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thirty-six dollars ; each share was valued at seven thousand eight hundred fifty-four dollars; of how many men did the company consist ?

Ans. 1234 men. 42. A tax of thirty millions fifty-six thousand four hundred sixty-five dollars is assessed equally on four thousand five hundred ninety-seven towns; what sum must each town pay?

Ans. 65381479 dollars. Art. 55. Method of operation, when the divisor is a composite number.

Ex. 1. A merchant bought 15 pieces of broadcloth for 1440 dollars, what was the value of each piece ?

Ans. 96 dollars.

The factors of 15 are 3 3) 1 4 40 dolls., cost of 15 pieces. and 5. Now, if we divide

the 1440 dollars, the cost 5) 480 dolls., cost of 5 pieces. of 15 pieces, by 3, we ob96 dolls., cost of 1 piece. evidently the cost of 5

tain 480 dollars, which is pieces, because there are 5 times 3 in 15. Then, dividing 480 dollars, the cost of 5 pieces, by 5, we get the cost of piece. Hence we deduce the following

Rule. Divide the dividend by one of the factors, and the quotient thus found by another, and thus proceed till every factor has been made a divisor, and the last quotient will be the true quotient required.

OPERATION.

EXAMPLES FOR PRACTICE.

Quotients. 2. Divide 765325 by 25 = 5 X 5.

30613 3. Divide 123396 by 84 = 7 X 12.

1469 4. Divide 611226 by 81, using its factors.

7546 5. Divide 987625 by 125, using its factors.

7901 6. Divide 17472 by 96, using its factors.

182 7. Divide 34848 by 132, using its factors.

264 Art. 56. Method of finding the true remainder, when here are several in the operation.

Ex. 1. How many months of 4 weeks each are there in 298 days, and how many days remaining?

Ans. 10 months and 18 days.

QUESTIONS. — Art. 55. What are the factors of 15? What do you get the cost of, in this example, when you divide by the factor 3 ? What, when you divide by 5? Why?' What is the rule for dividing by a composite number ?

OPERATION.

Since there are 7 days in 1 7) 298

week, we first divide the 298

days by 7, and have 42 weeks 4) 42 — 4 days

and a remainder of 4 days.

18 days. Then, since 4 weeks make 10- 2 weeks

1 month, we divide the 42 weeks by 4, and have 10 months and a remainder of 2 weeks. Now, to find the true remainder in days, it is evident that we must multiply the 2 weeks by 7, because 7 days make a week, and to the product add the 4 days; thus 2X7= 14, and 14+4=18 days for the remainder. Hence the propriety of the following

RULE. — Multiply each remainder by all the divisors preceding the one which produced it; and the first remainder being added to the sum of the products, the amount will be the true remainder.

Note. — There will be but one product to add to the first remainder, when there are only two divisors and two remainders.

Ex. 2. Divide 789 by 36, using the factors 2, 3, and 6, and find the true remainder.

Ans. 33.

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2) 789
3) 394-1, 1st Rem.
6) 131-1, 2d Rem.

5 X 3 X 2= 30, 1st Product.
1 X 2 2, 2d Product.

21-5, 3d Rem.

1, Ist Remainder. 33, true Rem.

EXAMPLES FOR PRACTICE. 3. Divide 934 by 55, using the factors 5 and 11, and find the true remainder.

Ans. 54. 4. Divide 5348 by 48, using the factors 6 and 8, and find the true remainder.

Ans. 20. 5. Divide 5873 by 84, using the factors 3, 4, and 7, and find the true remainder.

Ans. 77. 6. Divide 249237 by 1728, using the factors 12, 6, 6, and 4, and find the true remainder.

Ans. 405. Art. 57. To divide by 10, 100, &c., or 1 with ciphers at the right.

Ex. 1. Divide 356 dollars equally among 10 men, what will each man have ?

Ans. 35% dollars.

QUESTIONS. — Art. 56. When there are several remainders, what is the rule for finding the true remainder? Will you give the reason for this rule ?

OPERATION.

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Rem.

It will be remembered, that, to mul110) 3 516

tiply by 10, we annex one cipher, which

removes the figures one place to the Quotient 35 - 6 Rem. left, and thus increases their value ten

times. Now it is obvious, that, if we Or thus, 3 516.

reverse the process and cut off the righthand figure by a line, we remove the remaining figures one place to the right, and consequently diminish the value of each ten times, and thus divide the whole number by 10. The figures on the left of the line are the quotient, and the one on the right is the remainder, which may be written over the divisor and annexed to the quotient. Hence the share of each man is 359 dollars.

RULE. Cut off as many figures from the right hand of the dividend as there are ciphers in the divisor, and the figures on the left hand of the separatrix will be the quotient, and those on the right hand the remainder. EXAMPLES FOR PRACTICE.

Quotient. 2. Divide 6892 by 10.

689

2 3. Divide 4375 by 100.

75 4. Divide 24815 by 1000.

815 5. Divide 987654321123 by 100000000.

54321123 Art. 58. Method of operation, when the divisor has ciphers on the right.

Ex. 1. If I divide 5832 pounds of bread equally among 600 soldiers, what is each one's share ? Ans. 943 pounds. OPERATION.

The divisor, 600, may 100) 58132

be resolved into the fac

tors 6 and 100. We first 6) 58-32, 1st Rem. divide by the factor 100,

by cutting off two fig9 4, 2d Rem.

ures at the right, and Or thus, 600) 5832

get 58 for the quotient

and 32 for a remainder. 9-432

We then divide the quo

tient, 58, by the other factor, 6, and obtain 9 for the quotient and 4 for a remainder. The last remainder, 4, being multiplied by the divisor, 100, and 32, the first remainder added, we obtain 432 for the true remainder (Art. 56). Hence each soldier receives 943% pounds.

QUESTIONS. — Art. 57. How do you divide by 10? How does it appear that this divides the number by 10? What is the rule for dividing by 10, 100, &c.?

Art. 58. How do you divide by 600 in the example ? How does it appear that this divides the number?

RULE. Cut off the ciphers from the divisor, and the same number of figures from the right hand of the dividend. Then divide the re maining figures of the dividend by the remaining figures of the divisor, and the result will be the quotient. To complete the work, annex to the last remainder found by the operation the figures cut off from the div idend, and the whole will form the true remainder.

EXAMPLES FOR PRACTICE.

Quotients.

Rem. 2. Divide 3594 by 80.

44

74 3. Divide 79872 by 240.

332

192 4. Divide 467153 by 700.

667

253 5. Divide 13112297 by 8900.

2597 6. Divide 71897654325 by 700000000. 497654325 7. Divide 3456789123456787 by 990000.

306787 8. Divide 4766666000000 by 55550000000. 44916000000

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2157 27666t 185

60

§ VI. CONTRACTIONS IN MULTIPLICATION AND

DIVISION.*

CONTRACTIONS IN MULTIPLICATION.

OPERATION.

Art. 59. To multiply by 25.
Ex. 1. Multiply 876581 by 25.

Ans. 21914525.

We first multiply by 100, by 4) 87658100

annexing two ciphers to the mul

tiplicand, and since 25, the mul2191 4525 Product.

tiplier, 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, and the quotient is the product required.

* If the principles on which these contractions depend are considered too difficult for the young pupil to understand at this stage of his prog. ress, 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. 59. 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.

OPERATION.

ART. 60. To multiply by 331.
Ex. 1. Multiply 87678963 by 331. Ans. 2922632100.

We first multiply by 100, 3) 8767896300

as before, and since 33}, the

multiplier, is only one third 2922 632 100 Product.

of 100, we divide by 3 to ob

tain the true product. RULE. — Annex two ciphers to the multiplicand, and divide it by 3, and the quotient is the product required.

EXAMPLES FOR PRACTICE.

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

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

OPERATION.

Art. 61. To multiply by 125.
Ex. 1. Multiply 7896538 by 125. Ans. 987067250.

We multiply by 1900, by 8) 789 6538000

annexing three ciphers to the

multiplicand, and since 125, 987067250 Product.

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 by 8, and the quotient is the product.

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. 60. What is the rule for multiplying by 334 ? What is the reason for this rule ? – Art. 61. What is the rule for multiplýing by 125 ? Give the reason for the rule ?

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