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11. Count back by 10's from 107 to 7; thus: 107, 97, etc. 12. Count back by 10's from 95 to 5. From 79 to 9. 13. Count back by 10's from 83 to 13. From 98 to 18. 14. Subtract by 20's from 106 to 26; thus: 106, 86, etc.

WRITTEN EXERCISES,

PROCESS.

65. 1. From 643 subtract 456.

ANALYSIS.—We write the numbers as in the previous 6 43

case and begin at the right to subtract. 456

Since 6 units can not be subtracted from 3 units, we

unite with the 3 units a unit of the next higher order, 187 which is equal to 10 units, making 13 units: 6 units from

13 units leave 7 units, which we write under the units. Since one of tens was united with the units, there can be but 3 tens left. Because 5 tens can not be subtracted from 3 tens, we unite with the 3 tens as before, a unit of the next higher order, which is equal to 10 tens, making 13 tens: 5 tens from 13 tens leave 8 tens, which we write under the tens.

Since one of the hundreds was united with the tens, there are but 5 hundreds left: 4 hundreds from 5 hundreds leave 1 hundred, which we write under the hundreds. Hence the result is 187.

PROOF.—187, the remainder, plus 456, the subtrahend, equals 643, the minuend. Hence the result is correct.

66. RULE. — Write the subtrahend under the minuend, units under units, tens under tens, etc.

Begin at the right and subtract each figure of the subtrahend from the corresponding figure of the minuend, writing the result beneath.

If a figure in the minuend has a less value than the corresponding figure in the subtrahend, increase the former by ten, and subtract; then diminish by one, the units of the next higher order in the minuend, and subtract as before.

PROOF.-Add together the remainder and subtrahend. If the result be equal to the minuend the work is correct.

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33. A man set out on a journey of 861 miles. During the first day he traveled 297 miles, and during the second day 308 miles. How many miles had he yet to travel ?

34. A merchant deposited in a bank on Monday $584; on Tuesday, $759; on Wednesday, $463. He drew out $1298 during that time. How much did his deposits 'exceed what he drew out?

35. A grocer had 3715 pounds of sugar on hand. On one day he sold 1235 pounds, on the next 1317; the third day he sold to C all the sugar that remained. How many pounds did o buy?

36. I bought a horse for $637, and a cow for $317. I sold the horse for $729, and the cow for $356. How much did I gain by the sale?

37. In the first of three pavements there are 1317 bricks, in the second there are 2357, in the third there are 1719 less than in both the others. How many bricks in the third pavement?

38. In 1869 there were 264,146,900 bushels of wheat raised in the United States, and 874,120,005 bushels of corn. How much more corn than wheat was produced ?

39. A bought 351 acres of land, and B bought 27 acres more than A; B sold his land to C, who then had 537 acres. How many acres did C have at first?

40. A grocer retailed a quantity of sugar for $308.40, and so gained $106.28. How much had he paid for it?

41. The year 1870 was just 378 years after the discovery of America by Columbus. In what year did that event take place?

42. On Monday morning a bank had on hand $1826. During the day $2191 were deposited and $3412 drawn out; on Tuesday $3256 were deposited and $2164 drawn out. How many dollars were on hand Wednesday morning?

43. R. S. Hill is worth $15795, of which $2895 is invested in bank stock, $3864 in mortgages and the rest in land. How much has he invested in land?

44. Of the two pumbers 89346 and 56849, how much nearer is the one than the other to 68754?

45. The number of pupils who attended school in Boston in 1870 was 38944, and of this number 35442 attended the public schools. How many attended the other schools?

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67. 1. How many books are there in 2 piles containing 3 books each?

2. If you place 4 apples in a group, how many apples are there in 3 such groups? In 4 groups ?

3. When there are 3 roses in a cluster, how many are there in 3 clusters? In 4 clusters? In 5 clusters?

4. How many are 3+3+3+3, or four 3's?
5. How many are 4+4+4, or three 4's?
6. How many are four 4’s? Four-5's? Four 6's?

7. James bought 5 pencils at 5 cents each. How much did they cost him? How many cents are 5 times 5 cents ? How many are five 5's?

8. An orchard contains 5 rows of 6 trees each. How many trees are there in the orchard? How many trees are 5 times 6 trees? How many are 5 times 6?

9. James piled his blocks in 3 piles, each containing 5 blocks. How many blocks had he? How many are 3 times 5 blocks? How many are 3 times 5?

10. A boy earned $4 a week for 6 weeks. How much did he earn in all? How many dollars are 6 times $4? How

many are 6 times 4? 11. Harry played 5 hours per day. How many hours did he play in 6 days? How many are 6 times 5 hours? How many are 6 times 5?

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12. How does 5 times 4 compare with 4 times 5? 5 times 6 with 6 times 5?

13. When numbers are used without reference to any particular thing, they are called Abstract Numbers.

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DEFINITIONS,

68. Multiplication is a short process of finding the sum of equal numbers; or,

The process of repeating one number as many times as there are units in another.

69. The Multiplicand is the number to be repeated or multiplied.

70. The Multiplier is the number showing how many times the multiplicand is to be repeated.

71. The Product is the result obtained by multiplying.

72. The multiplicand and multiplier are called the factors of the product.

73. The Sign of Multiplication is an oblique cross: X. It is read, multiplied by, or times. When placed between two numbers it shows that they are to be multiplied together.

Thus, 4 X 3 is read, 4 multiplied by 3, or 3 times 4.

74. PRINCIPLES.—1. The multiplier must be regarded as an abstract number.

2. The multiplicand and product must be like numbers.

3. Either of the factors may be used as multiplicand or multiplier when both are abstract.

In practice, for convenience, the smaller number is generally used as multiplier.

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