Introduction to The National Arithmetic, on the Inductive System

Second Method.

OPERATION.

160
Tg
168
1926

3 x 6 x 8 = 144 numerator for
5 X 4 X 83

= 160 numerator for
7 x 4 x 6 = 168 numerator for s

4 X 6 X 8 = 192 common denominator. Note. It will be perceived, that this method does not express the fractions in so low terms as the other.

From the above illustration we deduce the following.

RULE.

Let compound fractions be reduced to simple fractions, mixed numbers to improper fractions, and whole numbers to improper fractions, by writing a unit under them; then find the least common multiple of all the denominators by the last rule, and it will be the denominator required. Divide the common multiple by each of the denominators, and multiply the quotients by the respective numerators of the fractions, and their products will be the numeralors required.

Or, multiply each numerator into mil te denominators except its own for a new numerator ; 1.nu • ' nominators into each other for a common deno 02. 2. Reduce and to a common denominator,

Ans. 18. 3. Reduce ], ts, and H.

Ans. 140, 1 4. Reduce #, P, and St.

Ans. 44, s, 19. 5. Reduce , and 1.

Ans. 1, 2, 6. Change t, 1, $, and 15. Ans. Pri 18, 198, 19 7. Change , , , and . Ans.

120, 120, 120, 121. 8. Change , %, ĝ, and . Ans. 1986, 1980, 1980, 1980° 9. Reduce }, Yo, and 7.

Ans. 16, 18,4

46,300. 10. Reduce 4, 1, 5, and 54. Ans. 43, 44, 45, 232 11. Reduce 1, 1, 1, $, , and I's. Ans. 14, 18, 31, 34, 31, 31. 12. Change $, , }, , t, and I's. Ans. 38, 38, 18, 36, 386,36

, • 13. Reduces, $, and Tz.

Ans. 38, 36, 35. 14. Change 78,541,7, and 8. Ang. 341, 24,00 32. 15. Change 4, 4, 5, 7, and 9. Ans. 4, 4, 2, 4, 6.

.
160

84.

792

360

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VIII. To reduce fractions of a lower denomination to a higher. 1. Reduce of a farthing to the fraction of a pound.

Ito Ans.

OPERATION.

1 x fqr. == d. X

This question may be anIk X Jd. = 17gs.

alyzed thus ; since 4 far

things make a penny, there 25 X iðgs. = zito £.

will be as many pence as farthings; therefore ţ of of a farthing is = f of a penny. Again, as 12 pence make a shilling, there will be tz as many shillings as pence, therefore tz of f of a penny is idg of a shilling. As 20 shillings make a pound, there will be at as many pounds as shillings, therefore ab of tóg of a shilling is also of a pound. Q. e. d.

The operation of this question may be abridged thus :

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ona

ke given fraction be reduced to a compound one by com; it with all the denominations between the given

the one to which it is required to reduce it ; then reduce fis compound fraction to a simple one. 2. Reduce 4 of a grain Troy to the fraction of a pound. 4 X 1 X 1 X 1

Ans. 7 x 24 x 20 x 12 10080 3. What part of an ounce is of a scruple ? 8 X1 X1 1

Ans. 10 X 8 X 8 80 4. What part of a ton is of an ounce ? 4 X1 X1 X1 X 1 1

Ans. 5 X 16 X 28 X 4 X 20 44800

5. What part of a mile is of a rod ? 8 X 1 X 1 1

Ans. 9 X 40 X 8 360 6. What part of 3 acres is of a square foot ? 4 X 1 X 1 X1 X1 1

Ans. 9 X 2721 X 40 X 4 X 3 294030 7. What part of 3hhds. is # of a quart? 4X1 X 1 X 1 1

Ans. 7 X 4 X 63 X 3 1323 8. What part of 3 yards square, are 3 square yards ?

Ans. si 9. What part of of a solid foot is $ of a foot solid ?

Ans. .

OPERATION.

IX. To reduce fractions of a higher denomination to a lower. 1. Reduce Tato of a pound to the fraction of a farthing.

Ans. 35 We explain this question in the following manner.

As shillings are twenutto X2=tdo=.

ths. tieths of a pound, there to X *=*= d.

will be 20 times as many 385 X *= 3tqr. Ans.

parts of a shilling in

Tato of a pound, as there are parts of a pound; therefore rado of a pound is equal to ttoo of 2 = tfdo=% of a shilling. And as

76 pence are twelfths of shillings, there will be twelve times as many parts of a penny in 7 of a shilling, as there are parts of a shilling ; therefore 7 of a shilling is equal to 7 of 1=+= of a penny. Again, as farthings are fourths of a penny, there will be 4 times as many parts of a farthing in 5 of a penny, as there are parts of a penny ; therefore 35 of a penny are equal to 35 of 4

of a farthing. Q. e. d. The operation of this question may be facilitated by the following manner.

Tavo XIX 12 X= 1966 = *tqr. Ans.

OPERATION.

Hence the following

RULE.

Let the given numerator be multiplied by all the denom. inations between it and the one to which it is to be reduced ; then place the product over this denominator, and reduce the fraction to its lowest terms. 2. What part of a grain is geto of a pound Troy?

goto X 1 X 1 X 2 =3748 = Ans. 3. Reduce tozo of a furlong to the fraction of a foot.

TITT X 4 X = 1988 = Ans. 4. What part of a square foot is 38080 of an acre ? 88050 X X4 X 272+ =#8888 = Ans.

$ 5. What part of a peck is it of a bushel ? 6. What part of a pound is ado of a cwt. ?

660

Ans. . Ans.

X. To find the value of a fraction in the known parts of the integer.

RULE.

Multiply the numerator by the next lower denomination of the integer, and divide the product by the denominator; if any thing remains, multiply it by the next less denomination, and divide as before, and so continue, as far as may be required ; and the several quotients will be the answer. 1. What is the value of 3t of a pound? Ans. 5s. 10d.