right, and proceeding towards the left, and draw a line below them. 2. Add the right hand column, and if the sum be less than ten, write it below the line at the foot of the column; but if it be ten, or more than ten, write down the excess of tens, and carry as many units as there are tens to the next column, with which proceed as before; and so on till the whole is finished, remembering at the last column to set down the whole amount. PROOF.--Cut off the upper line of figures, and find the sum of the rest; add this sum to the upper line, and, if their sum be the same as the first amount, or sum total, the work is right; ; or, which is com-monly practised, begin at the top and reckon downwards, and if it be right, this sum will be the same as the first amount. Addition and Subtraction 'Table. 245 1 2 3 4 5 6 7 8 9 | 10 | 11 | 12 356 4 6 7 8 8 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |17|18|19|20 same way with all other numbers. Then to prove Addition by casting out the 9's, observe the following RULE. - Cast the 9's out of each line of figures in the given sum, and write the excesses in a column at the right hand. Add the excesses, and the 9's being cast out of this sum and the sum of the given numbers, if the two excesses are equal, the Addition is supposed to have been correct. EXAMPLE. 16423 21230 90418 128071 Excesses. Here the excess of the first line is 7, the second 8, the third 4, and the sum of the excesses is 19. Then casting 7 the 9's out of 19, and also out of the sum 128071, and the 8 excess, in both cases is 1; therefore, the work is supposed 4 to be right. This method of proof is in all cases subject to the inconvenience that a wrong operation may sometimes 1 appear to be right, for if we change the places of any two figures in the sum, the result will still be the same. This method depends upon a property of the number 9, which belongs to no other number, except 3, namely, that any number divided by 9 leaves the same remainder as the sum of its digits divided by 9. Thus 436 divided by 9, the remainder is 4; the sum of the digits in 436 is (4+3+6=)13 and 13 divided by 9, the remainder is 4 as in the former case, and the same is true universally, as may be analytically demonstrated. It may be proved by Subtraction, thus; Beginning at the left, add again each of the several columns, subtract the sums from the sums at the foot of the columns respectively, and write down the remainders, which must be joined each as so many tens to the sum of the column on the right; if the work be right, there will be no remainder under the last column. See Example above. Here we add the left hand column, and find the sum 12, this taken from 12 leaves 0; then add the next, and find it 7; this from 8 leaves 1: the sum of the next is 10, this from 10 leaves 0; the next is 6; this from 7 leaves 1; the next is 11, this from 11 leaves 0; it is therefore right. 1 Examples. Thous. Hund. Tens. Units. 8224 101 81 1. What is the sum of 6432, 8224, 101 and 81, when added together? Having written the numbers according to the rule, and drawn a line below them, begin with the right hand column and say, 1 to 1 is 2, and 4 is 6, and 2 is 8, which 6432 being less than ten, write it below the line at the foot of the column. Then proceed to the next column and say, 8 to 0 is 8, and 2 is 10, and 3 is 13; 13 being one 10 and 3 over, write down 3, the excess, and carry 1 to the next column, saying 1 to 1 is 2, and 2 is 4, and 4 is 8, which write down. Then 8 to 6 is 14; this being the last column, write down the whole number by placing 4 the excess of tens, under the column, and 1 the number of tens, at the left hand. Ans. 14838 8406 Proof. 14838 To prove that the operation has been rightly performed, cut off the upper line of figures, and add the three lower lines as already taught, setting their sum, 8406, below a line drawn under the first amount, with each figure directly under the line which produced it; add this last sum to the upper line, and their sum, 14838, being the same as the answer, or amount of all the given numbers, the work is considered to be right. By careful attention to this rule and its illustration by the preceding example, the student will find but little difficulty in working the examples which follow. As arithmetick is usually studied on account of its practical utility in transacting the business of life, it is important that the scholar acquire such a knowledge of each rule as he proceeds, as will enable him readily to apply it in practice, whenever occasion shall require. To aid him in obtaining this knowledge, and to give a scope for the exercise of his judgment, a great variety of such questions will be given under each rule as will be most likely to occur in the transaction of business. 7 5. Sir I. Newton was born in the year 1642, and was 85 years old when he died; in what year did he die ? Ans. 1727. 6. 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 8th 38; how many scholars in all the schools ? Ans. 251. 7. How many days in the 12 calendar months? Ans. 365. 8. I have 100 bushels of wheat worth 125 dollars, 150 bushels of rye worth 90 dollars, and 90 bushels of corn, worth 45 dollars; how much grain have I, and what is it worth ? Ans. 340 bushels worth 270 dollars. 9. There are two numbers, the least is 2575, and the difference is 1448, what is the greater? Ans. 4023. 10. A man killed 4 hogs weighing as follows; one 371, one 510, one 472, and one 396 pounds; what did they all weigh ? Ans. 1749 lbs. 11. A man killed an ox, the meat weighed 642, the hide 105, and the tallow 92 pounds; what did they all weigh? 4. When are they simple? When 5. What is Simple Addition? compound ? 6. What is the whole or total num ber called? 10. Ans. 859 lbs. How do you write down the numbers to be added? Where do you begin the addition? How is the amount of each column to be set down? What do you observe respecting the sum of the last column ? 11. How is Addition proved? 12. Why do you carry for 10 rather than any other number 2. Simple Subtraction. SIMPLE SUBTRACTION is the method of taking a less number from a greater of the same denomination, so as to find the difference between them; as, 5 taken from 8, there remains 3, the difference. The greater of the given numbers is called the Minuend, the less, the Subtrahend, and the difference, the Remainder. Rule.* Write the less number under the greater, with units under units, tens under tens, and so on, and draw a line below them. Beginning at the right hand, take each figure of the subtrahend from the figure over it in the minuend, and set the remainder directly below. If the figure in the lower line be greater than the one over it, suppose 10 to be added to the upper figure, always remembering, in such cases, to carry 1 to the next figure in the lower line, with which proceed as before; and so on till the whole is finished. Proof. Add the remainder to the subtrahend, and if their sum be equal to the minuend, the work is right. Examples. 1. From 6485 subtract 4293. Minuend. 6485 Having placed the numbers as the rule directs, Subtrahend. 4293 begin at the right hand, and say, 3 from 5 there remains 2, which write down, and proceed to the Remainder. 2192 next figure, and say, 9 from 8; but 8 being less than 9, you must suppose 10 to be added to 8, mak6485 ing it 18, then say 9 from 18 there remains 9, which write down. Proceed to the next figure, but because you borrowed 10 in the last place, you must carry 1, saying 1 carried to 2 is 3, and 3 from 4 there remains 1, which write down, and proceed again; 4 from 6, there remains 2, which set down, and the work is done. Proof. PROOF. To know whether you have performed the operation correctly, * When the figures in the subtrahend are all less than their correspondent figures in the minuend, the sum of the several differences is evidently the true difference between the numbers; for as the sum of the parts is equal to the whole, so is the sum of the differences of the similar parts equal to the difference of the whole. And when the figure in the subtrahend is greater than its correspondent figure in the minuend, borrowing ten, which is the value of a unit in the next higher place, is in fact employing a unit of the next left hand figure of the minuend, before you arrive to it. But as this figure is not actually diminished, the true balance is preserved, by increasing its correspondent figure in the subtrahend by 1. If, when we borrow 10, we diminished the next figure in the minuend by 1, we should proceed more agreeably to truth, and the result would be the same as by the rule. The truth of the method of proof is obvious; for it is plain that the difference of two numbers added to the less must equal the greater. To prove Subtraction by casting out the 9's, subtract the excess of 9's in the subtrahend from the excess in the minuend, and if the remainder be equal to the excess of 9's in the remainder of the given sum, the work is supposed to be right. Ν. Β. When the excess of the remainder is less than the excess in the subtrahend, 9 must be added to it before subtracting the excesses. add the remainder 2192, to the subtrahend, 4293, and the sum 6485 being equal to the minuend, the work is right. 7. What number is that to which if you add 645, it will become 1826? Ans. 1183. 8. How many years from the flight of Mahomet in 622, to A.D. 1826? Ans. 1204 years. 9. America was discovered by Columbus in 1492, how many years since ? 10. A owed B 4850 dollars, of which he paid at one time 200 dollars, at another 475, at another 40, at another 1200, and at another 156; what remains due ? Ans. 2779 dollars. 11. The sum of two numbers is 64892, and the greater number is 46234, what is the smaller QUESTIONS. 1. What is Simple Subtraction? 2. What are the given numbers called? Ans. 1620. 5. Where do you begin to subtract? 6. What is to be done when the figure in the lower line is larger than the one over it? 3. What is the difference between them called? 4. How do you write down the numbers for Subtraction? 7. In such cases what do you do to the next figure in the lower line ? 8. How do you prove Subtraction? |