QUESTIONS FOR REVIEW. ORAL. -1. If 9 men can build a wall in 12 days, how long will it take 1 man to build it ? ANALYSIS.-It will take I man 9 times as long as 9 men, and 9 times 12 days are 108 days. Therefore, it will take I man 108 days. 2. If a jar of butter will last a family of 8 persons 6 weeks, how long will it last 1 person? 3. Henry can read a book through in 11 days by reading 6 hours each day: how long will it take him if he reads 1 hour a day? 4. If 12 men can frame a house in 8 days, how long will it take I man to frame it? 5. If I buy 4 barrels of apples at 3 dollars a barrel, and 4 barrels of pears at 5 dollars, what will be the cost of both? 6. A farmer having 15 bushels of wheat, sold 9 bushels at 2 dollars a bushel, and the remainder at 3 dollars a bushel: how much did he get for his wheat? WRITTEN.-1. If it takes 285 laborers 18 months to build a railroad, how long would it take 1 man to build it? 2. A ship of war has provisions to last a crew of 625 men 90 days: how long would they last 1 man? 3. If a clerk has 36 dollars a month for the first 4 months; 48 dollars a month for the next 4; and 60 dollars a month for the next 4; what will he receive for the year? 4. A man having 1000 dollars in his pocket, gave 45 dollars each to 12 poor persons: how much had he left? 5. If I receive 150 dollars a month, how much shall I have at the end of the year, after deducting 28 dollars a month for board? DIVISION. MENTAL EXERCISES. To TEACHERS.-The object of this preliminary Exercise is to develop he idea of "times," as used in Division, preparatory to learning the Table. 1. If I have 9 pencils, how many boys can I supply with 3 pencils each? ANALYSIS.—If I give one boy 3 pencils, how many will be left? "Six pencils." If I give another boy 3, how many pencils will be left? "Three." "None." If I give another 3, how many will be left? "Three." How many times are 3 pencils contained in 9 pencils? "Three times." 2. How many peaches, at 2 cents each, can you buy for 8 cents? Show this by counters. 3. How many oranges, at 4 cents each, can you buy for 12 cents? How many times 4 make 12? Show this. 4. In 1 gallon there are 4 quarts: how many gallons are there in 8 quarts? How many times 4 make 8? Show this by unit marks. 5. If a lad earns 5 dollars a week, how long will it take him to earn 25 dollars? How many times 5 make 25? Show this. 6. At 6 cents an ounce, how many ounces of candy can you buy for 18 cents? Show this by unit marks. 7. How many lambs, at 2 dollars apiece, can be had for 20 dollars? Show this. 8. At 4 dollars a pair, how many pair of boots can I buy for 16 dollars? Show this. 9. If I have 20 pounds of flour, how many poor per sons can I supply with 5 pounds each? * After 2, 3, 4, cc., in the second column, "times" is under stood. II is in II, once. 22, 33, 44, 55, 66, 77, 2345678 36, 48, 60, 72, 84, 88, 8 96, 12 is in 12, once. 24, DEFINITIONS. 1. What is Division ? Division is finding how many times one number is contained in another. 2. What is the number to be divided called? The Dividend. 3. The number to divide by? The Divisor. 4. What is the number obtained by division called? The Quotient. 5. What is the number left called? The Remainder. When it is said that 3 is contained in 13, 4 times and I over, which is the dividend? The divisor? The quotient? The remainder? REMARKS.-I. The remainder is always the same denomination as the dividend; for, it is a part of the dividend not yet divided. 2. A proper remainder is always less than the divisor. 6. When the dividend contains only one denomination, what is the operation called? Simple Division. 7. How is Division denoted? By a short horizontal line between two dots (÷), called the Sign of division. 8. When placed between two numbers what does it show? It shows that the number before it is to be divided by the one after it. Thus, 213, shows that 21 is to be divided by 3, and is read " 21 divided by 3." 9. How else is division denoted? By writing the divisor under the dividend with a short line between them; as 21. Read the following: 93 = 3; 24÷45+ 1; 39÷ 3 = 10 + 3; 5 + 4 =36÷4; 28 = 7; 35 = 4 + 3. OBJECTS OF DIVISION. 1. A lad having 6 cents wishes to buy pears, which are 2 cents apiece: how many can he buy? ANALYSIS.-He can buy as many pears as there are times 2 cents in 6 cents. The object then is to find how many times 2 is contained in 6; and 2 is in 6, 3 times. ILLUSTRATION. * * * * * * 2. A lad has 6 pears, which he wishes to divide equally between 2 companions: how many can he give to each? ANALYSIS.-The object of this example is to divide 6 pears into 2 equal parts. Dividing 6 by 2, the quotient is 3, which shows that there are 3 pears in each part. ILLUSTRATION. * * * * * * 10. What is the object or office of Division? Its object or office is twofold: First, To find how many times one number is contained in another. (Ex. 1.) Second, To divide a number into equal parts. (Ex. 2.) REMARK.-The two preceding examples are representatives of the two classes of problems to which Division is applied. In the first class, the divisor and dividend are always of the same denomination, and the quotient is times, or an abstract number. In the second, the divisor and dividend are of different denominations, and the quotient is always of the same denomination as the dividend. This class involves the idea of Fractions, and will receive further attention under that branch of the science. NOTE. The process of reasoning in the solution of these two classes of examples is somewhat different; but the practical operion is the same, viz.: to find how many times one number is contained in another, which accords with the definition of Division. 11. How divide a number into two, three, four, etc., equal parts? Divide the number by 2, 3, 4, 5, etc., respectively. 12. When a thing is divided into 2, 3, 4, etc., equal parts, what are the parts called? If divided into two equal parts, the parts are called halves; into three, the parts are called thirds; into four, they are called fourths; into five, fifths; etc. |