and you yourself can now put the third, or any other example, in the same form, and can read it when you have done You must really attend to these signs of addition and of equality, because they will be often used hereafter. SO. SUBTRACTION. (15.) We now come to the second rule of arithmetic, which is called the rule of Subtraction: it teaches us how to take, that is, to subtract, the smaller of two numbers from the greater, and so to find the difference or remainder. If both numbers are numbers of single figures only, you do not feel the want of a particular rule: if you are asked to find the difference between the number 8 and the number 5, that is, if you are asked to subtract 5 from 8, you can easily see that the difference or remainder must be 3; and you also know, that if 2 be subtracted from, that is, taken away from, 9, the remainder will be 7; that if 4 be subtracted from 6, the remainder will be 2 ; and so on : but if each be a number of several figures, you will want a rule to guide you to the correct remainder: this rule is as follows: Rule 1. Write the smaller number under the greater, taking care, as in addition, to place units under units, tens under tens, hundreds under hundreds, &c. 2. Then, beginning with the units, as in addition, subtract the lower figure from the upper : this you can easily do if the upper figure be the greater of the two; but if it be the less, you must fancy 10 added to it, which is sure to make it greater, so that you can now subtract the lower figure, and put the remainder underneath. 3. If you have increased the upper figure by 10, you must carry 1 to the next figure to be subtracted; if this next figure, thus increased by 1, be greater than the figure above it, 10 must be added to the upper figure, as before ; and after the remainder is put down, 1 must be carried to the next figure; and so on. You see, therefore, that the method is to subtract the units of the lower or smaller number from the units of the upper, the tens from the tens, the hundreds from the hundreds, and so on, always adding 10 to the upper figure when it is less than the figure under it, and taking care, in such a case, to carry 1 to the next figure: it is only when 10 is thus added to an upper figure that 1 is to be carried to the next lower one. The following examples will explain the operation : 1. Subtract 625 from 6879. Placing the smaller number under the 6879 greater greater, as in the margin, we say, 5 from 625 less 9, and 4 remain ; 2 from 7, and 5 remain ; 6 from 8, and 2 remain ; and as nothing 6254 remainder is taken from the upper figure 6, the complete remainder is 6254, or 6 thousand 2 hundred and 54. As in this example each of the lower figures is less than the figure above it, the subtraction is performed without adding 10 to any upper figure: in the next example such is not the case. 2. Subtract 13758 from 23596. The numbers being written as before, we see 23596 that the figure 6 in the units' place of the upper 13758 number is less than the figure 8 below it; we therefore fancy 10 to be added to the 6, making it 9838 16, and say, 8 from 16, and 8 remain, carry 1; 6 from 9, and 3 remain ; 7 from 15, and 8 remain, carry 1; 4 from 13, and 9 remain, carry 1; 2 from 2, and nothing remains; therefore the remainder is 9838. 3. Subtract 3506285 from 72311075. Instead of repeating the word remain at 72311075 every subtraction, it is better to proceed as fol- 3506285 lows: 5 from 5, nought; 8 from 17, 9, carry 1; 3 from 10, 7, carry 1; 7 from 11, 4, carry 1; 68804790 1 from 1, nought; 5 from 13, 8, carry 1; 4 from 12, 8, carry 1; 1 from 7, 6. (16.) The truth of this rule for subtraction may be shown in a few words :- Adding 10 to any figure of a number is the same as adding 1 to the figure before it: thus, the number 75, is 7 tens and 5; if I add 10 to the 5, I make it 7 tens and 1 ten and 5, that is 85. Again ; the number 623, is 6 bundreds, 2 tens, and 3; if I add 10 to the 2, I make it 6 hundreds, 12 tens, and 3; or 6 hundreds, 10 tens (which is another hundred), 2 tens, and 3, that is 723, and so in other cases ; so that adding 10 to a figure is, in fact, adding 1 to the figure before it. The rule tells us, that whenever we add 10 to an upper figure, we must subtract an additional 1 from the figure before it; therefore, the 1 that has been added to an upper figure, for convenience, is immediately afterwards taken away, so that all is brought right again. I shall add two or three examples more, with the remainders put down for you to look over; and shall then give some exercises in subtraction for you to find the remainders yourself. 6803029 34510381 8057130600 148112354 4287012 28278336 7909018246 (17.) You can prove whether the subtraction is correctly performed, by adding the remainder to the number wbich has been subtracted; the sum ought to be the top number : thus, taking the first of the three examples just given, you would say, 2 and 7 are 9; 1 and 1 are 2 ; 0 and 0 are 0; 7 and 6 are 13, 3 and carry 1; 1 and 8 are 9, and I are 10, O and carry 1; 1 and 2 are 3, and 5 are 8 ; and 2 are 6. And as the figures thus obtained are those of the top number, we conclude that the work is right. Exercises in Subtraction. 1. Subtract 375 from 846, and 1237 from 2865. 2. What is the difference between 36207 and 72098 ? 3. Take 7992 from 18097, and 300043 from 1001251. 4. Subtract seven thousand and fifty-three, from a hundred and eleven thousand and two. 5. Subtract thirteen thousand one hundred and seventeen, from twenty-two thousand and five. 6. What is the difference between one million three hundred and two thousand and forty-two, and three million one hundred and eleven? 7. The three greatest generals in modern times—the Duke of Wellington, Napoleon Bonaparte, and Marshal Soult-were all born in the same year, 1769; the last died in November, 1851 : how old was the Duke of Wellington then ? 8. The population of Ireland in the year 1841 was 8175124, and in 1851 it was 6515794: find by how many people the population had decreased in these ten years. * 9. The population of Ireland in the year 1821 was 6801827, and in the year 1831 it was 7767401 : what was the increase in these ten years ? 10. The population of Great Britain and its adjacent Islands * The learner should be required to state the numbers in these exercises in words; and to write his results both in figures and words : he may divide the figures into veriods, as explained at page 5. 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 Custom House 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 7530 + 1623 + 87 + 3406 ? 20. What is the difference between 112104 + 3820 + 3268, and 2389 + 103403 + 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, 24 + 36-17-41 + 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. |