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§ 99. To reduce a denominate fraction from a higher to a lower denomination.
RULE. I. Consider how many units of the next lower denomination make one unit of the given denomination, and place 1 under that number forming a second fraction.
II. Then consider how many units of the denomination still lower make one unit of the second denomination and place 1 under 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.
1. Reduce 1 of a £ to the fraction of a penny.
In this example of a pound is equal to 1 of 20 shillings. But of 40 of 1=240d. 1 shilling is equal to 12 pence; hence of a £=of 2 of Y=240d. Hence the reason of the rule is manifest.
Q. What do you first do in reducing a denominate fraction to a lower denomination? What next ?
What next? 2. Reduce cut. to the fraction of a pound.
Ans. 44816. 3. Reduce 15 of a £ to the fraction of a penny.
Ans. 31 d. 4. Reduce } of a day to the fraction of a minute.
Ans. 480m. 5. Reduce of an acre to the fraction of a pole.
Ans. P 6. Reduce of a £ to the fraction of a farthing;
Ans. 57 60 far. 7. Reduce 584 of a hogshead to the fraction of a gallon.
Ans.gal 8. Reduce to of a bushel to the fraction of a pint.
Ans. 25 pt. 9. Reduce ; of a day to the fraction of a second.
10. Reduce of a tun to the fraction of a gill.
Ans. 40320 gill.
CASE III. $ 100. To find the value of a fraction in integers of a less denomination.
RULE. I. Reduce the numerator to the next lower denomination, and then divide the result by the denominator.
II. If there be a remainder, reduce it to the denomination still less, and divide again by the denominator. Proceed in the same way to the lowest denomination. The several quotients, being connected together, will form the equivalent denominate number.
EXAMPLES. 1. What is the value of of a £?
Q. How much is one-half of a £? One-third of a shilling? One. half of a penny? How much is one-half of a lb. Avoirdupois? One. fourth of a ton? One-fourth of a cwt. ? One-half of a quarter ? One. fourth of a quarter ? One-seventh of a quarter ? One-fourteenth of a quarter ? One-twenty-eighth of a quarter? How do you find the value of a fraction in terms of integers of a less denomination ?
2. What is the value of lb. troy? Ans. 9oz. 12pwt. 3. What is the value of 6 of a cwt.? Ans. Iqr. 716. 4. What is the value of of an acre ? Ans. 2R. 20P. 5. What is the value of 1 of a £? Ans. S d. 6. What is the value of of a hogshead?
Ans. 52gal. 2qt.
7. What is the value of 39 of a hogshead ?
Ans. gal. qt. 8. What is the value of 2 of a guinea. Ans. 4s. 8d. 9. What is the value of of a lb. Troy?
Ans. 02. pwt. 10. What is the value of of a tun of wine ?
Ans. 3lhd. 31gal. 2qt.
CASE IV. $ 101. To reduce a denominate number to a fraction of a given denomination
RULE. Reduce the number to the lowest denomination mentioned in it : then if the reduction is to be made to a denomination still less, reduce as in Case II.; but if to a higher denomination reduce as in Case I.
1. Reduce 4s 7d to the fraction of a £. We first reduce the
OPERATION. given number to the
4s 7d=55d. lowest denomination Then, 55 of 1 of zb=£240 named in it, viz. pence. Then as the reduction
Ans. £ is to be made to pounds, a higher denomination, we reduce by Case I.
2. What part of a bushel is 2pk. 3qt.
We first reduce to quarts, this OPERATION. being the lowest denomination.
2pk. 3qt.=1921. We then reduce to bushels by Case I.
19 of į of 1= bu. 3. Reduce 2 feet 2 inches to the fraction of a yard.
Ans. i yd. 4. Reduce 3 gallons 2 quarts to the fraction of a hogshead.
Ans. Ehhd. 5. Reduce lqr. 71b. to the fraction of a hundred.
Ans. cwt. 6. What part of a hogshead' is 3qt. 1 pt.? Ans. ta. 7. What part of a mile is 6ft. 7in. ?
Ans. 63360° 8. What part of a mile is 1 inch ? Ans. 73360
9. What part of a month of 30 days, is 1 hour 1 minute 1 second ?
Ans. 10. What part of 1 day is 3hr. 3mi.?
1440 11. What part is 3hr. 3m. of two days ? Of 3 ? Of 4? Of 10 ? Of 25 ?
ADDITION OF VULGAR FRACTIONS.
§ 102. Addition of integer numbers teaches how to express all the units of several numbers by a single number.
Addition of fractions teaches how to express the value of several fractions by a single fraction.
It is plain, that we cannot add fractions so long as they have different units : for, } of a £ and į of a shilling make neither £1 nor 1 shilling.
Neither can we add parts of the same unit unless they are like parts; for } of a £ and of a £ make neither of a £ nor of a £. But } of a £ and of a £ may be added: they make of a £. So, of a £ and of a £ make of a £.
Hence, before fractions can be added, two things are necessary
1st. That the fractions be reduced to the same denomination.
2nd. That they be reduced to a common denominator.
Q. What does addition of integer numbers teach? What does addition of fractions teach? What two things are necessary before fractions can be added ? Can one-half of a £ be added to one-half of a shilling without reduction? Can one-half be added to one-fourth without reduction ?
§ 103. When the fractions to be added are of the same denomination and have a common denominator.
Add the numerators together, and place their sum over the common denominator: then reduce the fraction to its lowest terms, or to its equivalent mixed number.
1. Add }, , , and together.
It is evident, since all the parts are halves, that the true sum will
1+3+6+3=13 be expressed by the number of
=sum. halves: that is by thirteen two's.
Q. When the fractions are of the same denomination and have a common denominator, how do you find their sum? What is the sum of one-third and two-thirds ? Of three-fourths, one-fourth, and fourfourths? Of three-fifths, six-fifths, and two-fifths! Of three-sixths, seven-sixths, and nine-sixths? Of one-eighth, three-eighths and foureighths ? 2. Add } of a £, of a £, and į of a £ together.
'Ans. \ of a £.=£21. 3. What is the sum of ;+*+&+13+1.9. Ans. =*4. 4. What is the sum of 1+4+4+k+. Ans. 2.
§ 104. When the fractions are of the same denomination but have different denominators.
RULE. Reduce compound fractions to simple ones, mixed numbers to improper fractions, and all the fractions to a common denominator. Then add them as in Case I.
EXAMPLES. 1. Add , , and together. By reducing to a com
OPERATION. mon denominator the new 6x3x5=90 1st numerator. fractions are
4X2X5=40 2nd numerator.
2x3x2=12 3rd numerator. 20+40+1=1,
2x3x5=30 the denominator. which, by reducing to the lowest terms becomes 415.
Q. How do you add fractions which have different denominators ? How do you reduce fractions of different denominators to equivalent fractions having a common denominator? 2. Add of a £, of a £, and 5 of a £ together.
Ans. £158=£135=£1126 3. Add together, ), 41, and 6. Ans. 10318