460 310 First add the units' column: 5 units and 4 units are 264 9 units, which write under the units' column. Then 85 add the tens: 6 tens and 8 tens are 14 tens, and 6 tens are 20 tens, and 1 ten are 21 tens, equal to 2 hundreds and 1 ten; write the 1 ten in the tens' place, and carry the 2 hundreds to be added with the hundreds. 2 hundreds and 4 hundreds are 6 hundreds, and 2 hundreds are 8 hundreds, and 3 hundreds are 11 hundreds, equal to 1 thousand and 1 hundred, which write down, making the sum 1119. From this example we may derive the following 1119 RULE FOR ADDITION. Write the numbers under each other, so that units shall stand under units, tens under tens, tenths under tenths, &c. Then add the figures in the right hand column. If the amount does not exceed 9, write it under that column. If it is 10 or more, write the units' figure of the amount under the column added, and add the tens with the figures of the next column. In the same manner add the figures in each succeeding column, writing down the whole amount of the last column. PROOF. Add the columns both upwards and downwards. The amounts should be alike. The population of each of the United States was, New England States. Middle States. In 1840. In 1830. In 1840. 1,918,608 2,428,921 320,823 373,306 1,348,233 1,724,033 76,748 78.085 11. How many inhabitants in the New England States in 1830? In 1840? 12. How many in the Middle States in 1830? In 1840? 13. How many in the Southern States in 1830? In 1840? 14. How many in the Western States in 1830? In 1840? 15. How many in all the states in 1830? In 1840? . 16. A farmer sells 5 loads of potatoes; viz., one load containing 37 bushels, one 54, one 46, one 25, and another 17 bushels. How many bushels are there in the five loads? 17. A butcher buys 4 hogs, one weighing 324 pounds, another 287 pounds, another 409 pounds, and another 310 pounds. How many pounds in all? 18. A farmer buys a yoke of oxen for $87.26, an ox wagon for $125.17, a horse for 75 dollars, 3 cows for $24.25 apiece. How much did they all come to? 19. A man owes to A $341.17, to B $4016.35, to C $3101.01, to D $184.16, and to E $907.40. How much does he owe them all? 20. Add $38.017, $401.601, $3918.48, $4197, $50375.18, and $0.375. 21. Add 39.018, 401.007, 8160.1, 501.6803. 50.16803. 5016.803 and 501680.3. 22. Add twenty thousand two hundred and two; three hundred and eighty thousand and thirty-eight; fifty-seven million, five hundred and seventy thousand, and five hundred and seventy; and four millions and one hundred. 23. Add three hundred and seven, -and eighty-four hundredths; five thousand and twenty-one, and ninety-three thousandths; six million, and eighty-five ten thousandths; one hundred and ten thousand, nine hundred and four, -and seven hundred and one ten thousandths; forty-eight thousand, and forty-eight thousandths. 24. Add fifteen dollars and seven cents; eighty-nine dollars and forty cents; sixty-seven dollars and eight mills; three hundred and eight dollars, nine cents and one mill. 25. A farmer carried a load of potatoes to market and peddled them as follows: 3.5 bushels for two dollars and sixty cents; 8.75 bushels for six dollars, fifty cents; 15 bushels for eleven dollars, thirty-seven cents, five mills; and 7.25 bushels for five dollars, sixty-two cents, five mills. How many bushels were there, and how much did his load come to? 26. There are three hundred and sixty-five, and twenty-five hundredths days in a year. How many days are there in 4 years? In 8 years? In 10 years? In 14 years? In 20 years? 27. A man bought 4 barrels of flour at $5.25 a barrel, 5 barrels at $5.375 a barrel, and 3 barrels at $6.31 a barrel. How many barrels did he buy, and how much did he give for them? 28. A gentleman owns a house worth $4567, a store worth $2584, 6 acres of land worth $175 per acre; he has notes amounting to $3594, railroad and bank stock worth $2106.75, and other personal property worth $2184.50. How much is he worth? 19. Add the following numbers: (1.) 134 +27. Say 130 and 20 are 150; 4 and 7 are 11, which is equal to 10 and 1; 150 and 10 are 160, and 1 are 161; therefore, 134 and 27 are 161. (2.) 138+89. 30 and 80 are 110, which added to 100 makes 210; 8 and 9 are 17, which is equal to 10 and 7; 210 and 10 are 220, and 7 are 227; therefore, 138 and 89 are 227. (3.) 275348. 200+300 are 500; 70 and 40 are 110, which added to 500 make 610; 5 and 8 are 13, equal to 10 and 3; 610 and 10 are 620, and 3 are 623; therefore, 275 and 348 are 623. (4.) 165 32. 2841165. (5.) 310+675. 844148. (6.) 697-249. (7.) 638+286. (8.) 564379. NOTE. The teacher will best judge whether such mental operations are at present too difficult for the pupil or not. They are introduced merely as hints to the teacher, and it is hoped that he will not allow the pupil to pass over such exercises without a fair trial. A few similar exercises should, if possible, form a part of every recitation in arithmetic, at least till the learner can perform them with facility. QUESTIONS. What is addition? What is the sign of equality? The sign of addition? Give an example. What is the rule for addition? What is the method of proof? SECTION III. — SUBTRACTION. 20. SUBTRACTION is the method of finding the difference between two numbers, by taking the smaller from the greater. Sign of subtraction, or minus sign, written between two numbers, shows that the latter is to be taken from the former; as 154, which means that 4 subtracted from 15 leaves 11. It is read 15 less 4 is equal to eleven; or 15 minus 4 is equal to 11. The less number, or number to be subtracted, is called the subtrahend; that from which it is to be subtracted is called the minuend. The difference is called the remainder. 12 1. How much is 1 less 1? 2 less 1? 3 12 4- 1? 5 6-1? 7-1? 8-12 9 1? 10 - 1? 11 1? 12 1? 13-1 14-1? 15- 1? 161 17-1? 18 -12 19 20-1? 1? Continue this exercise, using all the numbers from 4 to 20, as subtrahends. 21. Subtract 543 from 864, thus: Minuend, 864 Subtrahend, 543 Remainder, 321 EXAMPLES FOR PRACTICE. (1.) 459-237. (2.) 6849-3526. (3.) 7684 5322. (4.) 69184-20032. (5.) 410647-200645. (6.) 84675 — 32423. (7.) 8547016 3333016. 8. 7840196-3430096. William had 15 cents, and Henry had 9. How many more had William than Henry? Their father gave each of them 4 cents more. How many more had William than Henry then? William has spent 7 cents for paper, and Henry has spent 7 cents for pens. How many more cents has William than Henry now? From these examples we see that if the same quantity be either added to, or subtracted from, two numbers, their difference will remain the same as before. 3617 2518 From 6135 subtract 3617. In this example we cannot take 6135 7 units from 5 units; we therefore add 10 units to the 5, making 15 units; and subtracting 7 units from 15 units, we write the remainder 8. Then, as we added 10 units to the minuend, we must add 1 ten to the subtrahend, which makes 2 tens; 2 tens from 3 tens leaves 1 ten. Again, we cannot take 6 hundreds from 1 hundred; adding 10 hundreds to the 1 hundred in the upper line, makes 11 hundreds; 6 hundreds from 11 hundreds leaves 5 6135 hundreds. Then adding 10 hundred, or 1 thousand to the lower line, makes 4 thousands. 4 thousands from 6 thousands leaves 2 thousands. The remainder is 2518. The difference between two numbers shows how much must be added to the less to make the greater. We may therefore prove the above remainder to be correct, by adding it to the smaller number. · The sum just equals the larger. RULE FOR SUBTRACTION. Write the numbers as in addition, units under units, &c., placing the subtrahend under the minuend. Beginning at the right hand, subtract each lower figure from the one above it, where it can be done. Where this cannot be done, add 10 to the figure in the upper line, and subtract; and add one to the next figure of the lower line. PROOF. Add the remainder and subtrahend together. should be equal to the minuend. EXAMPLES FOR PRACTICE. The sum NOTE. Naughts may be annexed to the minuend when necessary. 20. How much is $18.00 $8.25? .108? 3487 18.3684? 21. How much is 175-18.003? 71090.007-3898.1049? $1561.001-$316 3001.01-185.0304? 22. How many more inhabitants were there in the New England States in 1840 than in 1830? In the Middle States? In the Southern States? In the Western States? 23. How much more is 816841 than 610489? 24. What must be added to 35618 to make 195816? 25. What is the difference between fifty million, three hundred and one, and seven hundred and eighty-seven thousand and ninety? 26. Subtract five hundred thousand and seventy-five,— and one hundred and eighty-three ten thousandths, from four million two hundred thousand, and sixteen hundred thousandths. |