APPLICATIONS. 1. If a man travel 212 miles a week, how far will he travel in 52 weeks ? SOLUTION. If a man travel 212 miles in a week, he will travel in 52 weeks 52 times 212 miles, or 11024 miles. Therefore, if a man travel 212 miles in a week, he will travel in 52 weeks 11024 miles. 2. If an orchard containing 313 trees produce 15 bushels of apples to a tree, how many bushels is the produce of the whole orchard ? Ans. 4695. 3. How many lights of glass in 18 windows, if each window contains 24 ? 4. How many bushels of corn will grow on 160 acres, at the average rate of 45 bushels to an acre ? Ans. 7200 bushels. 5. What cost 2463 barrels of flour at 9 dollars a barrel ? 6. If a man can earn 83 dollars in one month, how many can he earn in 12 months ? Ans. 996 dollars. 7. What will be the cost of a farm containing 684 acres, at 57 dollars per acre ? Ans. 38988 dollars. 8. A certain field has 625 hills of potatoes, and each hill will average 8 potatoes; how many potatoes at that rate in the field? 9. If 17 men can do a piece of work in 91 days, how long will it take one man alone to do it? Ans. 1547 days. 10. An army consists of 6 brigades, each brigade of 4 regiments, and each regiment of 613 men, rank and file; required the number of men in the army? Ans. 14712 men. 11. If a saw-mill can produce 4360 feet of boards a day, how many can it produce in 106 days ? Ans. 462160 feet. 12. If the earth moves around the sun at the rate of 68000 miles an hour, how far will it move in 365 days of 24 hours each ? Ans. 595680000 miles. REVIEW QUESTIONS. What is the Rule in Addition ? (40) In Subtraction ? (45) In Multiplication ? (51) DIVISION. 56. 1. If 4 cents will buy one orange, how many oranges will 12 cents buy ? SOLUTION. If 4 cents will buy one orange, 12 cents will buy as many oranges as 4 cents are contained times in 12 cents, or 3. Therefore, if 4 cents will buy one orange, 12 cents will buy 3 oranges. 2. At 7 cents a pound, how many pounds of rice can be bought for 21 cents ? 3. If 2 boys share equally 8 marbles, how many will each have ? SOLUTION. If 2 boys share equally 8 marbles, each will have 1 of the 2 equal parts of 8 marbles ; 1 of the 2 equal parts of 8 marbles is 4 marbles. Therefore, if 2 boys share equally 8 marbles, each will have 4 marbles. One of the two equal parts of a number is called one half of the number; one of the three equal parts, one third of the number; one of the four equal parts, one fourth of the number; one of the five equal parts, one fifth of the number; one of the ten equal parts, one tenth of the number; one of the hundred equal parts, one hundredth of the number, and so on. Hence, A whole number has two halves, three thirds, four fourths, five fifths, ten tenths, one hundred hundredths, etc. 4. When 4 apples cost 16 cents, what is the cost of one apple? SOLUTION. When the cost of 4 apples is 16 cents, the cost of one apple is one fourth of 16 cents, or 4 cents. Therefore, when 4 apples cost 16 cents, the cost of one apple is 4 cents. 5. If 20 dollars are shared equally between 5 boys, how many does each boy receive ? The preceding operations illustrate what is called DIVISION. Hence, What is one of two equal parts of a number called? One of three equal parts? One of four equal parts ? 57. Division is the process of finding how many times one number is contained in another; or of finding one of the equal parts of a number. The DIVIDEND is the number to be divided. The QUOTIENT is the result or number obtained by the division. The REMAINDER is that part of the dividend which is left after finding the exact whole number of the quotient. Thus, 3 is contained in 7, 2 times and 1 as a remainder. The division is said to be exact, when there is no remainder. 58. The SIGN OF DIVISION is a short horizontal line with a dot above and below it, :, read, divided by. Thus, 6:2 is read, six divided by two. Sometimes, in place of the dots, the number divided is written above the line, and the number which divides it is written below. Thus, is read six divided by two. Division, also, may be indicated by a curved line, ), the divisor being written before, and the dividend after it. Thus, 2) 6 indicates that 6 is to be divided by 2, and is so read. 59. One or more equal parts of a unit are called fractions, to distinguish them from unbroken or whole numbers, which are called integers, or integral numbers. }, read one divided by two, or one half, , read two divided by three, or two thirds, etc., are fractional expressions. What is Division? The Dividend? The Divisor ? The Quotient ? The Remainder ? When is the division said to be exact? What are the Signs of Division? What are one or more equal parts of a unit called ? 5 in 5, 1 time 66 66 66 66 66 66 66 66 66 Division Table. 4 in 1, 1 time 2, 1 time 3, 1 time 4, 1 time 2, 2 times 4, 2 times 6, 2 times 8, 2 times 10, 2 times 3, 3 6, 3 9, 3 12, 3 4, 4 15, 3 16, 4 5, 5 20, 4 25, 5 18, 6 24, 6 30, 6 7, 7 14,7 « 21, 7 28, 7 35, 7 8, 8 16, 8 24, 8 32, 8 9, 9 40, 8 18, 9" 27,9 36, 9 45, 96 10,106 20, 10 « 30, 10 « 40,10 50, 10 10 in 6, 1 time 7, 1 time 8, 1 time 9, 1 time 12, 2 times 14, 2 times 16, 2 times 18, 2 times 20, 2 times 10, 1 time 18, 3 21, 3 24, 3 27, 3 24, 4 30, 35 28, 4 32, 4 36, 4 65 30, 5 6 40, 4 « 35, 5 40, 5 « 45, 5 36, 6 66 50, 5" 42, 6 48, 6 54, 6 42, 7 60, 666 49,7 56, 7 63,7 48, 8 70, 7 66 56,8 64, 8 72, 8 54, 9 80, 8 6 63, 9 72, 9 81, 9 60,10 66 90, 96 70,10 « 80,10 90, 10 1100,10 66 60. The process of Division is based upon the following PRINCIPLES. 1. The quotient will be an abstract number, when the divisor and dividend are like numbers. For, the quotient will denote how many times the divisor is contained in the dividend. 2. The divisor must be regarded as an abstract number, when the dividend is concrete, and the divisor not a like number. 3 in 3. 4 in 4. 5 in 5, etc. What is Repeat the line 1 in 1. 2 in 2. the first Principle? The second ? For, the divisor must then denote the number of equal parts into which the dividend is to be divided. Hence, 3. If the divisor and dividend are not like numbers, the quotient and dividend will be like numbers. For, the quotient will denote one of the equal parts into which the dividend is divided. 4. The remainder and dividend must always be like numbers. For, the remainder is evidently a part of the dividend. 5. Division may be regarded as the reverse of multiplication. For, the dividend corresponds to the product, and the divisor and quotient to the two factors. 61. From the foregoing principles, it follows that there may be two kinds of division : FIRST KIND when the size or value of the equal parts of a quantity is given, to find their number; and SECOND KIND when the number of equal parts of a quantity is given, to find their size or value. CA SE I. 62. To divide by Short Division. OPERATION. For convenience, we write the Divisor, 7) 1702 Dividend. divisor at the left of the dividend, and begin at the left to divide. 2437 Quotient. 7 is contained in 1 thousand no thousands times; therefore, there will be no thousands in the quotient. Try 17 hundreds ; 7 is contained in 17 hundreds, 2 hundreds times, and 3 hundreds, equal to 30 tens, remaining. We write the 2 hundreds, and add the 30 tens to the O tens, making 30 tens. What is the third Principle? The fourth? The fifth ? Name the two kinds of Division. |