27. 53642076 3 1 0 2 3. 28 1 2 3 4 5 6 7 29 48 6057042087094 31 6 2 8 3 5 9 0 6 7 1 8 760428 6 5 3 7 6 9 2 28. 9023754 6 8 2135 28 3 4 9 6 7 326 708 9306 34 2 16 7321 2 3 6 5 4 780 24369 3050 60 70 80 900 29. What is the amount of 46723, 6742, and 986 dollars? 80. A man has three orchards; in the first there are 140 es that bear apples, and 64 trees that bear peaches; in we second, 234 trees bear apples, and 73 bear cherries; in the third, 47 trees bear plums, 36 bear pears, and 25 bear cherries; how many trees in all the orchards ? SUPPLEMENT TO NUMERATION AND ADDITION. QUESTIONS. 1. What is a single or individual thing called? 2. What is notation : 3. What are the methods of notation now in use? 4. How many are the Arabic characters or figures? 5. What is numeration? 6. What is a fundamental law in notation? 7. What is addition? 8. What is the rule for addition? 9. What is the result, or number sought, called? 10. What is the sign of addition? equality? 12. How is addition proved? 11. EXERCISES. 1. Washington was born in the year of our Lord 1732; he was 67 years old when he died; in what year of our Lord did he die. 2. The invasion of Greece by Xerxes took place 481 years before Christ; how long ago is that this current year 1827 3. There are two r.umbers, the less number is 8671, the difference between the numbers is 597; what is the greater number, 4. A man borrowed a sum of money, and paid in part 684 dollars; the sum left unpaid was 876 dollars; what was the sun borrowed? 5. There are four numbers, the first 317, the second 812, the third 1350, and the fourth as much as the other three; what is the sum of them all? 6. A gentleman left his daughter 16 thousand. 16 hundred and 16 dollars; he left his son 1800 more than his daughter; what was his son's portion, and what was the amount of the whole estate ? Ans. Son's portion, 19,416. 7. A man, at his death, left his estate to his four children, who, after paying debts to the amount of 1476 dollars, received 4768 dollars each; how much was the whole estate? Ans. 20548. 8. A man bought four hogs, each weighing 375 pounds; how much did they all weigh? Ans. 1500. 9. The fore quarters of an ox weigh one hundred and eight pounds each, the hind quarters weigh one hundred and twenty-four pounds each, the hide seventy-six pounds, and the tallow sixty pounds; what is the whole weight of the ox? Ans. 600. 10. A man, being asked his age, said he was thirty-four years old when his eldest son was born, who was then fifteen years of age; what was the age of the father? 11. A man sold two cows for sixteen dollars each, twenty bushels of corn for twelve dollars, and one hundred pounds of tallow for eight dollars; what was his due ? SUBTRACTION OF SIMPLE NUMBERS. fi 6. 1. Charles, having 18 cents, bought a book, for which he gave 6 cents; how many cents had he left? 2. John had 12 apples; he gave 5 of them to his brother; how many had he left? 3. Peter played at marbles; he had 23 when he began but when he had done he had only 12; how many did he lose? 4. A man bought a cow for 17 dollars, and sold her again for 22 dollars; how many dollars did he gain? 5. Charles is 9 years old, and Andrew is 13; what is the difference in their ages? 6. A man borrowed 50 dollars, and paid all but 18; how many dollars did he pay? that is, take 18 from 50, and how many would there be left? 7. John bought a book and slate for 33 cents; he gave 8 cents for the book; what did the slate cost him? 8. Peter bought a waggon for 36 cents, and sold it for 45 cents; how many cents did he gain by the bargain? 9. Peter sold a waggon for 45 cents, which was 9 cents more than he gave for it; how many cents did he give for the waggon? 10. A boy, being asked how old he was, said that he was 25 years younger than his father, whose age was 33 years; how old was the boy? The taking of a less number from a greater (as in the foregoing examples) is called Subtraction. The greater number is called the minuend, the less number the subtro hend, and what is left after subtraction is called the differ ence, or remainder. 11. If the minuend be 8, and the subtrahend 3, what is the difference or remainder? Ans. 5. 12. If the subtrahend be 4, and the minuend 16, what is the remainder? 13. Samuel bought a book for 20 cents; he paid down 12 cents; how many cents more must he pay? SIGN. A short horizontal line, —, is the sign of subtrac tion. It is usually read minus, which is a Latin word signi fying less. It shows that the number after it is to be taken from the number before it. Thus, 8-35, is read 8 minus or less 3 is equal to 5; or, 3 from 8 leaves 5. The latter expression is to be used by the pupil in committing the following OPERATION. From 237 the minuend, 123 the remainder. 5-5=0 6 7 8 9 -5=4 10 5=5 6=0 6 1 6=2 3 6=4 7 8 9 10 - 2 3 7-7=0 9-7-2 18-7 how 28 -7= how 13: how many? 10 - 8-8=0 9 8-1 10-8=2 9 -9=0 10 9=1 many ? many ? T7. When the numbers are small, as in the foregoing examples, the taking of a less number from a greater is readily done in the mind; but when the numbers are large, the operation is most easily performed part at a time, and therefore it is necessary to write the numbers down before performing the Gegration. 14. A farmer, having a flock of 237 sheep, lost 114 of them by disease; how many had he left? Here we have 4 units to be taken from 7 units, 1 ten to be taken from 3 tens, and 1 hundred to be taken from 2 hundreds. It will therefore be most convenient to write the less number under the greater, observing, as in addition, to place units under units, tens under tens, &c. thus: We now begin with the units, saying, 4 (units) from 7, (units,) and there remain 3, (units,) which we set down directly under the column in unit's place. Then, proceed ng to the next column, we say, 1 (ten) from 3, (tens,) and here remain 2, (tens,) which we set down in ten's place. Proceeding to the next column, we say, 1 (hundred) from 2, (hundreds,) and there remains 1, (hundred,) which we set down in hundred's place, and the work is done. It now ap pears, that the number of sheep left was 123; that is, 237 - 114123. After the same manner are performed the following examples: 15. There are two farms; one is valued at 3750, and the other at 1500 dollars; what is the difference in the value of the two farms? 16. A man's property is worth 8560 dollars, but he has debts to the amount of 3500 dollars; what will remain after paying his debts? 17. James, having 15 cents, bought a pen-knife, for which he gave 7 cents; how many cents had he left? OPERATION. 15 cents. 7 cents. A difficulty presents itself here; for we cannot take 7 from 5; but we can take 7 from 15, and there will remain 8. 8 cents left. 18. A man bought a horse for 85 dollars, and a cow for 27 dollars; what did the horse cost him more than the cow? OPERATION. Horse, 85 The same difficulty meets us here as in the last example; we cannot take 7 from 5; but in the last example the larger number consisted of 1 ten ar 5 units, which Difference, 58 together make 15; we therefore took from 15. Here we have 8 tens and 5 units. We can now, in the mind, suppose 1 ten taken from the 8 tens, which would leave 7 tens, and this 1 ten we can suppose joined to the 5 units, making 15. We can now take 7 from 15, as before, and there will remain 8, which we set down. The taking of 1 ten out of 8 tens, and joining it with the 5 urits, is called borrowing ten. Proceeding to the next higher or der, or tens, we must consider the upper figure, 8, from which we borrowed, 1 less, calling it 7; then, taking 2 (tens) from 7, (tens,) there will remain 5, (tens,) which we set down, making the difference 58 dollars. Or, instead of making the upper figure 1 less, calling it 7, we may make the lower figure one more, calling it 3, and the result will be the same; for 3 from 8 leaves 5, the same as 2 from 7. |