Example III. A person spent $56.40 in buying geese and ducks; if each goose cost 7 dimes, and each duck 3 dimes, and if the total number of birds bought was 108, how many of each did he buy? In questions of this kind it is of essential importance to have all quantities expressed in the same denomination; in the present instance it will be convenient to express the money in dimes. Let x be the number of geese, then 108-x is the number of ducks. Since each goose costs 7 dimes, x geese cost 7x dimes. And since each duck costs 3 dimes, 108-x ducks cost 3(108-x) dimes. Therefore the amount spent is 7x+3(108-x) dimes. But the question states that the amount is also $56.40, that is 564 dimes. and Hence 7x+3(108-x)=564; 4x=240, .. x=60, the number of geese, 108-x=48, the number of ducks. Note. In all these examples it should be noticed that the unknown quantity x represents a number of dollars, ducks, years, etc.; and the student must be careful to avoid beginning a solution with a supposition of the kind, "let x=A's share" or let x=the ducks," or any statement so vague and inexact. It will sometimes be found easier not to put x equal to the quantity directly required, but to some other quantity involved in the question; by this means the equation is often simplified. Example IV. A woman spends $1 in buying eggs, and finds that 9 of them cost as much over 25 cents as 16 cost under 75 cents; how many eggs did she buy? Let x be the price of an egg in cents; then 9 eggs cost 9x cents, and 16 eggs cost 16x cents; ... 9x-25-75-16x, 25x=100; .. x=4. Thus the price of an egg is 4 cents, and the number of eggs =100 4 25. 10. A sum of $30 is divided between 50 men and women, the men each receiving 75 cents, and the women 50 cents; find the number of each sex. 11. The price of 13 yards of cloth is as much less than $10 as the price of 27 yards exceeds $20; find the price per yard. 12. A hundredweight of tea, worth $68, is made up of two sorts, part worth 80 cents a pound and the rest worth 50 cents a pound; how much is there of each sort? 13. A man is hired for 60 days on condition that for each day he works he shall receive $2, but for each day that he is idle he shall pay $1 for his board: at the end he received $90; how many days had he worked? 14. A sum of $6 is made up of 30 coins, which are either quarters or dimes; how many are there of each ? 15. A sum of $11.45 was paid in half-dollars, quarters, and dimes; the number of half-dollars used was four times the number of quarters and ten times the number of dimes; how many were there of each ? 16. A person buys coffee and tea at 40 cents and 80 cents a pound respectively; he spends $15.10, and in all gets 24 lbs. ; how much of each did he buy? 17. A man sold a horse for a sum of money which was greater by $68 than half the price he paid for it, and gained thereby $18; what did he pay for the horse ? There are 36 18. Two boys have 240 marbles between them; one arranges his in heaps of 6 each, the other in heaps of 9 each. heaps altogether; how many marbles has each ? 19. A man's age is four times the combined ages of his two sons, one of whom is three times as old as the other; in 24 years their combined ages will be 12 years less than their father's age; find their respective ages. 20. A sum of money is divided between three persons, A, B, and C, in such a way that A and B have $42 between them, B and C have $45, and C and A have $53; what is the share of each? 21. A person bought a number of oranges for $3, and finds that 12 of them cost as much over 24 cents as 16 of them cost under 60 how many oranges were bought? cents; 22. By buying eggs at 15 for a quarter and selling them at a dozen for 15 cents a man lost $1.50; find the number of eggs. 23. I bought a certain number of apples at four for a cent, and three-fifths of that number at three for a cent; by selling them at sixteen for five cents I gained 4 cents; how many apples did I buy? 24. If 8 lbs. of tea and 24 lbs. of sugar cost $7.20, and if 3 lbs. of tea cost as much as 45 lbs. of sugar, find the price of each per pound. 25. Four dozen of port and three dozen of sherry cost $89; if a bottle of port costs 25 cents more than a bottle of sherry, find the price of each per dozen. 26. A man sells 50 acres more than the fourth part of his farm and has remaining 10 acres less than the third; find the number of acres in the farm. 27. Find a number such that if we divide it by 10 and then divide 10 by the number and add the quotients, we obtain a result which is equal to the quotient of the number increased by 20 when divided by 10. 28. A sum of money is divided between three persons, A, B, and C, in such a way that A receives $10 more than one-half of the entire amount, B receives $10 more than one-third, and C the remainder, which is $10; find the amounts received by A and B. 29. The difference between two numbers is 15, and the quotient arising from dividing the greater by the less is 4; find the numbers. 30. A person in buying silk found that if he should pay $3.50 per yard he would lack $15 of having money enough to pay for it; he therefore purchased an inferior quality at $2.50 per yard and had $25 left; how many yards did he buy? 31. Find two numbers which are to each other as 2 to 3, and whose sum is 100. 32. A man's age is twice the combined ages of his three sons, the eldest of whom is 3 times as old as the youngest and 3 times as old as the second son; in 10 years their combined ages will be 4 years less than their father's age; find their respective ages. 33. The sum of $34.50 was given to some men, women, and children, each man receiving $2, each woman $1, and each child 50 cents. The number of men was 4 less than twice the number of women, and the number of children was 1 more than twice the number of women; find the total number of persons. 34. A man bought a number of apples at the rate of 5 for 3 cents. He sold four-fifths of them at 4 for 3 cents and the remainder at 2 for a cent, gaining 10 cents; how many did he buy? 35. A farm of 350 acres was owned by four men, A, B, C, and D. B owns five-sixths as much as A, C four-fifths as much as B, and D one-sixth as much as A, B, and C together; find the number of acres owned by each. CHAPTER XII. ELEMENTARY FRACTIONS. Highest Common Factor of Simple Expressions. 88. DEFINITION. The highest common factor of two or more algebraical expressions is the expression of highest dimensions [Art. 68] which divides each of them without remainder. The abbreviation H.C.F. is sometimes used instead of the words highest common factor. 89. In the case of simple expressions the highest common factor can be written down by inspection. Example 1. The highest common factor of a1, a3, a2, a® is a2. Example 2. The highest common factor of ab1, a2b3c2, ab7c is a2b; for a is the highest power of a that will divide a3, a2, a*; b4 is the highest power of b that will divide b1, b3, b7; and c is not a common factor. 90. If the expressions have numerical coefficients, find by Arithmetic their greatest common measure, and prefix it as a coefficient to the algebraical highest common factor. Example. The highest common factor of 21a13y, 35a2x1y, 28a3xy is 7a xy; for it consists of the product of (1) the greatest common measure of the numerical coefficients; Lowest Common Multiple of Simple Expressions. 91. DEFINITION. The lowest common multiple of two or more algebraical expressions is the expression of lowest dimensions which is divisible by each of them without remainder. The abbreviation L.C.M. is sometimes used instead of the words lowest common multiple. 92. In the case of simple expressions the lowest common multiple can be written down by inspection. Example 1. The lowest common multiple of a1, a3, a2, a6 is ao. Example 2. The lowest common multiple of a3b4, ab5, a2b7 is a3b7; for a3 is the lowest power of a that is divisible by each of the quantities a3, a, a2; and b7 is the lowest power of b that is divisible by each of the quantities b1, b3, b7. 93. If the expressions have numerical coefficients, find by Arithmetic their least common multiple, and prefix it as a coefficient to the algebraical lowest common multiple. Example. The lowest common multiple of 21a*x3y, 35a2x*y, 28a3xy is 420atxty; for it consists of the product of (1) the least common multiple of the numerical coefficients; Find both the highest common factor and the lowest common multiple of 17. 2ab2, 3a2b3, 4a1b. 18. 15x3y2, 5x2yz5. 20. 57axy, 76xy2z7. 22. 51m3p2, pn, 34mnp1. 19. 2a+, 8a2b3c7. 21. 32a4b3c, 48a7bc5. 23. 49a4, 56b1c, 21ac3. 24. 66a2bcx, 55ab'xyz, 121x3yz7. |