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§ 94. To reduce fractions of different denominators to equivalent fractions having a common denominator.
RULE. I. Reduce compound fractions to simple ones, and whole or mixed numbers to improper ones.
II. Then multiply each one of the numerators by all the denominators except its own, for the new numerators, and multiply all the denominators together for a common denominator : the common denominator placed under each of the new numerators will form the several fractions sought.
1. Reduce 1, ž, and to a common denominator.
1x3x5=15 the new numerator of the 1st.
3rd. and 2x3x5=30, the common denominator. Therefore, 16, 30, and 36, are the equivalent fractions. It is plain, that this reduction does not alter the values of the several fractions, since the numerator and denominator of each are multiplied by the same number. (See Proposition V.)
2. When the numbers are small the work may be performed mentally.
Thus, } } }=26, 18, 18Here we find the first numerator by multiplying 1 by 4 and 5; the second, by multiplying 1 by 2 and 5; the third, by multiplying 2 by 4 and 2; and the common denominator by multiplying 2, 4 and 5 together.
Q. What is the first step in reducing fractions to a common denominator ? What is the second ? Does the reduction alter the values of the several fractions ? Why not? When the numbers are small, how may the work be performed ? 3. Reduce 2}, and į of } to a common denominator.
2}=; and of ?=
4. Reduce 5*, , of }, and 4, to a common denominator.
Ans. 5. Reduce , 135, and 37, to a common denominator.
Ans. 525, 1080, and 22200. 6. Reduce 4, 31, 62, to a common denominator.
Ans. 200, 62, and 1550. 7. Reduce 71, 1, 61, to a common denominator.
Ans. 1080, 248, and 920
144 8. Reduce 45, 87, and 21 of 5, to a common denominator.
Ans. $ 95. Note 1. It is often convenient to reduce fractions to a common denominator by multiplying the numerator and denominator of each fraction by such a number as shall make the denominators the same in both.
EXAMPLES. 1. Let it be required to reduce į and } to a common denominator.
We see at once that if we multiply the numerator and denominator of the first fraction by 3, and the numerator and denominator of the second by 2, that they will have a common denominator.
The two fractions will be reduced to i and . 2. Reduce } and } to a common denominator.
If we multiply both terms of the first fraction by 3 and both terms of the second by 5, we have
}=is, and =1s 3. Reduce à, la, and to a common denominator.
Ans. 1, la, 12. 4. Reduce , to a common denominator.
Ans. § 96. Note 2. To reduce fractions to their least common denominator, we have the following
RULE. I. Find the least common multiple of the denominators as in $ 87 and it will be the least denominator sought.
II. Multiply the numerator of each fraction by the quotient which arises from dividing the common multiple by the de
nominator, and the products will be the numerators of the required fractions ; under which write the least common denominator.
1. Reduce 1, ã and to their least common denominator.
X3=24x3=72 1st numerator.
x5=21x5=105 2nd numerator.
x2=28X2=56 3rd denominator.
Ans. 1, 185 and 56 2. Reduce , and is to their least common denominator.
Ans. 36, 49 and is 3. Reduce 14, 63 and 51, to their least common denominator.
Ans. 4. Reduce 3 15, 24 and , to their least common denominator.
360o 3 609 360 5. Reduce 1920L , to their least common denominator.
120, 120, 120 6. Reduce 41 390, 41 and 8 to a common denominator.
100% 100% 100% 100* 7. Reduce 33, 412, 8106, 146, to their least common denominator.
Ans. 8. Reduce j, k, l, and to fractions having the least common denominator.
Ans. 1, iz, 12, 19. 9. Reduce }, &s, and io to fractions having the least common denominator.
10. Reduce }, á, ž, 1b, and 37 to equivalent fractions having the least common denominator.
Ans. 18, 39, 48, 43, 23, 24. Q. How do you reduce fractions to their least common denominator ? Does this reduction affect the values of the fractions ?
605 450 800
Ans. 36, 90, 90
60 50 63
REDUCTION OF DENOMINATE FRACTIONS. § 97. We have seen $ 45, that a denominate number
one in which the kind of unit is denominated or expressed. For the same reason, a denominate fraction is one which expresses the kind of unit that has been divided. Such unit is called the unit of the fraction. Thus, į of a £ is a denominate fraction. It expresses that one £ is the unit which has been divided.
The fraction of a shilling is also a denominate fraction, in which the unit that has been divided is one shilling. These two fractions are of different denominations, the unit of the first being one pound, and that of the second, one shilling
Fractions, therefore, are of the same denomination when they express parts of the same unit, and of different denominations when they express parts of different units.
REDUCTION of denominate fractions consists in changing their denominations without altering their values.
Q. What is a denominate number? What is a denominate fraction ? What is the unit called ? In two-thirds of a pound, what is the unit ? In three-eighths of a shilling, what is the unit? In one-half of a foot, what is the unit ? When are fractions of the same denomination? When of different denominations ? Are one-third of a £ and onefourth of a f of the same or different denominations? One-fourth of a £ and one-sixth of a shilling? One-fifth of a shilling and one-half of a penny? What is reduction? How many shillings in a £? How many in £2? In 3? In 4? How many pence in 1s? In 2? In 3? In 28 8d? In 38 6d? In 58 8d? How many feet in 3 yards 2ft. ? How many inches?
$ 98. To reduce a denominate fraction from a lower to a higher denomination.
RULE. I. Consider how many units of the given denomination make one unit of the next higher, and place l over that number forming a second fraction.
II. Then consider how many units of the second denomination make one unit of the denomination next higher, and place 1 over that number forming a third fraction ; and so on, to the denomination to which you would reduce.
III. Connect all the fractions together, forming a compound fraction ; then reduce the compound fraction to a simple one by Case V.
EXAMPLES. 1. Reduce of a penny to the fraction of a £.
The given fraction is į of a 1 OPERATION. penny. But one penny is equal of 1 of =£ o to of a shilling: hence of a penny is equal to į of of a shilling. But one shilling is equal to 1 of a pound: hence of a penny is equal to ļof of z of a £=£10. The reason of the rule is therefore evident.
2. Reduce of a barleycorn to the denomination of yards. Since 3 barleycorns
OPERATION. make an inch, we first of ļof of į=Xã yards. place 1 over 3 : then as 12 inches make a foot, we place 1 over 12, and as 3 feet make a yard, we next place 1 over 3.
Q. How do you reduce a denominate fraction from a lower to a higher denomination? What is the first step? What the second ? What the third ? 3. Reduce oz avoirdupois to the denomination of tons.
Ans. 14336T. 4. Reduce of a pint to the fraction of a hogshead.
Ans. hhd. 5. Reduce of a shilling to the fraction of a £.
Ans. £o 6. Reduce } of a farthing to the fraction of a £.
Ans. £2880 7. Reduce z of a gallon to the fraction of a hogshead.
Ans. å hhda 8. Reduce of a shilling to the fraction of a £.
Ans. 1828 oda. 10. Reduce of a pound to the fraction of a cwt.
Ans. cut. 11. Reduce 1 of an ounce to the fraction of a ton.