and 5 units. We can now, in the mind, suppose I ten taken from the 8 tens, which wonld leave 7 tens, and this i teu we can suppose joined to the 5 units, making 15. We can now take 7 from 15, as before, and there will remain o, which we set down. The taking of i ten out of 8 tens, and joining it with the 5 units, is called bor. rowing 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 m'.king the upper figure i less, calling it 7, we may make the lover rigere i 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? 713 -- 471 how many ? Ans. 242 dollars. 20. 1612 – 463 = how many ? Ans. 1147. 21. 43751 - 6732 = = how many? Ans. 36969. 18. The pupil will readily perceive, that subtraction is the rererse of addition. 22. A man bought 40 sheep, and sold 18 of them ; how many had he leit? 40 – 13 = how many ? Ans. 22 sheep. 23. A man sold 18 sheep, and had 22 left; how many had he at first ? 18 +- 22 how many ? 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 ? Hence, sul traction 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 minuind. 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 +6=21; proof, 21 - 6= 15, and 15 -8=7, and 7 -1=0, Froin the remarks and illustrations now given, we deduce the following RULE. 1. Write down the numliers, the less under the greater, placing units under units, tens under tens, &c., and draw a line under them. Il. Beginning with units, take successively each figure in the loider numer from the figure over it, and wriie the remainder direcily !ielow. 11. When the figure in the lower number exceeds the figure over it, suppose 10 to be al:led to the upper figure; boue in this case we niust a.14 I to the lower figure in the next coluinn, before subtraci. This is called borrowing 10. - how many 5 years? EXAMPLES FOR PRACTICE. 27. It 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; ín 1920 it was 1,659,854 ; what was the increase in 20 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 numher? 4. What is the result or answer called ? 5. What is the sign of subtraction ? 6. 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 hy 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 year 1826, in what vear was he born ? 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 6330. 7. In New York, hy the census of 1820, there were 123,706 inhahitants; 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 iwo daughters, and the remainder to his son; what wils his son's share? 9. From seventeen million take filig six thousand, and what will remain ? 10. What number, together with these three, viz. 1301, 2561, and 3120, will make ten ihousand ? il. 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 lo cancel the debt? Ans. 2ten doll. Lilis and 8 one doll. 13. The greater of two numbers is 24, and the less is 16; whai is their diference ? 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 greiller ? 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 pious 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 35 dollars, and his butter for 29 dollars; he receivedl, 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 hoy bouglit a sled for 28 cents, and gave 14 cents to have it repaireit; hie sold it for 40 cents; did he gain or lose by the bargain ? and how inoch? 20. One man travels 67 miles in a day, another man follows at the rate of 42 miles a day; if they lioth start from the saine place at the same time, how far will they be apart at the close of the first dar? of the second ? of the third ? of the fourth ? 21. One man starts from Boston Monday morning, and travels at the rate of 40 miles a day; another starts from the some place Tuesday morning, and follows on at the rate of 70 miles a day; how far are ihey apart Tuesday night? Ans. 10 miies. 22. Amiin, owing 379 dollars, paid at one time 47 dollars, at another time 84 dollars, at another time 23 dollars, and at another time 143 dollars; how !noch did he then owe? Ans. 82 dollars. 23. A man has property, 10 the amount of 34764 dollars, but thiere are demands against him to the amount of 14297 dollars; how many doliars will be left after the payment of his delis? 24. Four men benght a lot of land for 482 dollars; the first man paid 274 dollars, the second man 194 dollars less than the first, and ihe thiri mau 20 dollars less than the second ; how much did the second, the third, and the fourth man pay ? The second paid 80. Ans. The third paid 60. The fourth paid 63. 23. A man, having 10,000 dollars, gave away 9 dollars; how many had he left ? Ans. 9991. MULTIPLICATION 19. 1. If one orange costs 5 cents, how many cents must I give for 2 oranges ? how many cents for 3 oranges ? for 4 oranges ? 2. One lushel of apples costs 20 cents ; how many cents must I give for 2 bushels ? -for 3 hushels ? 3. One gallon contains 4 quarts; how many quarts in 2 gallons ? in 3 gallons ? in 4 gallons ? 4. Three men bought a horse ; each man paid 23 dollars for his share ; how many dollars did the horse cost them? 5. A man has 4 farms worth 324 dollars each; how many dollars are they all worth ? 6. In one dollar there are one hundred cents; how many cents in 5 dollars? 7. How much will 4 pair of shoes cost at 2 dollars a pair ? 8. How much will two pounds of tea cost at 43 cents a pound ? 9. There are 24 hours in one day; how many hours in 2 days ? in 3 days? in 4 days? in 7 days? 10. Six boys met a beggar, and gave him 15 cents each; how many cents did the beggar receive ? When questions occur, (as in the above examples,) where the same number is to be added to itself several times, the operation may he inuch facilitated by a rule called Multiplication, in which the number to be repeated is called the multiplicand, and the nuinher which shows how many times the multiplicand is to be repeated is called the multiplier. The multiplicand and multiplier, when spoken of collectively, are called the factors, (producers,) and the answer is called the product. 11. There is an orchard in which there are 5 rows of trees, and 27 trees in each row; how many trees in the orchard ? In this example, it is eviIn the first row, 27 trees. dent that the whole number second 27 of trees will be equal to the third 27 amount of five 27s added fourth 27 together. fifth 27 In adding, we find that 7 taken five times amounts to In the whole orchard, 135 trees. 35. We write down the five units, and reserve the 3 lens; the amount of 2 taken five times is 10, and the 3, which we reserved, makes 13, which, written to the left of units, makes the whole nuinber of trees 135. If we have learned that 7 taken 5 times amounts to 35, and that 2 taken 5 times amounts to 10, it is plain we need write the number 27 hut once, and then, setting Multiplicand, 27 trees in each row. the multiplier under it, we Multiplier, may say, 5 times 7 are 35, writing down the 5, and reProduct, 135 trees, Ans. serving the 3 (tens) as in ad lition. Again, 5 times 2 (tens) are 10, (tens,) and 3, (tens,) which we reserved, make 13, (tens,) as before, . . 5 rows. 110. 12. There are on a board 3 rows of spots, and 4 spots in each row; how many spots on the board? * * * * A slight inspection of the figure will show, that the number of spots inay be found either by taking 4 three times, (3 times 4 are 12,) or by taking 3 four times, (4 tiines 3 are 12 ;) for we may say there are 3 rows of 4 spots each, or 4 rows of 3 spots each; * * * * therefore, we may use either of the given nuinhers for a multiplier, as best suits our convenience. We generally write the numbers as in subtraction, the larger uppermost, withi units under units, tens under tens, &c. Thus, Multiplicand, 4 spots. Note. 4 and 3 are the factors, which Multiplier, 3 rows. produce the product 12. Product, 12 Ans. Hence,-Multiplication is a short way of performing many additions; in other words - It is the method of repeuting any number any given number of times. Sign. Two short lines, crossing each other in the form of the letter X, are the sign of niultiplication. Thus, 3 X 4 = 12, signifies that three times 4 are equal to 12, or 4 times 3 are 12. Note. Before any progress can be made in this rule, the following table must be committed perfectly to memory. MULTIPLICATION TABLE 2 times) arc 0 4 X 3= 12,6 X 6= 36 8 X 9= 72 10 X 12=120 2 24 X 4 = 16,6 X 7= 42 8 X 10 = 80 2 2= 44 X 5 = 206 X 8=:48 8 X11= 85 1= 11 3= 64 X 6= 246 9= 54 8 X 12 961 2 = 22 = 84 X 7= 256 X 10 = 60 0111 3 = 33 104 8 = 32 6 1166 4 X 9=366 X 12= 72 18 5 55 144 X 10= 40.7 0 66 8=164 X I1=44 x = 7 9 4= 3611 = 4511 X 8= 88 9 = 22 X 4= 28 9 X 7= 6311 X 10=110 12 = 245 2 = 35 9 X.83 720 X 11==121 3 = = 42 9 X 9 = 8111 X 12 = 132 3 35 4 207 X 7=4 9 X 10= 90 12 X 1 = = 65 X 5= 257 X 8= 50 9 X 11 12 X 1= 12 = 95 X 6=307 63 9 X 12 = 108 12 X 2 24 125 X 7 357 10 = 70 10 X 0 012 X 3 = 155 X 8=407 X 1177 X 1= 1012 48 185 X 9 45.7 X 12 = 54 2= 2012 3 2115 X 10 =508 X0= 010 3 = 3012 72 245 X 11=55 8 X 1 4= 4012 S4 27,5 x 12 = 60 216/10 96 30 X 0= 3= 2410 9=103 33 6 X 1 32 10 70:2 X 10 120 30 6 X 2=128 5 - 40110 8 = 8012 XII = 132 90112 x 12 = 11 == 416 X 4 755010 X 10 =100 4 X 2 SIC X 8 = 64.10 X 11=110 0 2= 77 6 99 7 05 99 9 36 10 60 3 3 50112 |