in the year 1841 was 18664761, and in 1851 it was 20936468: find the increase in these ten years. 11. The number of Season Tickets for the Great Exhibition, sold before the building was opened, was nineteen thousand five hundred and seven; of these, eight thousand six hundred and fifteen were Ladies' Tickets: how many of them were Gentlemen's Tickets? 12. The number of visits paid to the British Museum in the year 1850 was 1098863, and to Hampton Court Palace 221119: how many visits were paid to the former place more than to the latter? 13. The Gross Revenue of the Post Office* for the year ending on the 5th of January, 1851, was 2264684 pounds, and the cost of management was 1460785 pounds: what was the Net Revenue for the year? 14. The total number of passengers conveyed on the Railways of the United Kingdom in the half-year ending on the 30th of June, 1850, was 31766503; and in the halfyear ending on the 31st of December, 1850, the total number was 41087919: what was the increase in the number of passengers in the last half-year? 15. The gross receipts of the London and Brighton Railway during the week ending Nov. 22, 1851, were, for Passengers, 6217 pounds; for Goods, 2135 pounds. The gross receipts for Passengers and Goods, during the corresponding week of the preceding year, were 8149 pounds: find how much the receipts had increased. 16. The population of Great Britain and the neighbouring Islands in 1851 was 20936468; the population of England and Wales alone was 17922768, and the population of the British Islands alone was 142916: what was the population of Scotland? 17. The salaries paid to the officers employed by the CustomHouse in 1849 were as follows: salaries in England, 550236 pounds; salaries in Scotland, 62115 pounds; salaries in Ireland, 57903 pounds. The amount of Custom-House duty, collected in that year, was 22481339 pounds: what was the net amount received after these salaries were paid? * By revenue of the Post Office is meant income of the Post Office; and gross revenue, or gross receipts, means the money received before the expenses of management are subtracted; when these expenses are taken from the gross income, the remainder is called the net income.-See Exercise 17. 18. In the year 1849 there were 578159 children born in England and Wales; of these 295158 were males. In the same year 440853 persons died; of these 221801 were males; you are required to find how many females were born in 1849, and how many died. 19. What is the difference between 365 + 2041 + 109, and 75301623+ 87 + 3406? 20. What is the difference between 112104 +3820 + 3268, and 2389103403 + 13400? 21. Find the difference between 462873 + 5962 + 304 + 19871, and 1735 + 902603 + 72 + 139. (18.) The last three exercises bring into use the sign of addition, explained at page 9. There is also a sign of subtraction, which it is equally necessary that you should remember; it is the little mark -. This sign, placed before a number, means that the number is to be subtracted. By using this sign, which is called minus, we may express an example in subtraction without words, the sign of equality, =, being placed before the remainder: thus, the first example, page 11, may be written 6879-625 = 6254; the second example may be written 23596-13758 = 9838; the third may be written 72311075-3506285 = 68804790; and so of the other examples. If you were asked to read the first of these, you would say, 6879 minus 625, equals 6254: you can from this read the others without any help; and I dare say you could even read the following, namely, 2436-17-41 + 13—11—2 = 2; but in case you should be puzzled, I will read it for you: it is 24 plus 36 minus 17 minus 41 plus 13 minus 11 minus 2, equals 2; the meaning of which is, that if from the sum of 24, 36, and 13, the sum of 17, 41, 11, and 2, be subtracted, the remainder will be 2. MULTIPLICATION. (19.) WE now come to the third rule in Arithmetic,-the rule for multiplication,—which teaches us how to find the sum of a set of equal numbers without our taking the trouble to put them all down, and add them together, as in addition. If we have to find the sum of two equal numbers, we put down only one of those numbers, and multiply it by 2, according to the rule to be given presently; if we have to find the sum of three equal numbers, we put down one of them, and multiply by 3; and in like manner if we have to find the sum of eight, or nine, &c., equal numbers, we multiply one of them by 8, or 9, &c. In this manner we discover the sum required very soon. The number we multiply another number by is called the multiplier; and the other number the multiplicand: the result of the operation, and which in addition would be called the sum, is here called the product. Whatever be the multiplicand, and whatever be the multiplicr, the operation could be described in a single rule; but it will be easier for you if I divide the general rule into two particular rules: I shall therefore do this; but before you can use either rule, you must learn the Multiplication Table, which I here give. This table you must repeat in this way: Twice 1 are 2; twice 2 are 4; twice 3 are 6, &c. Three times 1 are 3; three times 2 are 6; three times 3 are 9, &c. Four times 1 are 4; four times 2 are 8; four times 3 are 12, &c. &c. When you say twice any number, you are said to multiply that number by 2; when you say 3 times, you are said to multiply by 3, and so on; and the number that results is the product: thus, when you say 4 times 6 are 24, the multiplier is 4, the multiplicand is 6, and the product is 24. You must remember this. When you say 4 times 8 are 32, if you were asked what is the multiplier, what is the multiplicand, and what is the product,—what would you answer? * * The learner may, if he please, commit to memory, at first, only a part of the table on the next page, and may select from the exercises that follow, such of them as require only those multipliers within the range of his knowledge of the table. After some practice in these, another portion of the table may be learnt. Simple as the Multiplication Table appears to the arithmetician, it should be regarded by every teacher as a thing of no small labour and difficulty to a mere beginner. MULTIPLICATION TABLE. Twice 3 times 4 times 5 times 7 times 8 times 9 times 1 are 2 1 are 3 1 are 4 1 are 5 1 are 6 1 are 7 1 are 8 6 times 10 times 11 times 12 times 1 are 9 1 are 10 1 are 11 1 are 12 (20.) You can easily satisfy yourself of the truth, and also of the use of this table, by taking out of it any multiplicand, and any multiplier,— by writing down the multiplicand as often as there are units in the multiplier, and then adding all these equal numbers together. You will find that the sum of them is always equal to the product put down in the table. Thus, the table tells us that 6 times 7 are 42; and we find, by addition, that six 7's are 42: thus, 7 +7 +7 +7 +7 + 7 = 42. In like manner, we are told by the table that 5 times 8 are 40; and we know, by addition, that 8 +8 +8 +8 +8 = 40; and so in every other case in which the equal numbers are each not greater than 12, and not more than twelve of them are to be added together. When any multiplicand is greater than 12, and the multiplier not greater than 12, the table will still help us to the product by aid of the first of the two rules I promised; which is as follows: 1. When the Multiplier is not greater than 12. RULE 1. Place the multiplier under the multiplicand, units under units. 2. Then, commencing at the units-figure of the multiplicand, multiply each figure, in succession, by the multiplier, and put the product under that figure, taking notice, however, that whenever any of these products is a number of two or three figures, the right-hand figure only is to be put down, and the rest carried to the next product, as in addition. Ex. 1. Multiply 2683 by 2. Having placed the 2 under the 3, as in the margin, I say, twice 3 are 6; twice 8 are 16, 6 and carry 1; twice 6 are 12, and 1 carried are 13, 3 and carry 1; twice 2 are 4, and 1 carried are 5. Therefore the product is 5366. 2. Multiply 728365 by 3. 3 times 5 are 15, 5 and carry 1; 3 times 6 are Therefore the 3. Multiply 276023 by 8. 8 times 3 are 24, 4 and carry 2; 8 times 2 are 16, and 2 are 18, 8 and carry 1; 8 times 0 are 0, and 1 are 1; 8 times 6 are 48, 8 and carry 4; 8 times 7 are 56, and 4 are 60, 0 and carry 6; 8 times 2 are 16, and 6 are 22. Therefore the product is 2208184, 2683 2 5366 728365 3 2185095 276023 8 2208184 |