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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). 1. From subtract .
ANALYSIS. Reducing the 3-3= ' 11310=
given fractions to a common
denominator, the resulting fractions f; and is express fractional units of the same value, (185, I). Then 12 fifteenths less 10 fifteenths equals 2 fifteenths = 15,
2. From 2381 take 24
ANALYSIS. We first reduce the frac2381 = 238
tional parts, 1 and 5, to the common
denominator, 12. Since we cannot 24 = 24
take ifrom in, we add 1 = ti, to ja, 213. Ans.
making i. Then, i, subtracted from 1 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.
EXAMPLES FOR PRACTICE.
1 From 1 take 's 2. From za
take 3. From 35 take 3 4. From take. 5 From take zo 6. From 13 take 24.
Ans. 3 7 From take a
Ans. 15 8. From takes 9. From take is. 10. From ? take 5 11. From take 13
12 12. From take 30 13. From 16á take 76
Ans. 93. 14. From 36.11 take 8:14.
Ans. 273 15. From 2510 subtract 1411. 16. From 75 subtract 43.
Ans. 704 17. From 187 subtract 55. 48. From 26:14 subtract 251%. 19. From 286 subtract 3 4.
Anrs. 2415 20. From 78.145 subtract 323.
21. The sum of two numbers is 261, and the less is 713; what is the greater?
Ans. 1972 22. What number is that to which if
you will be 978?
23. Whát number must you add to the sum of 1264 and 2401, to make 5605 ?
Ans. 1936 24. What number is that which, added to the sum of s, in, and is, will make 36 ? 25. To what fraction must ž be added, that the sum may
be 26 From a barrel of vinegar containing 317 gallons, 147 gallons were drawn; how much was then left ? Ans. 16 gallons.
27. Bought a quantity of coal for $140%, and of lumber for $4563. Sold the coal for $775%, and the lumber for $516,6; how much was my whole gain?
THEORY OF MULTIPLICATION AND DIVISION OF FRACTIONS.
189. In multiplication and division of fractions, the various operations may be considered in two classes : - Ist. 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).
FII. 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).
III. Perform the required operation by the numerator and the opposite by the denominator.
In the first opera
192. 1. Multiply by 4.
by 4 by multiplying its nume
rator by 4; and in the second X4= = 1
operation, we multiply the frac
tion by 4 by dividing its denom5 4
inator by 4, (190, I or III). Х = 1;
In the third operation, we express the multiplier in the form
of a fraction, indicate the multiplication, and obtain the result by cancellation. 2. Multiply 21 by 4.
ANALYSIS. To multiply by 4, 21 x 4 = 84 =
12 we must multiply by 4 and di
vide by 7, (191, I or III).
In the first operation, we first 21 x 4 = 3x4=12 multiply 21 by 4, and then di
vide the product, 84, by 7.
In the second operation, we 3
first divide 21 by 7, and then RA 4 Х 12
multiply the quotient, 3, by 4. 1 方
In the third operation, we ex
press the whole number, 21, in the form of a fraction, indicate the multiplication, and obtain the result by cancellation. 3. Multiply by .
Analysis. To multiply by
3, we must multiply by 7 and 24 step, is 8= 36
divide by 8, (191, I or III).
In the first operation, we mul. 10 = Ans.
tiply s by 7 and obtain it;
5 112 16
we then divide ii by 8 and obtain XZ
ifa, which reduced gives ja, the required product. In the second
operation we obtain the same result 5 Х
by multiplying the numerators to14 8
gether for the numerator of the pro
duct, and the denominators together for the denominator of the product. In the third operation, we indicate the multiplication, and obtain the result by cancellation.
193. From these principles and illustrations we derive the following general
RULE. I. Reduce all integers and mixed numbers to improper fractions. II. Multiply together the numerators for a new numerator,
and the denominators for a new denominator.
NOTE8.—1. Cancel all factors common to numerators and denominators. 2. If a fraction be multiplied by its denominator, the product will be the
EXAMPLES FOR PRACTICE.
1. Multiply s by 8.
Ans. 23 2. Multiply by 27, i by 4, and to by 9. 3. Multiply is by 15. 4. Multiply 8 by :
Ans. 6. 5. Multiply 75 by * , 7 by 8, 756 by 5, and 572 by : 6. Multiply by 7. Multiply it by žy, and i5 by 5}. 8. Multiply izby 17, and 3 by zi. 9. Multiply 24 by 38.
Ans. 10. 10. Multiply 1by 115.
Ans. 21 11. Multiply | by 218 12. Find the value of ý x 6.X i Xž 13. Find the value of į xxx X i X 48 14. Find the value of 31 x 33 x 1's. 15. Find the value of 27 x 24 x î x 168 x 115 x 261.
Ans. 2. 16. Find the value of ý xi X 4 x 15 x 17. Find the value of Me Xo X 53430
Ans. 24 Ans. i's: