6 + Let weight of the cover. Then 12+x= weight of the first cup covered. And = weight of the second cup, &c. 2 8. Some persons agreed to give 6d. each to a waterman for carrying them from London to Gravesend ; but with this condition, that for every other person taken in by the way, three pence should be abated in their joint fare. Now the waterman took in three more than a fourth part of the number of the first passengers, in consideration of which he took of 'them but 5d. each. How many persons were there at first? Let x = the nurnber of passengers at first. Then +3=the number taken in, &c. 4 9. Four places are situated in the order of the four letters, A, B, C, D. The distance from A to D is 134 miles, the distance from A to B is to the distance from C to D, as 3 to 2, and one fourth of the distance from A to B, added to half the distance from C to D, is three times the distance from B to C. What are the respective distances? 10. A field of wheat and oats, which contained 20 acres, was put out to a laborer to reap for $ 20 ; the wheat at $ 1.20 and the oats $ 0.95 per acre. Now the laborer falling ill reaped only the wheat. How much money ought he to receive according to the bargain? 11. Three men, A, B, and C, entered into partnership; A paid in as much as B and one third of C; B paid as much as C and one third of A; and C paid in $ 10 and one third of A. What did each in? pay Let the sum A contributed. Then +10= с 3. and + 10 + ing B &c. 12. A gentleman gave in charity £ 46 ; a part of it in equal portions to 5 poor men, and the rest in equal portions to 7 poor Now the share of a man and a woman together amounted to £8. What was given to the men, and what to the women? women. Let the sum a man received. Then x = the sum a woman received, &c. 13. Suppose that for every 10 sheep a farmer kept, he should plough an acre of land, and should be allowed an acre of pasture for every 4 sheep. How many sheep may that person keep who farms 700 acres? Let x = the whole number of sheep. The number of acres ploughed will be o of the number of sheep ; and the number of acres of the pasture will be of the number of sheep; both these added together must be the whole number of acres, &c. 14. A, B, and C make a joint stock; A puts în $ 70 more than B, and $ 90 less than C; and the sum of the shares of A and B is of the sum of the shares of B and C. What did each put in? Let x= the sum that B put in, &c. 15. Divide the number 85 into two such parts that if the greater be increased by 7 and the less be diminished by 8, they will be to each other in the proportion of 5 to 2. 16. It is required to divide the number 67 into two such parts that the difference between the greater and 75 ray be to the excess of the less over 12 in the proportion of 8 to 3. 17. A man bought 12 lemons and a pound of sugar for 56 cents, afterwards he bought 18 lemons and a pound of sugar at the same rate for 74 cents. What was the price of the sugar, and of a lemon? Let the price of the sugar. Then 56 x = the price of 12 lemons. 56 And = the price of 1 lemon. 12 = the price of a lemon. 74 Hence &c. 12 18 18. A man bought 5 oranges and 7 lemons for 58 cents ; af. terwards he bought 13 oranges and 6 lemons at the same rate for 102 cents. What was the price of an orange, and of a lemon? eighths the value of the first. Required the value of each horse. and y Equations with two Unknown Quantities. VIII. Many examples involve two or more unknown quantities. In fact, many of the examples already given involve several unknown quantities, but they were such, that they could all be derived from one. When it is necessary to use two unknown quantities in the solution, the question must always contain two conditions, from which two equations may be derived. When this is not the case the question cannot be solved. 1. A boy. bought 2 apples and 3 oranges for 13 cents; he afterwards bought, at the same rate, 3 apples and 5 oranges for 21 cents. How much were the apples and oranges apiece? Let x = the price of an orange, = the price of an apple. 3 x + 2 y 13, 2. 5x + 3y 21. Multiply the first equation by 3, and the second by 2, 3. 9 x + 6 y 10 x + 6 y 42. Subtract the first from the second, because the y's being alike in each, the difference between the numbers 39 and 42 must depend upon the x's. 5. 3 cents, the price of an orange. Putting this value of x into the first equation, 6. 9 + 2y = 13 7. y= 2 cents, the price of an apple. Proof. 2 apples at 2 cents each come to 4 cents, and 3 oranges at 3 cents come to 9 cents. 9 + 4 = 13. So 3 apples and 5 oranges come to 21 cents. Note. In this example I observed, that the coefficient of y in the first equation is 2, and in the second, the coefficient of y is 3. I multiplied the whole of the first equation by 3, and the whole of the second by 2; this formed two new equations in which the coefficients of y are alike. If the first equation had been multiplied by 5 and the second by 3, the coefficients of a would have been alike, and x instead of y would have been = 39 4. made to disappear by subtraction, and the same result would have been finally obtained. It is evident, that the coefficients of either of the unknown quantities may always be rendered alike in the two equations, by multiplying the first equation by the coefficient which the quantity that you wish to make disappear has in the second equation; and the second equation by the coefficient which the same quantity has in the first equation. They may be rendered alike more easily, when they have a common multiple less than their product. 2. A person has two horses, and a saddle which of itself is worth £ 10; if the first horse be saddled, he will be worth, as much as the other, but if the second horse he saddled, he will be worth as much as the first. What is the value of each horse? A question similar to this has already been solved with one unknown quantity, but it will be more easily solved by using two of them. Let x = the value of the first horse, and y= the value of the second horse. 1. By the conditions, =x + 10 7 83 = 10 5 y = 70 4. y = 5 Multiply the 3d by 7, and the 4th by 5, to free them from denominators; 5. 73 + 6 y 6. 8x Multiply the 5th by 5 and the 6th by 6, in order to make the coefficients of y alike in the two; 7. 35 x +- 30 Y= 350 8. Add together 7th and 8th, 48 x 35 x + 30 y = 350 + 300 10. Uniting terms, 650 11. X = 50 48 X 30 y = 300 9. 30 Y 13 x 6 y — 350 Putting 50, the value of x, into the 5th, 12. 70 13 420 14. y 70 Ans. The first is worth £ 50, and the second £70. Note. In this example the 30 y in the 7th equation had the sign +, and in the 8th the sign before it, hence it was necessary to add the two equations together in order to make the Y disappear, or as it is sometimes called, to eliminate y. 3. A market-woman sells to one person, 3 quinces and 4 melons for 25 cents, and to another, 4 quinces and 2 melons, at the same rate, for 20 cents. How much are the quinces and melons apiece? 4. In the market I find I can buy 5 bushels of barley and 6 bushels of oats for 27s., and of the same grain 4 bushels of barley and 3 bushels of oats for 18s. What is the price of each per bushel ? 5. My shoemaker sends me a bill of $ 12 for 1 pair of boots and 3 pair of shoes. Some months afterwards he sends me a bill of $ 20 for 3 pair of boots and 1 pair of shoes. the boots and shoes a pair? 6. Three yards of broadcloth and 4 yards of taffeta cost 57s. and at the same rate 5 yards of broadcloth and 2 yards of taffeta cost 81s. What is the price of a yard of each? 7. A man employs 4 men and 8 boy's to labor one day, and pays them 40s.; the next day he hires, at the saine wages, 7 men and 6 boys, and pays them 50s. What are the daily wages of each? 8. A vintner sold at one time 20 dozen of port wine and 30 doz. of sherry, and for the whole received £ 120; and at another time, sold 30 doz. of port and 25 doz. of sherry at the same prices as before, and for the whole received £ 140. the price of a dozen of each sort of wine? 9. A gentleman bas two horses and one chaise. The first horše is worth $ 180. If the first horse be harnessed to the chaise, they will together be worth twice as much as the second horse, but if the second be harnessed, the horse and chaise will be worth twice and one half the value of the first. What is the value of the second horse, and of the chaise? What are What was |