5 x 6 To solve the equation another way, let us resume =15. We may first multiply both members by 6 and as a fraction is multiplied by dividing its denominato we have The latter mode of solution is generally to be preferred. 2. In a certain village, of the people are English, Irish, French, and the remainder, 50 in number, are Germans. What is the whole population of the village? Let x be the number of inhabitants. Then, 2x = x + 1/3/2 + +100; multiply this by 2, 4 4x=2x+x++200; multiply this by 2, 2 8x=4x+2x+x+400; reduce, 8x=7x+400; transpose 7x, 8x-7x=400; reduce, x=400, Ans. The process, in this question, would have been shorter, if we had multiplied the first equation by 8 at once, since we should then have obtained 8x 8x= + + +400; that is. 2 8x 4 8 x 8 8x=4x+2x+x+400; but it is best to remove one denominator at a time, until the learner has become quite familiar with the process. 3. The sum of 3, 2, §, and of a certain number is 150. Required the number. Let x represent the number. Then, We are now to multiply so as to remove, successively, the denominators. But it is sometimes convenient to represent the multiplication of certain numbers in an equation. Multiply the equation by 3. 2 8x+9x+10x+21=3.4. 150; multiply by 2, 16x+18x+20 x + 21 x = 3.4.2.150; reduce, 75 x 3.4.2.150. Divid Now, since dividing one factor divides a product, in the second member we have only to divide 150 by 75. ing both members, we have x=3.4.2.2=48, Ans. 4. Four fifths of a number, increased by 2, is the same of the number diminished by 4. Required the as number. 5. The cost of of an acre of land being $35, what will an acre cost at the same rate? 6. Two numbers are to each other in the ratio of 9 to 7, and their sum is 112. Required the numbers. Remark. The ratio of 9 to 7 means that the greater is of the less, or that the less is of the greater. 7. A man's age is to that of his wife as 3 to 2, and his age exceeds hers by 10 years. Required their ages. 8. A grocer bought some tea, at $0.50 per pound, and as much coffee, at $0.12 per pound, and paid for both $21.92. How many pounds of each did he buy? 9. In a mixture of gold and copper, 1 ounce more than of the whole was gold, and of the whole was copper. Required the weight of the whole, and that of each ingredient. 10. In a mixture of copper, tin, and zinc, 5 oz. more than of the whole was copper, 2 oz. less than of the whole was tin, and 1 oz. less than of the whole was zinc. Required the weight of the entire mixture, and that of each ingredient. 11. A man had passed of his life in Germany, 12 years more than of it in England, and 3 years less than of it in France. Required his age. 12. A trader bought a quantity of oats, at 2 s. a bushel, and twice as much corn, at 5 s. a bushel. He afterwards sold, at the same prices he gave, of his oats and of his corn for 79 s. How much of each did he buy? 13. If I multiply a certain number by 5, divide the product by 7, and diminish the quotient by 3, I shall obRequired that number. tain the original number. 14. A man's age, at the time of his marriage, was to that of his wife as 3 to 2; but after they had been married 10 years,. 3 times his age was equal to 4 times hers. Required their ages at the time of marriage. 15. When a barrel of apples was worth as much as a barrel of flour, a farmer gave 6 barrels of apples and $3 in money for 3 barrels of flour. Required the estimated price of each, per barrel. 16. Two boys set out, at the same time, and from the same place, to run to a certain goal. One could run only as fast as the other, and was 5 rods short of the goal at the time his companion had reached it. Required the distance which they proposed to run. 17. What number is that, of which, increased by 6, will be the same as 11 of it diminished by 3? SECTION V. EQUATIONS CONTAINING FRACTIONAL PARTS OF QUANTITIES CONSISTING OF SEVERAL TERMS. ART. 24. 1. A is 20 years older than B, and B's age is equal to 7 of A's. Required the age of each. Let x represent B's age in years; then x+20 will represent A's. One eleventh of A's is x+20 and is 7 times 11 11x=7x+140; transpose 7x, 11x-7x=140; reduce, 4x=140; .. x = 35 years, B's age, x+20=55 years, A's age; Ans. 2. A had twice as much money as B. A gained $500,and B $100; then of B's money was equal to of A's. How much had each at first? х Let x be the number of dollars B had; then A would have 2 x dollars. After gaining, B would have x+100, and A 2x+500; then of x+100 must be equal to of 2x+500. Hence, 10x+1000=8x+2000; transpose, 3. A man paid for a horse and chaise $300, and of the price of the horse was the same as the price of the chaise increased by $28. Required the price of each. Suppose the horse cost x dollars; then 300 the price of the chaise. Hence, x will be +28. Multiply by 5, +140; multiply by 7, 25x+980; transpose 25 x, } Ans. and of A's age is the 28x+25x=7500+980; reduce, 300 53x=8480; .. x= $160, price of the horse; x= $140, price of the chaise ;) 4. A is 15 years older than B, same as of B's. Required their ages. 5. What number is that to which if 6 be added, and from which if 5 be subtracted, 4 of the sum shall exceed of the difference by 5? 6. What number is that from which if 10 be subtracted, and to which if 11 be added, of the difference shall be the same as of the sum? 7. A drover, having a certain number of oxen, and twice as many sheep, bought 5 oxen and sold 2 sheep, after which he found that of his number of oxen was equal to of his number of sheep. How many of each had he at first? 8. A man agreed to work a year for 50 bushels of corn and $75 in money; but, on account of sickness, he wrought only 10 months, and received the 50 bushels of corn and $54 in money. What was the estimated price of the corn per bushel? 9. A man's age, 5 years before his marriage, was of his age 5 years after his marriage. Required his age at the time he was married. |