Note 1.-When only the first quotient has a remainder belonging to it, then this remainder will be the true or total remainder, the same as if the division had been performed all at once. But if there is any other remainder, then, to find the total remainder, proceed as follows: When only two divisors are used, multiply the last remainder by the first divisor; to the product add the first remainder, and the sum will be the total remainder. When more than two divisors are used, multiply the last remainder by the divisor used for finding the next foregoing quotient; to the product add the remainder, if any, belonging to that quotient; then multiply the sum by the next foregoing divisor, adding in the corresponding remainder, if any, and so on through all the divisors to the first; and the result will be the whole remainder required. After having found the total remainder, you may place it over the given divisor, and annex the fraction to the quotient, as usual.

Complete quotient, or Ans. 13803. 22 total remainder.

the method of finding the whole remainder from the several particular ones, will best appear from the nature of Vulgar Fractions. Thus, in the first example above, the first remainder being 1, when the divisor is 7, makes : this must be added to the second remainder, 3, making 3 to be divided by the second divisor, viz. by 5. But 34= 3×7+1 22

; and this divided by 5, gives

value of the remainders.

22

22

the

Note 2.-To find out the component parts of a number greater than 144, proceed as follows; viz. Find, by trial, whether any of the numbers between 1 and 13 will divide the given number without a remainder; and if any of them will, then divide, and set down the quotient. Divide this quotient in like manner by some number between 1 and 13, if it can be done without leaving a remainder; and so proceed, till you find a quotient less than 13. Then the several divisors and the quotient last found, will be the component parts required; that is, their product will be equal to the given number. Thus, it may be readily found that the given divisor in example 2d is divisable by 8, and the quotient thus obtained, by 7; then the last quotient will be 12, which being less than 13, the division need not be continued any farther. So, 8×7×12=672.

3. Divide 5210015 by 81.

4. Divide 81799 by 96.

5. Divide 538865 by 144.

6. Divide 52120 by 1728. 7. Divide 2179045 by 252.

Ans. 643214.
Ans. 852

Ans. 37421474.
Ans. 30 1728.

APPLICATION OF DIVISION.

280

Ans. 864752.

Note. When the value, or weight, &c, of any number of articles is given, to find the value of each one of them, when they are all alike, or the average value of each, when they are unlike: Divide the value, &c. of the whole, by the number of articles, and the quotient will be the answer.

*

*This rule is the converse of that given in the Note which precedes the practical questions in Multiplication; and the reason of it is very obvious.

EXAMPLES.

1. What is the value of a pound of flour, if 24 pounds be worth 96 cents?. Here 96-24-4 cents, the Ans. 2. If a farm containing 365 acres, be worth 8395 dollars, what is the value of each acre? Ans. 23 dollars. 3. A farmer butchered 25 hogs, which together weighed 6525 pounds: what was the average weight of each hog? Ans. 261 pounds. 4. The holy bible contains 1189 chapters. How many chapters must a man read in a day, to read the bible through in a year, or 365 days? Ans. 34 chapters.

94

5. The Hudson and Erie canal, in the State of NewYork, is 363 miles long; and the whole expense of making it was about 7260000 dollars: What was the average expense for each mile? Ans. 20000 dollars.

6. Supposing a man labors a year for 132 dollars; how much has he a month? Ans. 11 dollars.

7. If an estate, valued at 25340 dollars, be divided equally among 7 heirs, what will be the value of each portion? Ans. 3620 dollars.

8. It is calculated that the number of square miles of land in the whole world is about fifty millions, and that the whole number of inhabitants is six hundred and fifty millions: What is the average number to each square mile of land? Ans. 13. 9. A man intending to go a journey of 1200 miles, would complete the same in 50 days: I demand how many miles he must travel each day? Ans. 24.

QUESTIONS ON THE FOREGOING.

1. What is division? 2. What is the dividend? divisor? quotient? 3. What is simple division? 4. What is the first case in simple division? 5. How are the dividend and divisor to be set down? 6. How many figures of the dividend must be taken for the first dividual? 7. Where is the first quotient figure to be placed? 8. What do you take for the second dividual? 9. When the number thus formed is less than the divisor, how do you proceed? 10. What is the second case? 11. How do you place the dividend and divisor in this case; and where do you place the quotient, when found? 12. When you have found a quotient figure, how do you make out a subtrahend; and from what do you subtract it? 13. How then do you make out a new dividual?

14. If, after annexing a figure to the remainder, it is still less than the divisor, how do you proceed? 15. How do you perform division when there are ciphers on the right hand of the divisor; and how do you find the true remainder? 16. How do you perform division when the divisor is a composite number; and how do you find the true or total remainder? 17. How do you prove division? 18. When the amount or value of several articles is given, how do you find the average value of each?

SUPPLEMENT TO MULTIPLICATION.*

Note.-A Vulgar or Common Fraction, is expressed by two numbers, placed one above the other, with a line between them; as 1, 4, &c.; the upper number being called the numerator, and the lower number the denominator.-A mixed number, is composed of a whole number and a fraction; as 42, 75, &c.

PROBLEM I. To multiply a whole number by a fraction.

RULE.-Multiply the whole number by the numerator of the fraction, and divide the product by the denominator; and the quotient will be the required product of the whole number and fraction. When the numerator of the fraction is 1, the answer will be found by merely dividing the whole number by the denominator of the fraction.

*The rules which are given in the Supplements to Multiplication and Division, properly belong to Vulgar Fractions; but, as the learner will have occasion to use them before he comes to the sections on fractions, I think it necessary to insert them here. The reason of these rules will appear from the rules for the multiplication and division of vulgar frac tions.

PROBLEM II.-To find the product of a whole number and a mixed number.

RULE.-1. Find the product of the whole numbers, by multiplying them together as usual.

2. Multiply the factor which is a whole number by the fraction belonging to the other factor, as in Prob. I.; add the result to the product of the whole numbers, and the sum will be the total product required.

To divide a whole number by a mixed number.

RULE.

1. Multiply the integral part of the divisor by the denominator of the fraction, and take the sum for a new divi

sor.

2. Multiply the dividend by the fraction belonging to the given divisor, and take the product for a new dividend.

3. Divide the new dividend by the new divisor, and the quotient will be the answer sought.

1. Divide 415 by 21.

EXAMPLES.

Here 2×4+1-9, the new divisor; and 415×4=1660, the new dividend. Then 1660÷9-1844, the Ans.

2. Divide 630 by 8%.

3. Divide 4518 by 1251.

4. Divide 140 by 55.

Ans. 75.

Ans. 36.

Ans. 2718.