¶ 7. 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 operation. 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. 8 cents left. A difficulty presents itself here; for we cannot take 7 from 5; but we can take 7 from 15, and there will remain 8. 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 and 5 units, which ́together make 15; we therefore took 7 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 units, is called borrowing ten. Proceeding to the next higher order, 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. 19. A man borrowed 713 dollars, and paid 471 dollars; how many dollars did he then owe? many? 20. 1612-465 how many? 21. 43751- 6782 how many,? 713-471= how Ans. 242 dollars. Ans. 1147. Ans. 36969. T8. The pupil will readily perceive, that subtraction is the reverse of addition. 22! A man bought 40 sheep, and sold 18 many had he left? 40-18 how many? 23. A man sold 18 sheep, and had 22 left; he at first? 18+22= how many? of them; how Ans. 22 sheep. how many had Ans. 40. 24. A man bought a horse for 75 dollars, and a cow for 16 dollars, what was the difference of the costs? 75-16= how many? Reversed, 59 +16 how many? 25. 114-103 how many? Reversed, 11+ 103 = how many? 26. 143 -76 = how many? Reversed, 67 +76 = how many? Hence, subtraction may be proved by addition, as in the foregoing examples, and addition by subtraction. To prove subtraction, we may add the remainder to the⚫ subtrahend, and, if the work is right, the amount will be equal to the minuend. To prove addition, we may subtract, successively, from the amount, the several numbers which were added to produce it, and, if the work is right, there will be no remainder. Thus 7+ 8+621; proof, 21 6 = 15, and 15 — 8 — 7, and 7 — 7 = 0. From the remarks and illustrations now given, we deduce the following RULE. I. Write down the numbers, the less under the greater, placing units under units, tens under tens, &c. and draw a line under them. II. Beginning with units, take successively each figure in the lower number from the figure over it, and write the remainder directly below. III. When the figure in the lower number exceeds the figure over it, suppose 10 to be added to the upper figure; but in this case we must add 1 to the lower figure in the next column, before subtracting. This is called borrowing 10. EXAMPLES FOR PRACTICE. 27. If a farm and the buildings on it be valued at 10000, and the buildings alone be valued at 4567 dollars, what is the value of the land? 28. The population of 'New England, at the census in 1800, was 1,232,454; in 1820 it was 1,659,854; what was the increase in 20 years? 29. What is the difference between 7,648,203 'and 928,671 ? 30. How much must you add to 358,642 to make 1,487,945 ? 31. A man bought an estate for 13,682 dollars, and sold it again for 15,293 dollars; did he gain or lose by it? and how much? 32. From 364,710,825,193 take 27,940,386,574. SUPPLEMENT TO SUBTRACTION. QUESTIONS. 1. What is subtraction? 2 What is the greater number called? 3. the less nunber? 4. What is the result or answer called? 5. What is the sign of subtraction? What is the rule? 7. What is understood by borrowing ten? 8. Of what is subtraction the reverse? 9. How is subtraction proved? 10. How is addition proved by subtraction? EXERCISES. 1. How long from the discovery of America by Columbus, in 1492, to the commencement of the Revolutionary war in 1775, which gained our Independence? 2. Supposing a man to have been born in the year 1773, how old was he in 1827 ? 3. Supposing a man to have been 80 years old in the 1826, in what year was he born? year 4. There are two numbers, whose difference is 8764; the greater number is 15687; I demand the less? 5. What number is that which, taken from 3794, leaves 865? 6. What number is that to which if you add 789, it will become 6350? 7. In New York, by the census of 1820, there were 123,706 inhabitants; in Boston, 43,940; how many more inhabitants were then in New York than in Boston? 8. A man, possessing an estate of twelve thousand dollars, gave two thousand five hundred dollars to each of his two daughters, and the remainder to his son; what was his son's share? 9. From seventeen million take fifty-six thousand, and what will remain ? 10. What number, together with these three, viz. 1301, 2561, and 3120, will make ten thousand? 11. A man bought a horse for one hundred and fourteen dollars, and a chaise for one hundred and eighty-seven dollars; how much more did he give for the chaise than for the horse? 12. A man borrows 7 ten dollar bills and 3 one dollar bills, and pays at one time 4 ten dollar bills and 5 one dollar bills; how many ten dollar bills and one dollar bills must he afterwards pay to cancel the debt? Ans. 2 ten doll. bills and 8 one doll. 13. The greater of two numbers is 24, and the less is 16; what is their difference? 14. The greater of two numbers is 24, and their difference 8; what is the less number? 15. The sum of two numbers is 40, the less is 16; what is the greater? 16. A tree, 68 feet high, was broken off by the wind; the top part, which fell, was 49 feet long; how high was the stump which was left? 17. Our ancestors landed at Plymouth, Massachusetts, in 1620; how many years since ? 18. A man carried his produce to market; he sold his pork for 45 dollars, his cheese for 38 dollars, and his butter for 29 dollars; he received, in pay, salt to the value of 17 dollars, 10 dollars worth of sugar, 5 dollars worth of molasses, and the rest in money; how much money did he receive? Ans. 80 dollars. 19. A boy bought a sled for 28 cents, and gave 14 cents C |