SECTION VIII. Questions for Practice. 1. A vintner sold to one man 16 dozen of sherry wine and 19 dozen of port, for $382; and to another man, 24 dozen of sherry and 17 dozen of port, for $458; the prices being the same to both. What was the price of each kind of wine? 2. Two sisters having bought some lace, says Mary to Ann, "Give me a yard of yours, and I shall have as much as you." Ann replied, "If you will give me a yard of yours, I shall have twice as much as you." How many yards had each? 3. Required two such numbers, that, if of the first be added to g of the second, the sum shall be 66; but if of the first be added to of the second, their sum shall be 60. 4. Two persons, A and B, talking of their ages, says A to B, "12 years ago I was twice as old as you; and in 12 years my age will be to yours as 3 to 2." What is the age of each? 5. Three young men, A, B and C, speaking of their money, A says to B and C, "If each of you will give me $5, I shall have just half as much as both of you will have left." B says to A and C, "If each of you will give me $5, I shall have just as much as both of you will have left." C says to A and B, "If each of you will give me $5, I shall have twice as much as both of you will have left." How much has each? 6. If you add 2 to the numerator of a certain frac tion, its value becomes; but if you add 2 to its denominator, the fraction will be equal to . What is the fraction? 7. Three factory girls, A, B and C, weave 62 yards of cloth in a given time. A weaves 4 times as many yards as C, added to twice the yards woven by B; and twice A's part, added to 3 times B's, is equal to 17 times C's. How many yards does each weave? 8. Find three such numbers, that of the first, of the second, and of the third, shall be 50; of the first, and twice the difference of the third and second, shall be 40; and 10 less than of the sum of all the numbers shall be 30. 9. Two men, A and B, are employed to set up 220 rods of fence. If A work 9 days and B 8, the fence will not be completed by 2 rods; but if A work 8 days and B 9, they will be able to finish the fence. and 4 rods more. How many rods can each build in a day? 10. A farmer employed three laborers, A, B and C, to work at different times. At one time, A and B together earned $56 in 8 weeks; at another time, A and C earned $54 in 9 weeks; and, at another, B and C earned $50 in 10 weeks. What did he pay each man for a week's work? *11. If you divide the greater of two numbers by the less, the quotient will be 7; and the amount of the numbers is 1008. Required the numbers. 12. A farmer, to pay a debt to a trader, agrees to fill a certain chest with a mixture of corn and oats the corn being 5 shillings a bushel, and the oats 3 shillings. If he delivers 7 bushels of oats, and the balance of the debt in corn, the chest will not be full by 2 bushels; but if he delivers 6 bushels of corn, and then fills the chest with oats, 6 shillings of the debt will remain unpaid. What is the amount of the debt? How many bushels does the chest hold? How many bushels of each kind of grain must he deliver? 13. A boy bought, at one time, 5 apples, 6 pears and 4 peaches, for 44 cents; at another time, 7 pears, 5 peaches and 3 oranges, for 56 cents; at another, 8 apples, 12 peaches and 5 oranges, for 89 cents; and at another, 10 apples, 3 pears and 9 oranges, for 74 cents. What did he pay for each kind of fruit? 14. Three persons, A, B and C, talking of their money, A says to B and C, "Give me of your money, and I shall have $85." B says to A and C, "Give me of your money, and I shall have $85." C says to A and B, "Give me of your money, and I shall have $85." What has each? 15. A gentleman gave $4350 for a house-lot, the land being valued at $2 a foot. If it had been 6 feet wider, it would have cost $5394. What were the length and breadth of the lot? 16. I have a certain number of cents in each hand. If I put 10 out of my left hand into my right, there will be twice as many in my right as remain in my left; but if I put 10 out of my right hand into my left, there will be three times as many in my left hand as remain in my right. How many cents have I in each hand? 17. Three boys, A, B and C, play with marbles. First, A loses to B and C as many as each of them has. Next, B loses to A and C as many as each of them now has. Lastly, C loses to A and B as many as each of them has 16 marbles. now has. After all, each of them How many had each at first? 18. The fore-wheel of a coach makes 5 revolutions while the hind-wheel is making 4; but if the circumference of each were one yard greater, their revolutions would be to each other as 6 to 5. What is the circumference of each in feet? 19. Three sportsmen, before they separated in the fields, agreed to make an equal division of whatever game they might take. During the day, they bagged 96 birds; and, in order to divide them equally, A gave B and C as many as they took; next, B. gave A and C as many as they then had; and, finally, C shared in the same manner with A and B, when they all had the same number. How many birds were taken by each? 20. A merchant sent ventures to sea for his three children, A, B and C, upon the condition that they should make an equal division of the proceeds. When the ship arrived, A gave to B and C a sum equal to of their ventures. In like manner, B gave to A and Ca sum equal to of what they then had. found, if he gave to A and B a sum equal to they already had, that they would each have And C of what 64 dol lars. To how much did each of the ventures amount? CHAPTER IX. EXERCISES IN GENERALIZATION. We have already seen that Algebra enables us to solve various questions with facility, and to obtain their true answers in numbers. But this is not its only, nor even its highest office. It also enables us to deduce general truths from particular instances, and thus to form rules for conducting numerical calculations. This use of Algebra will be illustrated by a few miscellaneous examples. 1. Add together the sum and the difference of 294 and 175. 294 + 175 = 469 Their sum, 175 = 119 ANS. 588. To generalize this question, let a represent the greater quantity, and b, the less. Then a + b will be their sum, and ab, their difference; which we are required to add together. |