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7. Find the sum of 1 of 16, and {of of 27. Thus of

=1, and į of of 1=1, then and reduced to a common denominator, will become and 24 which added together, make 37=1514. Ans. 8. Add together 1 of 25, and of 15

Ans. 2012: 9. Find the sum of ã and of of 13.

Ans. 39. 10. Add together }, }, and of of 51.

Ans. 316 Fractions of different integers (as found in money, weight, &c.) must be reduced to those of the same, before adding ; or the value of each fraction may be found by Problem VIII. page 101, and then added together.

11. Add of a shilling to of a pound. Thus, ks. reduced to the fraction of a pound, by Prob. IV. page 171, equal to lo£; then, to £; and f£ added together=140 of a pound, whose value by Prob. VIII. page 94,=4s: 9d. 3qrs. Ans.

Or thus, s.=0s. 4d. 2qrs.

and j£=4 5 1f

Ans. as before 4 93} 12. Add á of a lb. Troy to g of a pwt.

Ans. 10oz. Opwt. Ogrs. 13. Add together of a hogshead, i of a gallon, and of á quart.

Ans. Ilgal. Oqt. Opt. 14. Find the sum of of a cwt. and is of a pound:

Ans. Iqr. 41b. 6oz. 2:dr. 15. Add 5 of a mile, and 15 of a yard together.

Ans. 4fur. 98yds. 2ft. 1 in. 16. Add 1 of a week, } of a day, } of an hour, and of à minute together.

Ans. 2d. 2h. 30min. 45sec.



Prepare the fractions as in addition ; then the difference of the numerators written over the common denominator, will be the difference of the fractions required.


1. From take of Thus, fof=s, then and is reduced to their least common denominator by Rule 2,


page 160, become 1 and , and the difference between the numerators, 21 and 12, is 9, which written over the common denominator, becomes = Ans. 2. From take of g.

Remains 3. From 15 take } of

Rem. 4. From 4take 3.

Rem. *=116 5. What is the difference between į and is? 6. From 14. take } of 17).

Ans. 10. 7. From 38 take

Ans. Note. To subtract a fraction from a whole number, take the numerator from its denominator, and place the remainder over the denominator, then take 1 froin the whole number.

To subtract mixed numbers without reducing them to improper fractions.

When the lower fraction is greater than the upper one; subtract its numerator from the common derominator, and to the difference add the upper numerator; then carry 1 to the lower whole number:

8. From 16 subtract. Thus, taken from leaves, that is, the numerator 5, subtracted from the denominator ; leaves 4, which placed over the denominator is g, then 1 taken from 16, leaves 15.

Ans. 15 9. From 25 take it.

Ans. 243. 10. From 29 take 15; thus, and reduced to a The lower fraction being common denominator. greater than the upper one,

make 12 and 12 we proceed thus ; subthen 293=2912 tracted from 13 leaves is 151 =15

and (the upper fraction)

added to i=ii (that is, we Ans. 1311

subtract the numerator of the lower fraction (9) from its denominator (12) and to the difference add the upper numerator (8) makes 11=t) we then carry 1 to the whole number, and subtract, and the rem. is 1311 11. What is the difference between 145 and 197.

Ans. 476 12. From 36 take 97.


Ans 26



Reduce Compound fractions to simple ones, and whole and mixed numbers to improper fractions. Then multiply the numerators together for a new numerator, and the denominators together for a new denominator


51 TO3

1. What is the product of 51 multiplied by } of ?

thus, 5}=10, and } of = foi then 10 x1=. Ans. 2. Multiply by

Ans. = 3. Multiply Hi by š.

Ans, 4. Multiply 41 by š.

Ans. 5. Multiply 1 of by off

Ans. 6. Multiply of 5 by of 8.

Product 21. 7. Multiply 7} by 91.

Product 695 8. Multiply 123 by z of 7.

Product 293



Prepare the fractions as before, then invert the divisot and proceed exactly as in multiplication ; and the products will be the quotient required.

How many times is į contained in ?
Thus, by the rule we invert the divisor, which becomes
; then sx g=16, or 21%, the answer.
Illustration of the Rule.-Had it been required to divide

by 2 a whole number, instead of 2 ninths, it is evident it would have given the quotient 2} eighths, or id; but the divisor, being ninths, will be contained in the dividend 9 times were the whole number is contained 1 time; there. fore the quotient is 9 times too small, and must be multiplied by 9, the denominator of the dividing fraction, and 9

=16, or 218 the answer as before. This process consists only in niultiplying the numerator of the divisor into the denominator of the dividend, and the denominator of the divisor into the numerator of the dividend.


1. Divide ž by

Thus, x=, Ans. 2. Divide ii by

Quotient=123 3. Divide 1 of by of

Quotient 18=

; 4. What is the quotient of 75 divided by of 7 ?

Åns. 488=136 5. Divide 9} by 16

Quotient 99%. 6. Divide 4 by of 4.

Quotient 220 7. Divide of 4 by 45.

Quotient 2 8. How many times is 3h contained in of 27?

Ans. 12195 9. How many times is a contained in ? Ans. 317 10. Divide 19 by 37.

Quotient 11. At g of a dollar per bushel, how many bushels of oats can be bought for $195, or $112? Ans. 44 bu.

12. At of a dollar per bushel, how much corn can be bought for 27 dollars ?

Ans. 31}} bushels.



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Prepare the fractions as in multiplication: iken state the question in the same manner as taught in the Rule of Three in whole numbers ; then invert* the first term and multiply all three of the terms continually together, and the prod. uct will be the answer in the same name of the second or middle term.


bbl. $


1. If of a barrel of flour cost of a dollar, what will of a barrel cost ?

Thus, į: Then the divisor or first term being inverted will stand thus i xx}=$70, and $2= 5dollars, the answer

The same, by analysis.—If cost of a dollar 1 barrel,

* The reason of inverting the first term in this rule is very evident, since fractions depend on the same principle as whole numbers, and this produces the same effect as that of multiplying the second and third terms together and divi. ding by the first; therefore, the first term being a divisor, we invert it as taught in Division of Vulgar Fractions



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RULE OF THREE DIRECT IN VULGAR FRACTIONS. 177 will cost 8 times as much. Thus, 8 times a=40; then of $4=$120, or 5 dollars, the answer as before.

2. If of a bushel of wheat cost of a dollar, what will 8} bushels cost ?

Thus, : 1 :: 5, and xx_1=9=$10%), Ans. 3. If 1081 dollars will buy 8} bushels of wheat, how much wheat will į of a dollar buy? Thus, 1080=867, and 8!=*; then 86 ::: :, Ans.

4. If of a yard cost of a pound, what will is of an Ell English come to?

fyd.=of 1 of }=40, or Ell English: Then, as 1 : 1:15; and 4 ***** = 13 8 21 5. If of a cwt. cost of a dollar, what will 15cwt. come to ?

Ans. $20, 30c. 86 m. + 6. If of of an acre of land cost $9, what will 301 acres come to ?

Ans. $2041,871 7. If of a vessel cost 1236 dollars, what are y of her worth?

Ans. $206. 8. A merchant sold 51 pieces of cloth, each containing 12 yards, at 9s. id. per yard ; what did the whole come to ?

Ans. £31 9s. 10d. 3}qrs. 9. At $35 per cwt., what will ozlbs. come to ?

Ans. $86, or 31cts. 2 m.+ 10. A person owning of a vessel sold of his share for 835 dollars ; what was the whole vessel worth at that raie ?

Ans. 1565 dollars 62 cents.



Prepare the given fractions, and state the question as in direct proportion ; then invert the third term and multiply all three of the terms together, the product will be the answer in the same name of the middle term.


1. How much flannel that is of a yard wide, will line 5 yards of cloth which is 1į yards wide ? yds. wide. yds.

yards. As 11:53 :: }; and x 22 x=3 = 143 Answer.

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