ber, and all those containing known quantities, into the other. If any commodity to be weighed, as coffee, for instance, were mixed with the weights of a grocer, although his scales might be perfectly balanced, it is evident that he could not determine how many pounds of coffee they contained. His only way would be, to collect all the coffee into one of the scales, and all the weights into the other. Now, an equation is not unlike the grocer's scales: the commodity to be weighed is the unknown, and the leaden weights, the known quantity. If 60 x be subtracted from each side of the equation, we shall obtain B. 75 x 60 x 60 x + 300 If we cancel + 60 x and member, the equation will be 60 x, in the second Compare equations A and c, where a term contain ng the unknown quantity, 60 x, is removed from the second to the first member; the sign + being chang ed to -. 9. The water had been flowing from a full cistern 6 hours, at the rate of 12 gallons an hour, when a pipe was conducted into it, which restored 21 gallons an hour. In how many hours was the cistern full again? 10. A man, being asked his age, answered, that if his father's age, which was 52 years, were added to three times his own, the sum would be five times his age. How old was he? *11. A father is three times as old as his son; but in 20 years he will be only twice as old. What is the age of each? 12. When a boy would buy a certain number of oranges at 6 cents apiece, he found they would come to 12 cents more than he had; he, therefore, bought the same number at 5 cents each, and had 6 cents left. How many oranges did he buy? * 13. Divide 64 into two such parts, that five times the first shall be equal to three times the second. And 5 x = (64x) x 3, by the question; A. or 5 x 192 - 3 x. It appears that 5 x is not equal to 192 by 3 x. If we add 3 x to each side of the equation, we have B. 5 x + 3 x = 192 3x + 3 x. And if we cancel the 3 x and 3 x, in the second member, the equation becomes c. 5 x + 3 x = 192; or 8 x = 192, by addition. and 64 x = 40, the second part. Observe, in the equations A and c, that- 3 x is removed from the second member to the first, the sign being changed to +. From the preceding examples and observations we derive the following principle: Any term may be transposed from one member of an equation to the other, if the sign be changed. 14. A pole is 4 feet in the ground, of its whole length under water, and above water. Required its length. 15. The head of a fish weighs 8 lbs.; his tail weighs as much as his head and half his body, and his body What is the weighs as much as weight of the fish? his head and tail. 16. A man, being asked his own and his wife's age, said, that his youngest child was 4 years old; that the age of his wife was twice the age of the child and of his own age; and that his own age was equal to the united ages of his wife and child. How old were they? * 17. A merchant has wines at 9 shillings, and at 13 shillings, per gallon; and he would make a mixture of 100 gallons, that shall be worth 12 shillings per gallon. How many gallons of each must he take? *18. How many gallons of wine, at 9 shillings a gallon, must be mixed with 20 gallons at 13 shillings, that the mixture may be worth 10 shillings a gallon? 19. A merchant, having mixed 10 gallons of wine, at 8 shillings a gallon, with 25 gallons at 10 shillings, wishes to add as much wine at 15 shillings as shall make the whole mixture worth 2 dollars a gallon. How many gallons must he take? 20. How many gallons of water must be mixed with 35 gallons of wine, at 9 shillings, and 45 gallons at 13 shillings, a gallon, that the whole mixture may be worth 10 shillings a gallon? 21. A clerk spends of his salary for board and lodging, of the remainder in clothes, and saves $150 per annum. What is his salary? 22. What is that number, and of which is 35 more than its sixth part? * 23. Two men, A and B, have each $80. A spends $5 more than twice as much as B, and has then half as much as B, wanting $13. How much did each spend ? 24. Divide 84 into two such parts, that if of the less be subtracted from the greater, and of the greater be subtracted from the less, the remainders shall be equal. tracted from the less; and, by the terms of the ques tion, their values are equal. 84 Observe that the sign-, before the fraction **, affects the value of the whole fraction, and not any particular term of it. We are not required to subtract half of 84 from the preceding term, but only half of the excess of 84 above x. Therefore, if the signis prefixed to 84, when the denominator is removed, the sign - before x must be changed to +; otherwise, too much, by the value of x, would be taken from the first term of the equation. Were the sign + before the fraction, it would not be ne 2 cessary to change the sign before x from to +. [See Chap. VI. Sec. III.] 25. Divide 72 into two unequal numbers, so that, if of the less be subtracted from the greater, the remainder may be equal to of the number which remains, after the excess of the greater above the less is subtracted from the less. 26. Two clerks, A and B, sent ventures in different vessels, A's being worth only gained and B lost $23; then tracted from A's, would leave venture. How much did each send? as much as B's. A of B's returns, subof the value of A's 27. A gentleman bought a watch and chain for $160. If of the price of the watch be subtracted from 6 times the price of the chain, and of the price of the chain be subtracted from twice the price of the watch, the remainders will be equal. What was the price of each? 28. A person, being asked the time of day, replied, "If to the time from noon be added its 1, 3, 1 and 1, |