1. What is the sum of 3 of a dollar and of a dollar? As both the fractions are 9ths of the same unit, the magnitude of the parts is the same in both-the number of parts, 3 and 4, may therefore be added as whole numbers, and their sum, 7, written over 9, thus 3, expresses the sum of two given fractions.

2. What is the sum of of a yard and of a yard? As the parts denoted by the given fractions are not similar, we cannot add them by adding their numerators, 3 and 2, because the answer would be neither nor ; but if we reduce them to a common denominator, & becomes and, (240) Now each fraction denotes parts of the same unit, which are of the same magnitude, namely, 24ths; their numerators, 8 and 9, may therefore beladded; and their sum, 17, being written over 24, we have of a yard for the sum of and of a yard.

8

250. To add fractional quantities.

RULE.-Prepare them, when necessary, by changing compound fractions to single ones, (224) mixed numbers to improper fractions,(218) fractions of different integers to those of the same,(247, 248) and the whole to a common denominator, (240); and then the sum of the numerators written over the common denominator, will be the sum of the fractions required

QUESTIONS FOR PRACTICE.

3. What is the sum of and of a dollar?

}+}=&+&=5, Ans.

4. What is the sum of of a cwt. ?

and

Ans. 24.

5. What is the sum of

8. What is the sum of 3, Ans. 3

of a week and of a day?
+26=1+8=w., and ?

2d. 14h. Ans.

251. SUBTRACTION OF FRACTIONS.

ANALYSIS.

1. What is the difference between

of a dollar?

of a dollar and

evidently expresses 2 tenths more than 3 tenths; then is the difference.

2. What is the difference between 2 of a yard and g of a yard?

Here we cannot subtract from 3, for the same reason that we could not add them,(49). We therefore reduce them to a common denominator, (4, 4) and then the difference of the numerators, (9-8=1) written over 24, the common denominator, gives for the difference of the fractions.

RULE.-Prepare the fractions as for addition,(250) and then the difference of the numerators written over the common denominator will be the difference of the fractions required.

252. RULE OF THREE IN VULGAR FRAC. TIONS.

and

RULE.-Prepare the fractions by reduction, if necessary, state the questionby the general rule (198); invert the first term, and then multiply all the numerators together for a new numerator, and all the denominators together for a new denominator; the new numerator, written over the new denominator, will be the answer required.

REVIEW.

11. How would you divide a whole number by a fraction?(225) -a fraction by a fraction?

12. How may you enlarge the terms of a fraction ?(229) How diminish them?

13. How would you find the greatest common divisor of two numbers? How reduce a fraction to its lowest terms?

14. How would you find a common multiple of two numbers?(236) -the least common multiple? 15. How are fractions brought to

a common denominator ?(239)-to the least common denominator?

16. How are fractions of a higher denomination changed to a lower denomination ?(243)-into integers of a lower?-a lower denomination to a higher?-into integers of a higher?

17. Is any preparation necessary in order to add fractions ?(243)why must they have the same denominator? How are they added? How is subtraction of fractions performed? How the rule of three?

SECTION VIII.

POWERS AND ROOTS.

1. Envolution.

ANALYSIS.

253. Let A represent a line 3 feet long; if this length be multiplied by itself, the product, (3x3=) 9, feet is the area of the square, B, which measures 3 feet on every side. Hence, if a line, or a number, be multiplied by itself it is said to be squared, or because it is used twice as a factor, it is said to be raised to the second power; and the line which makes the sides of the square is called the first power; the root of the square, or its square root. Thus the square root of B=9, is A=3.

A

B

254. Again, if the square, B, be multiplied by its root, A. the product, (93)27 feet, is the volume, or content, of the cube, A C E, p. 51. [61] which measures 3 feet on every side. Hence if a line or a number he multiplied twice into itself, it is said to be cubed, or because it is employed three times as a factor, (3><3×3=27) it is said to be raised to the third power, and the line or number which shows the dimensions of the cube, is called its cube root. Thus the cube root of A C E-27, is A-3.

255. Again, if the cube. D, be multiplied by its root, A, the product, (273) 81 feet, is the content of a parallelopipedon, A CE, whose length is 9 feet, and other dimensions, 3 feet each way, equal to 3 cubes, A C E, placed end to end. Hence if a given number be multiplied 3 times into itself, or employed four times as a factor, (3×3×3×3=81)

D

it is raised to the fourth power, or biquadrate, of which the given num ber is called the fourth root.