9. If light moves at the rate of 192000 miles a second, how many seconds is it in coming from the moon to the earth, the distance being 240000 miles ? Ans. 146.000 seconds. 10. Certain States and the District of Columbia furnished in the late war 2653062 soldiers; how many days would it take a person to count them at the rate of 15000 a day? Ans. 17613883 days. REVIEW EXERCISES. 1. If the multiplicand is 15607 and the multiplier 3094, what is the product ? Ans. 48288058. 2. If the dividend is 48288058 and the the divisor 3094, what is the quotient? 3. If two men start from the same point and travel in opposite directions, the one at the rate of 42 miles and the other 45 miles a day, how far apart will they be at the end of 12 days? SOLUTION. If one man travel at the rate of 42 miles a day, he will travel in 12 days 12 times 42 miles, or 504 miles. If the other man travel at the rate of 45 miles a day, he will travel in 12 days 12 times 45 miles, or 540 miles. If the one travel 504 miles and the other 540 miles, and in opposite directions, they must be apart, the sum of 504 miles and 540 miles, or 1044 miles. Therefore, if two men start from the same point and trarel in opposite directions, the one at the rate of 42 miles and the other 45 miles a day, they will be 1044 miles apart at the end of 12 days. 4. If two men start from the same point and travel in the same direction, the one at the rate of 512 miles and the other 540 miles a week, how far apart will they be at the end of 8 weeks ? REVIEW QUESTIONS. What is the Rule for Short Division ? (62) What is the Rule for Long Division ? (63) 5. I have two farms; the first contains 160 acres, worth 80 dollars an acre, and the second 220 acres, worth 65 dollars an acre ; how much are both worth? Ans. 27100 dollars. 6. What will be the cost of 103 barrels of flour at 7 dollars a barrel? Ans. 721 dollars. 7. A merchant bought 30 hogsheads of molasses at 45 dollars each, and paid 800 dollars down, and gave his note for the balance; for what amount was the note ? 8. If a man sell 90 acres of land at 38 dollars an acre, and divide the money equally among his 5 children, what is each child's share ? SOLUTION. If a man sell 90 acres of land at 88 dollars an acre, he vill receive for it 90 times 38 dollars, or 3420 dollars. If he divide 3420 dollars equally among his children, each chila will receive as a share one-fifth of 3420 dollars, or 684 dollars. Therefore, if a man sell 90 acres of land at 38 dollars, and divide the money equally among his 5 children, each child's share is 684 dollars. 9. If a man having 5500 dollars to invest should purchase 15 United States bonds, at 105 dollars each, how many shares of railroad stock, at 157 dollars each, could he purchase with the balance ? Ans. 25 shares. 10. William Miller bought some land for 18050 dollars. He sold 50 acres of it for 60 dollars an acre, and then found that the remainder cost him 50 dollars an acre; how many acres were there of the remainder ? Ans. 301 acres. 11. Bought 5 cows at 50 dollars each, and 7 horses at half the price each of the entire cost of the cows; how much was she cost of both ? Ans. 1125 dollars. 12. Smith has 168 acres of land, Johnson 4 times as much and 35 acres, and Wade 3 times as much as both of them less 1200 acres; how many acres in all have they? Review QUESTIONS. What is the Proof in Addition ? (40) In Subtraction ? (45) In Multiplication? (51) In Division ? (62) What is the answer in Multiplication called? (47) In Division ? (57), GENERAL PRINCIPLES AND APPLI CATIONS. 66. The Fundamental Operations or Processes of Arithmetic, or those upon which all others depend, are based upon Notation, and are ADDITION, SUBTRACTION, MULTIPLICA TION, AND DIVISION. 67. The SIGNs used to indicate processes, or to abbreviate xpressions, are called SYMBOLS. t, read plus, or added to. = read equals, or equal to. read minus, or less. read therefore, hence. X, read multiplied by. •., read since, because. • , read divided by. (), parenthesis. 68. Numbers in a parenthesis, or under a vinculum, are to be regarded as all subject to the same operation. Thus, 16—(3 X 2), denotes that the product of 3 multiplied by 2 is to be subtracted from 16. 16. (5+3) X 5, denotes 5 times the difference between 16 and the sum of 5 added to 3. Exercises. 1. What is the value of (31 X 6) — 86 ? SOLUTION. 31 X 6 equals 186, and 186 - 86 equals 100. Therefore, 31 multiplied by 6, in parenthesis, less 86, equals 100. 2. (18 ; 6) +13= how many ? Ans. 16. 3. (6 X 6) = (4 x 3) = how many ? Ans. 3. 14. (8+2 x 5) - 20 = how many ? Ans. 30. (160—70) + 18 5. = how many ? Ans. 9. 12 6. (3+9) X (13 — 5 X 2) = how many ? Upon what are the Fundamental Operations of Arithmetic based ? Name them. What are Symbols of Operation? How are numbers in a parenthe sis, or under a vinculum, to be regarded ? FORMULAS. 69. An Arithmetical Formula is an arithmetical expression of a general rule. 70. The following formulas, which include the fundamental operations of Arithmetic, follow from the preceding definitions, principles, and illustrations : 1. The Sum= all the parts added. 2. The DIFFERENCE = the Minuend the Subtrahend. 3. The MINUEND = the Subtrahend + the Difference. 4. The SUBTRAHEND = the Minuend the Difference. 5. The PRODUCT = the Multiplicand X the Multiplier. 6. The MULTIPLICAND = the Product = the Multiplier. 7. The MULTIPLIER = the Product - the Multiplicand. 8. The QUOTIENT = the Dividend — the Divisor. 9. The DIVIDEND = the Quotient X the Divisor. 10. The DIVISOR = the Dividend ; the Quotient. These ten Formulas, from their general nature and importance, may be regarded as Fundamental. The ninth and tenth formulas are general, as will hereafter appear; but, when there is a Remainder to be considered, they may for present applications be given thus : 11. The DIVIDEND= (the integral part of the Quotient X the Divisor) + the Remainder; 12. The DIVISOR = (the Dividend the Remainder) : the integral part of the quotient. What is an Arithmetical Formula? To what is the Sum equal ? The Difference? The Minuend? The Subtrahend? The Product? The Multiplicand ? The Multiplier? The Quotient ? The Dividend ? The Divisor ? Which of the formulas may be regarded as fundamental ? When there is a Remainder to be regarded, how may the ninth and tenth formulas be given ? The sixth and seventh formulas furnish reliable methods of proving Multiplication by Division. Thus, the multiplication is proved, when, 13. The Product = the Multiplier = the Multiplicand; or, 14. The Product = the Multiplicand = the Multiplier. SOME PRINCIPLES OF DIVISION. 4; and = 8. 4 4 71. Multiplying the dividend, or dividing the divisor, by any number, multiplies the quotient by the same number. Thus, 16 16 X 2 16 8, or 422 72. Dividing the dividend, or multiplying the divisor, by any number, divides the quotient by the same number. Thus, 16 16:52 2. 4 X 2 73. Dividing or multiplying both the dividend and divisor by the same number will not change the quotient. Thus, 16 16 - 2 16 X 2 4, or 4 x 2 16 4; and 4 = 4; and REVIEW EXERCISES. 1. If the items of a certain debt are 12 dollars, 106 dollars, and 112 dollars, what is its entire sum? SOLUTION. Since the sum is equal to all the parts added, if the items of a certain debt are 12 dollars, 106 dollars, and 112 dollars, its entire sum must be equal to 12 dollars + 106 dollars + 112 dollars, or 230 dollars. Which formulas furnish a proof of Multiplication ? What is the first Principle of Division ? The illustration? The second Principle? The illustration? The third Principle? The illustration ? |