ADDITION.

185. The denominator of a fraction determines the value of the fractional unit, (165); hence,

I. If two or more fractions have the same denominator, their numerators express fractional units of the same value.

II. If two or more fractions have different denominators, their numerators express fractional units of different values.

And since units of the same value only can be united into one sum, it follows,

III. That fractions can be added only when they have a common denominator.

ANALYSIS. We first reduce the given fractions to a common denominator, (III). And as the resulting fractions, 13, 25, and have the same fractional unit, (I), we add them by uniting their numerators into one sum, making 453, the answer.

186. From these principles and illustrations we derive the following general

RULE. I. To add fractions. When necessary, reduce the fractions to their least common denominator; then add the numerators and place the sum over the common denominator.

II. To add mixed numbers.

-Add the integers and fractions

separately, and then add their sums.

NOTE. All fractional results should be reduced to their lowest terms, and if improper fractions, to whole or mixed numbers.

4. What is the sum of 711, 83, 217, 51% and 43?

Ans. 283.

5. What is the sum of 37, 1237, 1387 and 58?

20. Add 41, 1053, 3003, 241 and 4721.

21. Add 4, 21, 116, 22, 516, 73, 4 and 65.

Ans. 188.

Ans. 116138.

22. Four cheeses weighed respectively 36, 423, 39,7% and 511 pounds; what was their entire weight? Ans. 16947 pounds. 23. What number is that from which if 4 be taken, the remainder will be 33?? Ans. 83.

5

54

from

24. What fraction is that which exceeds by ? 25. A beggar obtained of a dollar from one person, another, from another, and from another; how much did he

get from all?

26. A merchant sold 464 yards of cloth for $1277, 64 yards for $2265, and 765 yards for $3123; how many yards of cloth did he sell, and how much did he receive for the whole ?

Ans. 1877 yards, for $66618.

SUBTRACTION.

187. The process of subtracting one fraction from another is based upon the following principles:

I. One number can be subtracted from another only when the two numbers have the same unit value.

Hence,

II. In subtraction of fractions, the minuend and subtrahend must have a common denominator, (185, I).

fractions and express fractional units of the same value, (185, I). Then 12 fifteenths less 10 fifteenths equals 2 fifteenths, the

answer.

2. From 2381 take 248.

2381

OPERATION.

=

238

3

248-2419

213 Ans.

ANALYSIS. We first reduce the fractional parts, and, to the common denominator, 12. Since we cannot take 1 from, we add 1= 13, to 15, making 15. Then, 12 subtracted from

leaves; and carrying 1 to 24, the integral part of the subtrahend, (73, II), and subtracting, we have 213 for the entire remainder. 188. From these principles and illustrations we derive the following general

RULE. I. To subtract fractions. When necessary, reduce the fractions to their least common denominator. Subtract the numerator of the subtrahend from the numerator of the minuend, and place the difference of the new numerators over the common denominator.

II. To subtract mixed numbers. Reduce the fractional parts to a common denominator, and then subtract the fractional and integral parts separately.

NOTE. We may reduce mixed numbers to improper fractions, and subtract by the rule for fractions. But this method generally imposes the useless labor of reducing integral numbers to fractions, and fractions to integers again.

21. The sum of two numbers is 261, and the less is 7,3; what

is the greater?

Ans. 19

22. What number is that to which if you add 184, will be 978?

the sum

23. What number must you add to the sum of 1264 and 240}, to make 560§ ?

Ans. 193

24. What number is that which, added to the sum of and, will make ??

36

Ans. 3.

25. To what fraction must be added, that the sum may be § 26 From a barrel of vinegar containing 31

were drawn; how much was then left?

27. Bought a quantity of coal for $140%,

gallons, 143 gallons Ans. 16 gallons. and of lumber for

$456. Sold the coal for $7753, and the lumber for $516,3; how

much was my whole gain?

Ans. $6942.

THEORY OF MULTIPLICATION AND DIVISION OF FRACTIONS.

189. In multiplication and division of fractions, the various operations may be considered in two classes:

1st. Multiplying or dividing a fraction.

2d. Multiplying or dividing by a fraction.

190. The methods of multiplying and dividing fractions may be derived directly from the General Principles of Fractions, (174); as follows:

I. To multiply a fraction.-Multiply its numerator or divide its denominator, (174, I. and II).

II. To divide a fraction.-Divide its numerator or multiply its denominator, (174, I. and II).

GENERAL LAW.

III. Perform the required operation upon the numerator, or the opposite upon the denominator, (174, III).

191. The methods of multiplying and dividing by a fraction may be deduced as follows:

1st. The value of a fraction is the quotient of the numerator divided by the denominator (168, I). Hence,

2d. The numerator alone is as many times the value of the fraction, as there are units in the denominator.

3d. If, therefore, in multiplying by a fraction, we multiply by the numerator, this result will be too great, and must be divided by the denominator.

4th. But if in dividing by a fraction, we divide by the numerator, the resulting quotient will be too small, and must be multiplied by the denominator.

Hence, the methods of multiplying and dividing by a fraction may be stated as follows:

I. To multiply by a fraction. Multiply by the numerator and divide by the denominator, (3d).

II. To divide by a fraction.-Divide by the numerator and multiply by the denominator, (4th).