the columns of tens, hundreds, &c. always remembering, that ten units of any one order, are just equal to one unit of the next higher order. PROOF. 82. Begin at the top, and reckon each column downwards, and if their amounts agree with the former, the operation is supposed to have been rightly performed. NOTE.-No method of proving an arithmetical operation will demonstrate the work to be correct; but as we should not be likely to commit errors in both operations, which should exactly balance each other, the proof renders the correctness of the operation highly probable. QUESTIONS FOR PRACTICE. 5. According to the census of 1820, Windsor contained 2956 inhabitants, Middlebury 2535, Montpelier 2308, and Burlington 2111; how many inhabitants were there in those four towns? Operation. 2956 Windsor. 2535 Middlebury. 2308 Montpelier. 2111 Burlington. 9910 Total. 9910 Proof. . 6. A man has three fields, one contains 12 acres, anoTher 23 acres, and the other 47 acres; how many acres are there in the whole? Ans. 82. 7. A person killed an ox, the meat of which weighed 642 pounds, the hide 105 pounds, and the tallow 92 pounds; what did they all weigh? Ans. 839. 8. How many dollars are 2565 dollars, 7009 dollars, and 796 dollars, when added together? Ans. 10870 dolls. 9. In a certain town there are 8 schools, the number of scholars in the first is 24, in the second 32, in the third 28, in the fourth 36, in the fifth 26, in the sixth 27, in the seventh 40, and in the eighth 58; how many scholars in all the schools? Ans. 251. 10. Sir Isaac Newton was born in the year 1642, and was 85 years old when he died; in what year did he die? Ans. 1727. 11. I have 100 bushels of wheat, worth 125 dollars, 150 bushels of rye, worth 90 dol lars, and 90 bushels of corn, worth 45 dollars; how many bushels have I, and what is it worth? Ans. 340 bush. worth 260 dolls. 12. A man killed 4 hogs, one weighed 371 pounds, one 510 pounds, one 472 pounds, and the other 396 pounds; what did they all weigh? Ans. 1749 pounds. 13. The difference between two numbers is 5, and least number is 7: the greater? 14 The difference between two numbers is 1448, and the least number is 2575; what is the greater? Ans. 4023. 15. There are three bags of money, one contains 6462 dollars, one 8224 dollars, and the other 5749 dollars; how many dollars in the three bags? Ans. 20435 dolls. 16. According to the census of the United States in 1820, there were 3995058 free white males, 3866657 free white females, and 1776289 persons of every other description; what was the whole number of inhabitants at that time? Ans. 9637999. and 14 from Woodstock to Windsor; how far is it from Burlington to Windsor? Ans. 99 miles. 18. How many days in a common year, there being in January 31 days, in February 28, in March 31, in April 30, in May 31, in June 30, in July 31, in August 31, in September 30, in October 31, in November 30, and in Decem ber 31 days? Ans. 365. 19. A person being asked his age, said that he was 9 years old when his youngest brother was born, that his brother was 27 years old when his eldest son was born, and that his son was 16 years old; what was the person's age? Ans. 52 years. 8192735 16423 1234567 214268 7986698 40961359 25. 2746+390+1001+9976+4321+6633-25067, Ans 26. 39543216-4826382+19181716-63551264, Ans. 2. MULTIPLICATION. ANALYSIS. 83. We have seen that Addition is an operation by which several numbers are united into one sum. Now it frequently happens that the Dumbers to be added are all equal, in which case the operation may be abridged by a process called Multiplication. 1. If a book oost 5 cents, what will 4 such books cost? Addition. Ans. 20 cts. Four books will evidently cost four times as much as one book; and to answer the Multiplication. Ans. 20 cts. only in the form of expression; for we can arrive at the amount of 4 times 5 only by a mental process similar to that at the left hand. Hence, in order to derive any advantage from the use of Multiplication over that of Addition, it is necessary that the several results arising from the multiplication of the numbers below ten, should be perfectly committed to memory. They may be learned from the Multiplication table, page 19. (16) 2. If 1 pound of raisins cost 9 cents, what will 7 pounds cost? 84. 3. There are 24 hours in a day; how many hours are there in 3 days! Ans. 72 hours. Multiplication. 24 hours. Three days will evidently contain 2 3 hours. it, as at the right hand, and say 3 times 4 are 12, (the same as 3 fours added together) which are 1 ten and 2 units. We there fore write down the 2 units in the place of units, and reserving the 1 ten to be joined with the tens, we say, 3 times 2 tens are 6 tens, to which we add the 1 ten reserved, making 7 tens. We therefore write 7 at the left hand of the 2, in the place of tens, and we have 72 hours, the same as by Addition. In Multiplication the two numbers which produce the result, as 24 and 3 in this example, are called factors. The factor which is repeated, as the 24, is called the multiplicand; the number which shows how many times the multiplicand is repeated, as the 3, is called the mul tiplier; and the result of the operation, as the 72, is called the product. 4. There are 320 rods in a mile; how many rods in 8 miles? 85. 5. A certain orchard consists of 26 rows of trees, and in each row are 26 trees; how many trees are there in the orchard? 26 26 156 52 Here we find it impracticable to multiply by the whole 28 Operation. at once; but as 26 is made up of 2 tens and 6 units, we may separate them, and multiply first by the units, and then by the tens; thus, 6 times 6 are 36, of which we write down the 6 units, and reserving the 3 tens, we say 6 times 2 are 12, and 3, which was reserved, are 15, which we write down, the 5 in the place of tens, and the 1 in the place of hundreds, and thus find that 6 of the rows contain 156 trees. We now pro. ceed to the 2, and say 2 times 6 are 12; the 2 by which w multiply being 2 tens, it is evident that the 12 are so many tens; but 12 tens are 1 hundred and 2 tens; we therefore write the 2 ander the place of tens, which is done by putting it directly under the 2 in the multiplier, and reserve the 1 to be united with the hundreds. then say 2 times 2 are 4; both these 2's being in the tens' places, their 676 We + product 4 s nundreds, with which we unite the 1 hunded reserved, making 5 hundreds. The 5 being written at the left hand of the 2 tens, we have 5 hundreds and 2 tens, or 520 for the number of trees in 20 These being added to 156, the number in 6 rows, we have 676 for number of trees in 26 rows, or in the whole orchard. 86. 6. There are in a gentleman's garden 3 rows of trees, and 5 trees in each row; how many trees are there in the whole? 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, We will represent the 3 rows by 3 lines of l's, and the 5 trees in each row by 5 1's in each line. Here it is evident that the whole number of 1's are as many times 5 as there are lines, or 3 times 5-15, and as many times 3 as there are columns, or 5 times 3-15. This proves that 5 multiplied by 3 gives the same product as 3 multiplied by 5; and the same may be shown of any other two factors. Hence either of the two factors may be made the multiplicand, or the multiplier, and the preduct will still be the same. therefore prove multiplication by changing the places of the factors, and repeating the operation. We may SIMPLE MULTIPLICATION.. 87. Simple Multiplication is the method of finding the amount of a given number by repeating it a proposed number of times. There must be two or more numbers given in order to perform the operation. The given numbers, spoken of together, are called factors. Spoken of separately, the num ber which is repeated, or multiplied, is called the multiplicand; the number by which the multiplicand is repeated, or multiplied, is called the multiplier; and the number produced by the operation is called the product. RULE. 88. Write the multiplier under the multiplicand, and draw a line below them. If the multiplier consist of a single figure only, begin at the right hand and multiply each figure of the multiplicand by the multiplier, setting down the excesses an carrying the tens as in Addition. (84) If the multiplier con sists of two or more figures, begin at the right hand and mul tiply all the figures of the multiplicand successively by each figure of the multiplier, remembering to set the first figure of each product directly under the figure by which you are multiplying, and the sum of these several products will be the total product, or answer required. (85) PROOF. 89. Make the former multiplicand the multiplier, and the former multiplier the multiplicand, and proceed as before; if it be right, the product will be the same as the former. (86) QUESTIONS FOR PRACTICE. 7. In the division of a prize among 207 men, each man's are was 584 dollars; what was the value of the prize? 584 dolls. 3738 1068 Ans. 1105 38 dolls. 8. If a man earn 3 dolls. a week, how much will he earn in a year, or 52 weeks Ans. 156 dolls. 11. If a personant 180 in a minute, how many will he count in an hour? Ans. 10800. 12. A man had 2 farms, on one he raised 360 bushels of wheat, and on the other 5 times as much; how much did he raise on both? Ans. 2160 bush. 18. In dividing a certain sum of money among 352, each man received 17 dollars, what was the sum divided? Ans. 5984 dolls. 14. If a man's income be 1 dollar a day, what will be the amount of his income in 45 years, allowing 365 days to each year? Ans. 16425 dolls 15. A certain brigade con sists of 32 companies, and each company of 86 soldiers; how many soldiers in the bri gade? Ans. 2752. 16. A man sold 742 thousand feet of boards at 18 dol lars a thousand; what did they come to? Ans. 13856 dolls. 17. If a man spend 6 cents a day for cigars, how much will he spend in a year of 865 days? Ans. 2190 c cts. $21.90. 18. If a man drink a glass of spirits 3 times a day, and each glass cost 6 cents, what will be the cost for a year? Ans. 6570 cts.= $65.70. 19. Says Tom to Dick, you have 7 times 11 chesnuts, but I have 7 times as many as you, how many have I? Ans. 539. 20. In a prize 47 men shared equally, and received 25 dol lars each; how large was the prize? Ans. 1175 dolls. 21. What is the product, 308879 by twenty thousand five hundred and three? Ans 6382946137. 22. What will be the cost of 924 tons of potash at 95 dolls. ja ton? Ans. 87780 dolls. Product, 3400950961 23. Multiply 848329 by 4009. Prod. 170040 Ans. 149940 |