21. A farmer having 600 sheep, divide flocks of 75 each: how many flocks had he 294. When the product of three numbers the numbers are given, to find the other number. Divide the given product by the product of the tu numbers, and the quotient will be the other number. 270 22. There are three numbers whose product is 60; one of them is 3, and another 5: it is required to find the other? Solution. 5×3=15; and 60÷15=4, the number required. PROOF. 5×3×4=60, the given product. 23. The product of A, B, and C's ages is 210 years: the age of A is 5 years, and that of B, is 6 years: what is the age of C? 24. Three boys together had 1728 marbles; two of them had a dozen apiece: how many had the other? SECTION XI. ANALYSIS. 295. The term analysis in physical science denotes the separation of a compound body into its elements or constituent parts. In arithmetic it signifies the resolving or separating numbers into the parts or the factors of which they are composed. (Arts. 26, 57. Obs.) Analysis may be applied with advantage not only to the development of mathematical principles, but also to QUEST.-294. When the product of three numbers and two of them are given, how is the other found? 295. What is meant by analysis in physical science? What in arithmetic? To what may analysis be advantageously applied? the solution of a great variety of examples, both in arithmetic and practical life. In the preceding sections the pupil has become acquainted with the method of analyzing particular principles, and thence deducing general truths or rules. We will now illustrate the application of analysis to the solution of examples or problems. OBS. Business men have a method of solving practical questions, which is frequently shorter and more expeditious than that of arithmeticians fresh from the schools. If asked, by what rule they perform them, their reply is, "they do them in their head," or by the "no rule method." Their method consists in analysis. It might with propriety be called the common sense rule. A little practice will give the learner great facility as well as satisfaction in its application. MENTAL EXERCISES. Ex. 1. If 8 barrels of flour cost 40, how much will 5 barrels cost? Analysis.-1 is 1 eighth of 8: therefore 1 barrel will cost 1 eighth as much as 8 barrels; and 1 eighth of $40 is $5. Now it is obvious that 5 barrels will cost 5 times as much as 1 barrel; and 5 times $5 are $25, the answer required. Or, we may reason thus: 5 barrels are of 8 barrels; 5 barrels will therefore cost as much as 8 barrels. Now 1 eighth of $40 is $5, and 5 eighths is 5 times $5, which is $25. 2. If 7 lbs. of tea cost 42 shillings, what will 10 lbs. cost? 3. If 9 sheep are worth $27, how much are 15 sheep worth? 4. If 10 barrels of flour cost $60, what will 12 barrels cost? 5. Suppose 30 gallons of molasses cost $15, how many dollars will 7 gallons cost? QUEST. Obs. How do business men solve practical questions? What do they call it? In what does it consist? What might it with propriety be called? 6. If a man earns 54 shillings in 6 days, how much can he earn in 15 days? 7. If 12 men can build 48 rods of wall in a day, how many rods can 20 men build in the same time? 8. A gentleman divided 90 shillings equally among 15 beggars how many shillings did 7 of them receive? 9. Suppose 75 pounds of butter lasts a family of boarders 25 days, how many pounds will supply them for 12 days? 10. If 7 yards of cloth cost $30, how much will 9 yards cost? 11. If 10 barrels of beef cost $72, how much will 8 barrels cost? 12. If 7 acres of land cost $50, what will 12 acres cost? 13. A farmer buying an ox cart, paid $15 down, which was of the price of it: what was the price of the cart; and how much does he owe for it? Analysis.-The question to be solved is simply this: 15 is of what number? If 15 is, is of 15, which is 5. Now if 5 is 1 tenth, 10 tenths is 10 times 5, which is 50. Ans. {$50 is the price of the cart, and $50-$15-35, the sum unpaid. 3 Note. In solving examples of this kind, the learner is often perplexed in finding the value of o, &c. This difficulty arises from supposing that if of a certain number is 15, of it must be 1 of 15. This mistake will be easily avoided by substituting in his mind the word parts for the given denominator. Thus, if 3 parts cost $15, 1 part will cost of $15, which is $5. But this part is a tenth. Now if 1 tenth cost $5, then 10 tenths will cost 10 times as much. 14. A man bought a yoke of oxen, and paid $56 cash down, which was of the price of them: what did they cost? 15. A merchant bought a quantity of wood and paid $45 in goods, which was of the whole cost: how much did he pay for the wood? 16. A whale ship having been out 24 months, the Cap tain found that his crew had consumed of his provisions how many month's provision had he when he embarked, and how much longer would his provisions last? 17. How many times 7 in of 35? Analysis of 35 is 7, and is 4 times 7, which is 28. Now 28 is 4 times 7. Ans. 18. How many times 6 in 3 of 45? 24. 25. ΤΟ of 64 are how many times 7? 26. of 100 are how many times 12? 10 27. of 110 are how many times 8? I 28. of 180 are how many times 10? 29.5 of 84 are how many times 9 ? 30. How many yards of cloth, at $7 per yard, can be bought for of $54? 31. How many barrels of flour, at $5 per barrel, can be bought for of $60? 32. A man had $64 in his pocket, and paid of it for 10 barrels of flour: how much was that per barrel? 33. 40 is of how many times 6? Analysis. Since 40 is, is of 40, or 8; and is 9 times 8, or 72. Now 6 is contained in 72, 12 times. Ans. 12 times. of how many times 12? 41. Frank sold 10 peaches, which was of all he had; he then divided the remainder equally among 5 companions how many did they receive apiece? 10 42. Lincoln spent 60 cents for a book, which was 12 of his money; the remainder he laid out for oranges, at 4 cents apiece: how many oranges did he buy? 43. A man paid away $35, which was of all he had; he then laid out the rest in cloth at $2 per yard: how many yards did he obtain ? 44. A farmer bought a quantity of goods, and paid $20 down, which was of the bill; how many cords of wood, at $3 per cord, will it take to pay the balance? 45. A man bought a horse and paid $60 in cash, which was of the price: how many barrels of flour, at $6 per barrel, will it take to pay the balance? 46. 3 of 27 is of what number? Analysis.-3 of 27 is 9. And if 9 is of a certain number, of that number is 3; and 4 is 4 times 3, which is 12, the number required. 47. of 30 is 48. of 40 is 49. 4 of 35 is 50. of 54 is of what number? of what number? of what number? of what number? EXERCISES FOR THE SLATE. 296. It will be seen from the preceding examples that no particular rules can be prescribed for solving questions by analysis. None in fact are requisite. The process will be easily suggested by the judgment and the conditions of the question. OBS. The operation of solving a question by analysis, is called an analytic solution. In reciting the following examples, the pupil QUEST.-296. Can any particular rules be given for solving questions by analysis? Is it necessary to give them? How then will you know how to proceed? Obs. What is the operation of solving questions by analysis, called? |