Then x sheep, at $7 apiece, will come to 7 x dolls., and 3 x calves, at $5 apiece, will come to 5 times 3 x dolls., that is, 15x dolls. These added together must make the whole price. 72 + 15 x=374 x= 17 = number of sheep. calves. 3x = The learner must have remarked by this time, that when a question is proposed, the first thing to be done, is to find, by means of the unknown quantity, an expression which shall be equal to a given quantity, and then from that, by arithmetical operations, to deduce the value of the unknown quantity. This expression of equality between two quantities, is called an equation. In the last example, 7 : + 15 x=374 is an equation. The quantity or quantities on the left of the sign= are called the first inember, those on the right, the second member of the equation. (70+ 15 x) is the first member of the above equation, and 374 is the second member. Quantities connected by the signs + and - are called terms. 7 x and 15 x are terms in the above equation. The figure written before a letter showing how many times the letter is to be taken, is called the coefficient of that letter. In the quantities 7 x, 15 x, 22 x'; 7, 15, 22, are coefficients of x. The process of forming an equation by the conditions of a question, is called putting the question into an equation. The process by which the value of the unknown quantity is found, after the question is put into an equation, is called solving or reducing the equation. No rules can be given for putting questions into equations ; this must be learned by practice; but rules may be found for solving most of the equations that ever occur. After the preceding questions were put into equation, the first thing was to reduce all the terms containing the unknown quantity to one term, which was done by adding the coefficients. As 7.30 + 15 x are 22 x. Then, since 22 x = 374, 1 x must be equal to z's of 374. That is, When the unknown quantity in one member is reduced to one term, and stands equal to a known quantity in the other, its value is found by dividing the known quantity by the coefficient of the un known quantity. 3 ore 10. A man bought some oranges, some lemons, and some pears, for 156 cents; the oranges at 6 cents each, the lemons at 4 cents, and the pears at 3 cents ; there was an equal number of each sort. Required the number of each. 12 Bro 11. In fencing the side of a field, the length of which was 450 yards, two workmen were employed; one fenced 9 yards, and the other 6 yards per day. How many days did they work ? 12. Three men built 780 rods of fence ; the first built 9 rods per day, the second 7, and the third 5; the second worked three times as many days as the first, and the third, twice as many days as the second. How many days did each work: 3:13. 2ne. 34. 320.58 13. A man bought some oxen, some cows, and some calves for $348 ; the oxen at $39 each, the cows at $18, and the calves at $4. There were three times as many cows as oxen, and twice as many calves as cows. How many were there of each sort ? 14, A merchant bought a quantity of flour for $132 ; for one half of it he gave $5 per barrel, and for the other half $7. How many barrels were there in the whole ? nets Biserna Let x denote one half the number of barrels. 15. From two towns, which are 187 miles apart, two travellers set out at the same time with an intention of meeting ; one of them travels at the rate of 8, the other of 9 miles each day. In how many days will they meet ? & Ifta y., virsunen II. 1. A cask of wine was sold for $45, which was only of what it cost. Required the cost. Let x denote the cost. 4 preferable. 3 x :45 Ans. $60 32 = 15 1 x If of x comes to 45, then must come to $ of 45, or 4 15, and x will be 4 times 15, or 60. A better method. 3 x = 45 x = 60 31 Observe, that is the same as į of 3x. Now if i of 3 x 4 is 45, 3 x itself must be 4 times 45, or 180; 3 x being 180, 4 must be į of 180, which is 60. 2. A man, being asked his age, answered, that if its half and its third were added to it the sum would be 88. What was his age? Let x denote his age; then, x + to =88 2 3 Reducing the terms to a com- 6 x 3 x + + =88 mon denominator, 6 6 6 Adding them together, =88 6 of 11 x being 88,11 x will be 6 times 98,11 x = 528 Dividing by 11, X = 45 2x } 11 x Ans. 48 years. 3. If of a hogshead of wine cost $65; what will a hogshead cost at that rate ? 4. There is a pole and į under water, and 5 feet out of water ; what is the length of the pole? Let « denote the whole length. Then + +5 must be 2 3 equal to the whole length. Hence, + +5 2. 3 Reducing to a common denominator, + + 5 6 6 6 x 3 x 2 x 6 x 5 x +5 6 Adding together, 6 Since the two members are equal, if 5.2 be subtracted from both, they will still be equal ; hence, 6 x = 5 2 x 6 and x = 30 Ans. 30 feet. Proof. One half of 30 is 15, and one third of thirty is 10. Now 30 = 15 + 10 + 5. There is another mode of reducing the above equation which in most cases is to be preferred. It is the same in principle. If both members of an equation be multiplied by the same number, they evidently will still be equal. In the equation, *. 2 First multiply both members by 2, the denominator of one or the fractions, and it becomes, 2 x= x + +10. 3 Next multiply both members by 3, the denominator of the other fraction, and it becomes, 6 x = 3 x + 2 x + 30 or 6 x = 5x + 30. Subtracting 5 x from both members, X = 30 as before. 5. In an orchard of fruit trees į of them bear apples, i of them pears, į of them plums, 7 bear peaches, and 3 bear cherries ; these are all the trees in the orchard. How many are there? 80 tons 6. A farmer, being asked how many sheep he had, answered, he had them in four pastures; in the first he had } of them, in the second į, in the third b, and in the fourth he had 24 sheep. How many had he in the whole ? 96 Ano. 7. A person having spent į and } of his money, had $263 left. How much money had he at first ? $759.998 hrswei 8. A man driving his geese to market, was met by another, who said good morrow, master, with your hundred geese ; said he, I have not a hundred, but if I had as many more, and half of my age š part added as many more, and two geese and a half, I should nave a hundred. How many had he ? 85 otrs. 9. A and B having found a bag of money, disputed about the division of it. A said that į and į and of the money made $130, and if B could tell how much money there was, he should have it all, otherwise none of it. How much money was there in the bag ? $62,40 Shuan 10. Upon measuring the corn produced in a field, being 96 bushels, it appeared that it had yielded only one third part more than was sown. How much was sown? 72 borkils Answer 11. A man sold 96 loads of hay to two persons ; to the first ž, and to the second of what his stack contained. How many loads did the stack contain at first ? 102,99 + su 12. A and B talking of their ages, A says to B if }, j, and be added to my age, and 2 years more, the sum will be twice my age. What was his age ? 13. What sum of money is that whose }, i, and together amount to £9 ? 14. The account of a certain school is as follows: jy of the boys learn geometry, learn grammar, learn arithmetic, an learn spelling, and 9 learn to read. What is the number of scholars in the school ? 15. There is a fish whose head weighs 9 lb. his tail weighs as much as his head and half his body, and his body weighs as much as his head and tail both. What is the weight of the fish ? Represent the weight of the body by x. 16. There is a fish whose head is 4 inches long, the tail is twice the length of the head, added to of the length of the body, and the body is as long as the head and tail both. What is the whole length of the fish ? 17. A and B talking of their ages, A says to B, your age is twice and three fifths of my age, and the sum of our ages is 54. What is the age of each? 18. A man divided $40 between two persons ; to the first he gave a certain sum, and to the second only as much. How much did he give to each ? 3 x Let æ denote the share of the first, s *will denote the share |