49. Examples.

Minuend, 359.7
Subtrahend, 186.3

1. From 359.7 take 186.3.

SOLUTION. We write units under units, tens under tens, tenths under tenths, and so on. Then 3 tenths from 7 tenths leaves 4 tenths, and we write 4 under the column of tenths; 6 units from 9 units leaves 3 units, and we write 3 under the column of units; since we cannot take 8 tens from 5 tens, we change one of the 3 hundreds to 10 tens and add them to the 5 tens, making 15 tens; then 8 tens from 15 tens leaves 7 tens, and we write 7 under the column of tens. As we have taken one of the 3 hundreds, we have only 2 hundreds remaining; and 1 hundred from 2 hundreds leaves 1 hundred. The remainder, therefore, is 173.4.

Remainder, 173.4

2. From 50 take 27.65.

(4)(9)(9)(10)

5 0.00

SOLUTION. Since the subtrahend contains tenths and hundredths, and the minuend has neither tenths nor hundredths, we put zeros in the place of tenths and hundredths in the minuend. As there are no hundredths, no tenths, and no units in the minuend, 1 of the 5 tens is taken, leaving 4 tens, and changed to 10 units; then 1 of the 10 units is taken, leaving 9 units, and changed to 10 tenths; then one of the 10 tenths is taken, leaving 9 tenths, and changed to 10 hundredths. That is, 50.00 is changed to 4 tens, 9 units, 9 tenths, and 10 hundredths. Then, subtracting, we have 22.35.

2 7.6 5

2 2.3 5

50. Hence, we have the following

RULE FOR SUBTRACTION. Write the subtrahend under the minuend, placing units of the same order in the same column.

Begin at the right and subtract each order of units of the subtrahend from the corresponding order of the minuend. Write the result beneath, step by step, and put in the decimal point when reached.

If any order of the minuend has fewer units than the same order of the subtrahend, increase the units of this order of the minuend by 10 and subtract; then diminish by one the units of the next higher order of the minuend.

PROOF. Add the remainder and subtrahend. If the sum equals the minuend, the work may be assumed correct.

1. 234-123. 11. 789-456.
2. 343-123. 12. 879-456.
3. 424-123. 13. 978-456.
4. 555-123. 14. 6378-456.
5. 676-123. 15. 6855-456.
6. 725-123. 16. 6853-456.
7. 839-123. 17. 7797-456.
8. 999-123. 18. 7006-456.
9. 1000123.
10. 5120-123.
31. $183.45-$76.47.

EXERCISE 10.

Find the remainder and prove :

21. 974-779.

22. 368249. 23. 2301-479. 24. 2731-929. 25. 708-394. 26. 1123-1072. 27. 891 773. 28. 8103-5621.

19.

20.

3542 - 456.
4000 - 456.

29. 19.001-3456.

32. $716.43-$628.74.
33. $647.51-$549.64.
34. $270.04 - $128.31.
35. $125-$101.50.
36. $247.93-$129.47.
37. $641.87 - $333.95.
38. $56.27 $29.89.

39. 3.1415927-2.7182818.

30. 2180-792.

51. 0.381966 -0.30103.
52. 3.1415927 — 0.7853982.
53. 2.3561945-0.7853982.
54. 1.5707963 -0.7853982.
55. 3.1415927 — 0.5235988.
56. 2.6179939 -0.5235988.
57. 2.0943951 -0.5235988.
58. 1.5707963 — 0.5235988.
59. 1.0471975-0.5235988.

40. 0.7853982 -0.5235988. 60. 1-0.381966.

41. 4.8104774-0.4342945.

61. 1.4142136 -0.618034.
0.618034 0.381966.
63. 9,873,210 - 8,765,420.
64. 8010.101 - 4187.94.
65. 1,000,000 - 817,259.
66. 729,434 613,488.
67. 6532.18-1916.47.
68. 1718.754-1389.328.
69. 21,205 — 1787.563.
70. 42,786.95-4278.695.

42. 2.5399772 — 0.3937043. 62.
43. 0.3937043-0.3047973.
44. 3.2808693-0.3047973.
45. 3.2808693 -1.6093295.
46. 3.785-0.6213768.
47. 15.4323487 -0.264.
48. 1.7320508-1.4142136.
49. 2.236068-1.7320508.
50. 2.236068-0.618034.

EXERCISE 11.

1. In a till are $391 in bills, $67.50 in gold, $39.75 in silver, and $2.77 in copper and nickel. How much money is in the till?

2. Starting out with $315.75 in one wallet and $54.37 in another, I pay the grocer $127.38; the butcher, $64.17; the shoemaker, $21.40; the landlord, $50; the tailor, $35. What ought I to have left?

3. On a bill of $753.43, I pay $517.87. How much do I still owe? If I owe $817.87, and have but $637.50, how much do I lack of being able to pay?

4. If a man was born January 1, 1812, how old was he January 1, 1878?

5. America was discovered in 1492. How many years after its discovery was each of the following events?

Settlement of Florida, 1565; of Virginia, 1607; of Massachusetts, 1620; of Quebec, 1608; French and Indian War, 1756; Declaration of Independence, 1776; Inauguration of Washington, 1789; War with England, 1812; Mexican War, 1846; Civil War, 1861.

6. The minuend is one hundred million two hundred fifty-six thousand three hundred seventy-two, and the subtrahend is nineteen million nine hundred thousand nine hundred and ninety-nine. Find the remainder.

7. If the minuend is 9874, and remainder 3185, what is the subtrahend? The subtrahend being 7659, and remainder 675.68, what is the minuend?

8. The smaller of two numbers is 7.95764328; their difference is 0.00087692. What is the larger number?

9. The larger of two numbers is 7.95764328, and their difference is 7.153485. What is the smaller number?

10. If the subtrahend is 10,542, and the difference 544.2, what is the minuend?

11. A man pumps out of a cistern in one hour 243.75 gallons; in the next hour, 227.5 gallons; in 45 minutes more, 137.75 gallons; and the cistern is empty. How many gallons of water were in it?

12. From what number must I subtract 5 to leave 7? 8 to leave 9? From what number must I subtract 5.1736 to leave 8.1964? 6.231 to leave 9.6648? 74.213 to leave 25.787 ?

13. What must be subtracted from 1 to leave 0.5? to leave 0.53? to leave 0.532 ? to leave 0.5236? to leave 0.5235988 ?

14. I start on a journey of 3433 miles. The first day I make 428 miles; the second day, 511 miles; the third, 497 miles; the fourth, 513. How many miles of my journey remained for me at the close of each day? How many miles had I gone at the close of each day?

15. Subtract 76,343 from the sum of 61,932, 51,387, 5193, 4674, and 8199; then subtract 23,657 from the remainder.

16. Jones bought a farm and stock for $7633.90; sold the stock for $305.75; then sold the farm for $7325. How much did he lose?

17. If I gave $4375 for my land, and paid for house, barn, sheds, and fences, $2789.50; also $973.75 for horses, cattle, tools, etc.; what did my farm and stock cost?

18. If I paid $8138.25 for land and cattle, and sold part of the land for $675, and part of the cattle for $217.50, what is the cost of the land and the cattle left?

19. John has 158 cents, James has 271 cents; James gives John 56 cents. Which has then more than the other, and how many cents more?

20. A cattle dealer had 228 oxen, 475 sheep, and 49 lambs; he sold 17 oxen, 64 sheep, and 7 lambs. How many animals of each kind did he then have, and how many all together?

CHAPTER III.

MULTIPLICATION.

51. Multiplication. The process of taking a number of units a number of times is called multiplication.

52. Multiplicand. The number of units taken is called the multiplicand.

53. Multiplier. The number that shows how many. times the multiplicand is taken is called the multiplier.

54. Product. The number found by multiplication is called the product.

55. The multiplier always signifies a number of times, and is, therefore, an abstract number.

56. The multiplicand and product are like numbers.

57. Factors. The numbers used in making a product are called factors of the product.

58. The product of two factors is the same whichever factor is taken as the multiplier.

Thus, 3 times 44 times 3. The dots in the mar

gin read across the page make 3 fours; read up and down the page they make 4 threes.

NOTE. The multiplicand always signifies a number of units, whether the kind of units is stated or not. The only difference between 15 and 15 horses is that in the first case the kind of units counted is not stated, and in the second case the kind is stated.

We may interchange the multiplicand and multiplier if we refer to the numbers only. Thus, in the example 3 times 4 horses, we cannot say 4 horses times 3, but we may interchange the 3 and 4, and have 4 times 3 horses. The product in either case is 12 horses. With this understanding, we may always use the smaller number as multiplier.