16y+64=16 y2—16y+64=16 Required two numbers whofe product is 108, and sum of their squares 369. 1. A man and his wife did ufually drink out a barrel of beer in 12 days; and they found, by often experience, that the wife being abfent, the man drank it out in 20 days. In how many days would the wife alone drink it out at her rate of drinking? Anf. 30 days. 2. Two fhips, A and B, loaded with the fame fort of wine, failing by a pass, they were obliged to pay toll according to the quantity each had on board. A had 250 hogfheads, out of which the paid 1 hogfhead, and 36 fhillings more. B had 400 hogfheads, out of which the paid 2 hogfheads, and received back 20 fhillings. Required at what rate the wine was valued per hogfhead. Anf. 41. 14s. Suppofe the minute and hour hands of a common clock to be be in conjunction, in how many hours will they be in conjunction again? Anf. In 1 hour. 4. Required two numbers, fuch that the quot of the greater divided by the leffer may be 2 lefs than their difference, and their product may exceed their fum by 20. Anf. 8 and 4. 5. A boy is offered 10 apples for a penny, and 25 pears for 2d: He agreed to buy 100 apples and pears together for 91d. Required the number of each Anf. 75 apples and 25 pears. 6. Required two numbers whofe fum is 108, and proportion as 5 to 4: Anf. 60 and 48, LITERAL EQUATIONS. In literal equations unknown quantities are reprefented by x, y, z, as before; known quantities by a, b, c, d, &c. The rules for tranfpofing and exterminating quantities are the fame as above. When the value of the unknown quantity is thus discovered, we obtain a general theorem, which will ferve for the folution of all queftions under the like conditions. EXAMPLE I. Required a theorem for determining two numbers whofe fum, s, and difference, d, are given. Let x be the greater and y the lefs. In words. From half the fum fubtract half the differ ence, the remainder will be the lefs. To half the fum add half the difference, the fum will be the greater. Ex. 2. The powers or forces of three different agents being given, to find a general theorem for determining the time in which they would, all three together, produce a given effect. Three day-labourers, A, B, and C, have undertaken a piece of work, which A could perform in a days, B in b days, and C in days. In what time will they perform it, if all the three work together? The rule obtained may be translated thus. Divide the product of the three given times by the fum of the products of each two taken separately. 3. Required a theorem for determining two numbers, whose fum (s) and fum of their fquares (q) are given. Anf. |