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ILLUSTRATION. — 29 pounds = 1 pounds. If 2 cost 60 cents, š will cost i of 60 cents = 5 cents; and , or a pound, will cost 5 times 5 cents = 25 cents.

Ans. $0.25. 50. How many times will 60 contain 2? ? 51. Paid $54 for 75 barrels of oil ; what cost 1 barrel ?

Ans. $7. 52. How many times is 75 contained in 54 ?

53. How many cords of wood, at $5 per cord, can be bought for $66 ?

54. How many times will 66 contain 54 ? 55. Gave $40 for 63 yards of broadcloth ; what cost 1 yard ? 56. How many times is 6 contained in 40 ?

57. The distance between two places is 110 rods. I wish to divide this distance into spaces of 54 rods each. Required the number of spaces.

VII. CONTRACTIONS IN MULTIPLICATION AND

DIVISION.*

CONTRACTIONS IN MULTIPLICATION.

Art. 61. To multiply by 25.

Ex. 1. Multiply 876581 by 25.

Ans. 21914525.

OPERATION.

We multiply by 100, by an4) 8 7 6 5 8100

nexing two ciphers to the multi

plicand ; and since 25, the multi2191 4 5 25 Product.

plier, is only one fourth of 100, we divide by 4 to obtain the true product.

RULE. Annex two ciphers to the multiplicand, and divide it by 4.

* If the principles on which these contractions depend are considered too difficult for the young pupil to understand at this stage of his progress, they may be omitted for the present, and attended to when he is further advanced.

QUESTIONS. – What is the rule for dividing when there are ciphers on the right of the divisor ? — Art. 61. What is the rule for multiplying by 25 ? What is the reason for the rule ?

EXAMPLES FOR PRACTICE.

2. Multiply 76589658 by 25.

Ans. 1914741450. 3. Multiply 567898717 by 25. Ans. 14197467925. 4. Multiply 123456789 by 25. Ans. 3086419725. Art. 62. To multiply by 334. Ex. 1. Multiply 87678963 by 337. Ans. 2922632100.

We multiply by 100, as be3 ) 8 7 6 7 8 9 6 3 0 0

and since 335, the mul

tiplier, is only one third of 292 2 6 3 2100 Product. 100, we divide by 3 to obtaiu

the true product. RULE. · Annex two ciphers to the multiplicand, and divide it by 3.

OPERATION.

fore ;

EXAMPLES FOR PRACTICE.

2. Multiply 356789541 by 33. 3. Multiply 871132182 by 331. 4. Multiply 583647912 by 337.

Ans. 11892984700.
Ans. 29037739400.
Ans. 19454930400.

Art. 63. To multiply by 125.
Ex. 1. Multiply 7896538 by 125.

Ans. 987067250.

OPERATION.

We multiply by 1000, by 8) 78 965 3 8 000

annexing three ciphers to the 9 8 706 7250 Product. multiplicand ; and since 125,

the multiplier, is only one eighth of 1000, we divide by 8 to obtain the true product.

RULE. Annex three ciphers to the multiplicand, and divide it by 8.

EXAMPLES FOR PRACTICE.

2. Multiply 7965325 by 125. 3. Multiply 1234567 by 125. 4. Multiply 3049862 by 125.

Ans. 995665625.
Ans. 154320875.
Ans. 381232750.

QUESTIONS. — Art. 62. What is the rule for multiplying by 33 ? What is the reason for this rule ? - Art. 63. What is the rule for multiplying by 125 ? Give the reason for the rule.

ART. 64. To multiply by any number of 9's.
Ex. 1. Multiply 4789653 by 99999. Ans. 478960510347.
OPERATION

By adding 1 to any num4 7 8 9 6 5 3 0 0 0 0 0

ber composed of nines, we 4 7 8 9 6 5 3

obtain a number expressed

by 1 with as many ciphers 47 8 9 6 0510 347 Product. annexed as there are nines in

the number to which 1 is added. Thus, 999+1= 1000. Therefore, annexing to the multiplicand as many ciphers as there are nines in the multiplier is the same thing as multiplying the number by a multiplier too large by 1, and subtracting the number to be multiplied from this enlarged product will give the true product.

RULE. - Annex as many ciphers to the multiplicand as there are 9's in the multiplier, and from this number subtract the number to be multiplied.

EAXMPLES FOR PRACTICE.

2. Multiply 1234567 by 999.
3. Multiply 876543 by 999999.
4. Multiply 999999 by 999999.

Ans. 1233332433. Ans. 876542123457. Ans. 999998000001.

CONTRACTIONS IN DIVISION.

OPERATION.

Art. 65. To divide by 25.
Ex.1. Divide 1234567 by 25.

Ans. 4938268

Multiplying the dividend by 4 makes 1 2 3 4 5 6 7

it four times too great ; therefore, to 4

obtain the true quotient, we must di

vide by 100, a divisor four times greater 49 3 8 216 8 Quotient.

than the true one. This we do by cut

ting off two figures on the right. RULE. — Multiply the dividend by 4, and divide the product by 100.

EXAMPLES FOR PRACTICE. 2. Divide 9876525 by 25.

Ans. 395061. 3. Divide 1378925 by 25.

Ans. 55157. 4. Divide 899999 by 25.

Ans. 35999208

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QUESTIONS. — Art. 64. What is the rule for multiplying by any number of 9's? What is the reason for the rule ? — Art. 65. What is the rule for dividing by 25 ? Give the reason for the rule.

OPERATION.

Art. 66. To divide by 331.
Ex. 1. Divide 6789543 by 33}.

Ans. 203686 2007
Multiplying the dividend hy 3 makes

it three times too great ; therefore, to 6 7 8 9 5 4 3

obtain the true quotient, we must divide by 100, a divisor three times greater

This is done by 203 6 8 612 9 Quotient. than the true one.

cutting off two figures on the right. RULE. Multiply the dividend by 3, and divide the product by 100

3

EXAMPLES FOR PRACTICE.

2. Divide 987654321 by 334.
3. Divide 8712378 by 331.
4. Divide 4789536 by 331.
5. Divide 89676 by 331.
6. Divide 17854 by 33.

Ans. 29629629 63

Ans. 261371 34
Ans. 143686187

Ans. 26908
Ans. 535.000

OPERATION.

ART. 67. To divide by 125.

. Ex. 1. Divide 9874725 by 125.

Ans. 78997

Multiplying the dividend by 8 makes 9 8 7 4 7 25

it eight times too great; therefore, to 8

obtain the true quotient, we must divide by 1000, a divisor eight times greater

than the true one. We do this by cut 7 8 9 9 718 0 0 Quotient.

ting off three figures on the right. RULE. - Multiply the dividend by 8, and divide the product by 1000.

EXAMPLES FOR PRACTICE.

2. Divide 1728125 by 125.
3. Divide 478763250 by 125.
4. Divide 591234875 by 125.
5. Divide 489648 by 125.
6. Divide 836184 by 125.

Ans. 13825. Ans. 3830106. Ans. 4729879. Ans. 3917 1846 Ans. 6689472

1000

QUESTIONS. — Art. 66. What is the rule for dividing by 33} ? Give the reason for the rule. · Art. 67. What is the rule for dividing by 125? What is the reason for the rule ?

Ø VIII. MISCELLANEOUS EXAMPLES,

INVOLVING THE FOREGOING RULES.

gain?

1. A BOUGIIT 73 hogsheads of molasses at 29 dollars per hogshead, and sold it at 37 dollars per hogshead; what did be

Ans. 584 dollars. 2. B bought 896 acres of wild land at 15 dollars per acre, and sold it at 43 dollars per acre; what did he gain ?

Ans. 25088 dollars. 3. N. Gage sold 47 bushels of corn at 57 cents per bushel, which cost him only 37 cents per bushel ; how many cents did he gain?

Ans. 940 cents. 4. A butcher bought a lot of beef weighing 765 pounds at 11 cents per pound, and sold it at 9 cents per pound; how many cents did he lose?

Ans. 1530 cents. 5. A taverner bought 29 loads of hay at 17 dollars per load, and 76 cords of wood at 5 dollars a cord; what was the amount of the hay and the wood ?

Ans. 873 dollars. 6. Bought 17 yards of cotton at 15 cents per yard, 46 gallons of molasses at 28 cents per gallon, 16 pounds of tea at 76 cents a pound, and 107 pounds of coffee at 14 cents a pound; what was the amount of my bill ?

Ans. 4257 cents. 7. A man travelled 78 days, and each day he walked 27 miles; what was the length of his journey ?

Ans. 2106 miles. 8. A man sets out from Boston to travel to New York, the distance being 223 miles, and walks 27 miles a day for 6 days in succession; what distance remains to be travelled ?

Ans. 61 miles. 9. What cost a farm of 365 acres at 97 dollars per acre ?

Ans. 35405 dollars. 10. Bought 376 oxen at 36 dollars per ox, 169 cows at 27 dollars each, 765 sheep at 4 dollars per head, and 79 elegant horses at 275 dollars each; what was paid for all ?

Ans. 42884 dollars. 11. J. Barker has a fine orchard, consisting of 365 trees, and each tree produces 7 barrels of apples, and these apples will bring him in market 3 dollars per barrel; what is the income of the orchard ?

Ans. 7665 dollars. 12. J. Peabody bought of E. Ames 7 yards of his best broadcloth at 9 dollars per yard, and in payment he gave

Ames

a

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