Charles Davies.

School arithmetic. Analytical and practical online

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2. The expressions, (12-r-3)xl, (12-7-6) x 5, (12-f-4)x3, indi-
cate that the quotients are to be multiplied by 1, 5, and 3.

EXAMPLES.

Reduce the following fractions to their least common
denominator.

2. Reduce f , f , T 3 T .

3. Reduce 14f, 6-f, 5J.

4. Reduce -^ -fa, f .

5. Reduce -flfr, ^, f.

6. Reduce , 3^, 4.



1. Reduce 3|,

8. Reduce J, , j, and .

9. Reduce 2J of , 3} of 2.

10. Reduce -f, f , , and T V

11. Reduce J, f, f, I .



171. Wliat is the least common denominator of several fractions?
How do you reduce fractions to their least common denominator V



COMMON FRACTIONS.



161



OPERATION.



OPERATION.



ADDITION OF FRACTIONS.

172. Addition of Fractions is the operation of finding the
number of fractional units in two or more fractions.

1. What is the sum of J, f , and f ?

ANALYSIS. The fractional unit is the same
in each fraction, viz. : ^ ; but the numerators
show how many such units are taken (Art. 148) ;
hence, the sum of the numerators written over
tJie common denominator, expresses the sum of Ans. f =4.
the fractions.

2. What is the sum of J and f ?

ANALYSIS. In the first, the fractional unit
is , in the second it is ^. These unite, not
being of the same kind, cannot be expressed in
the same collection. But the =f, and f =$,
in each of which the unit is : hence, their
sum is ^=1^.

NOTE. Only units of the same kind, whether fractional or inte-
gral, can be expressed in the same collection,

From the above analysis, we have the following

RULE. I. When the fractions have the same denominator,
add the numerators, and place the sum over the common deno-
minator.

II. When they have not the same denominator, reduce them
to a common denominator, and then add as before.

NOTE. After the addition is performed, reduce every result to
its lowest terms.



*-*



EXAMPLES.



1. Add J, f , f , and f .

2. Add |, f , and f

3. Addf, f,^,an

4. Add t^,^, an

5. Add f , .ft, and ft.

6. Add i, |, f, and ft.

7. Add |, I fc and ft.



8. Add |, I i and -ft.

9. Add 9, |, T V, f , and f .

10. Add J, f , f , 1, and f

11. Add f V, f , A, and f .

12. Add |, f , and f.

13. Add T V, f , f, and f .

14. Add -!%, f, f, and ^.



162 SUBTRACTION OF

15. What is the sum of 19}, 6, and 4|?

OPERATION.

Whole numbers. Fractions.

19 + 6+4=29^ ^ *++*=*=

17o. NOTE. When there are mixed numbers, add the uhole,
numbers and fractions separately, and then add their sums.

Find the sums of the following fractions :

16. Add 3J, 7y%, 12f, 1?. 20. Add 900 T V, 450,

17. Add 16, 9|, 25, T . 21. AddJof T 3 T of T to

18. Add | of |, 4. of 9, 14 T V 22. Add 17| to f of 27$.

19. Add 2 T 8 T , 6, and 12-if. 23. Add $, 7J, and 8|.

24. What is the sum cf | of 12 of 7|, and $ of 25 ?

25. What is the sum of -fa of 9f and -^ of 328f ?

174. 1. What is the sum of -J- and ?

NOTE. If each of the two fractions has OPERATION.

1 for a numerator, the sum of the frac- A- +1 c + 5 il
tions will be equal to the sum of their _5 + G _

denominators divided by their product. ~5 j^~ G " ao'

2. What is the sum of | and ^- ? of and T V ?

3. What is the sum of -f and -fa ? of T \j- and y 1 ^- ? of T ^
andi?

4. What is the sum of J and yV? f 1 and ? of J
and yV ?

SUBTRACTION OF FRACTIONS.

175. SUBTRACTION of Fractions is the operation of finding
the difference between two fractions.



173. What is addition of fractions ? When the fractional unit is the
same, what is the sum of the fractions ? What units may be expressed
in the same collection ? What is the rule for the addition of fractions ?

173. When there are mixed numbers, how do you add ?

174. When two fractions have 1 for a numerator, what is their sum
equal to ?

175. What is subtraction of fractions ?



COMMON FBACTIONS.



163



1. What is the difference between and f ?

ANALYSIS. In this example the fractional unit
is i : there are 5 such units in the minuend and
3 in the subtrahend : their difference is 2 eighths ;
therefore, 2 is written over the common denomi-
nator 8.



2. From J^. take -i

3. From -| take f .



4. From

5. From



OPERATION.



take
take



OPERATION.



i . 4

jj, __ y* _ g _
ttr ~T1F TT



6. What is the difference between and



ANALYSIS. Reduce both to the same frac-
tional unit -^ : then, there are 10 sucli units
in the minuend and 4 in the subtrahend:
hence, the difference is 6 twelfths.



From the above analysis we have the following

RULE. I. When the fractions have the same denominator,
subtract the less numerator from the greater, and place the
difference over the common denominator.

II. When they have not the same denominator, reduce them
lo a common denominator, and then subtract as before.

EXAMPLES.
Make the following subtractions :



1 . From -f- take f.

2. From f take f.

3. From - take -



4. From 1, take -fifo.

5. From of 12, take ff of J.

6. F'mf of 1J of 7, take j. off.

7. From f of J of J take -ft of of 1.

8. From of J of 6J, take f of f of f .

9. From T * T of f of J, take ^ of ^.

10. What is the difference between 41 and



OPERATION.



or >



16i MULTIPLICATION OF

176. Therefore : When there are mixed numbers, change
both to improper fractions and subtract as in Art. 11.5 ; or,
subtract the integral and fractional numbers separately, and
write the results.

11. From S4-& take 16J. | 12. From 246f take 164.

13. From 7 take 4} : ^ =1 ft. and 1=^.

NOTE. Since we cannot take & from -/,- we OPERATION.

borrow 1, or ||, from the minuend, which added 7*=7T&-

to ^r=H J then f f from f leaves f<f. We must 41 4 V
now carry 1 to the next figure of the subtrahend

and proceed as in subtraction of simple numbers. Ans. 2|-^

14. From 16* take 5f 16. From 36f take 27^.

15. From 26f take 19f It. From 400 T \ take 327*.

18. From J take ^.

NOTE. When the numerators are 1, OPERATION.

the difference of the two fractions is l_ T i_ = l-_.

equal to the difference of the denomina- i __ i _
tors divided by their product

19. What is the difference between ^ and J ? Between

iand T V? ^and^V? A and -



MULTIPLICATION OF FRACTIONS.

177. MULTIPLICATION of Fractions is the operation of taking
one number as many times as there are units in another,
when one of the numbers is fractional, or when they are both
fractional.

1. If one yard of cloth cost of a dollar, what will 4 yards
cost?

ANALYSIS. Four yards will cost 4 OPERATION.

times as much as 1 yard; if 1 yard J x4r=A ^ 2J
costs 5 eighths of a dollar, 4 yards will

cost 4 times 5 eighths of a dollar, which are 20 eighths ; equal to
2i dollars.

176. When there are mixed numbers, how do you subtract? Explain
the case when the fractional part of the subtrahend is the greater ?

177. What is multiplication of fractions ?



COMMON FKACTIOHS. 1G5



OPERATION.

W /\ ^ tj j -



2d. If we divide the denominator by 4, OR,

the fraction will be multiplied by 4 (Prop. o



II) : performing the operation, we obtain,
which 2i : hence,



To multiply a fraction by a whole number : Multiply the
numerator, or divide the denominator by the multiplier.



EXAMPLES.



1. Multiply -^ by 12.

2. Multiply | by 7.

3. Multiply iff. by 9.



4. Multiply 1 T ~JL by 5.

5. Multiply -J-ff by 49.

6. Multiply i^f by 26.

7. If 1 dollar will buy f of a cord of wood, how much will
15 dollars buy ?

8. At | of a dollar a pound, what will 12 pounds of tea
cost ?

9. If a horse cats J of a bushel of oats in a day, how much
will 18 horses eat ?

10. What will 64 pounds of cheese cost, at -^ of a dollar
a pound ?

11. If a man travel 2 of a mile an hour, how far will he
travel in 16 hours?

12. At f of a cent a pound, what will 45 pounds of chalk
cost?

13. If a man receive -^ of a dollar for 1 day's labor, how
much will he receive for 15 days ?

14. If a family consume ^ of a barrel of flour in 1 month,
how much will they consume in 9 months ?

15. If a person pays -j-J- of a dollar a month for tobacco,
how much does he pay in 1 8 months ?

181. To multiply a whole number by a fraction.
1. At 15 dollars a ton, what will |- of a ton of hay cost?

ANALYSIS. 1st. Four fifths of a ton will
cost 4 times as much as 1 fifth of a ton ; if OPERATION.

1 ton cost 15 dollars, 1 fifth will cost i of 15 (15-i-5)x4 = 12
dollars, or 3 dollars, and i will cost 4 times 3 v
dollars, which are 12 dollars.



180. How do you multiply a fraction by a whole number ?



166 MULTIPLICATION OF

OR : 2d. 4 fifths of a ton will cost 1 fifth

of 4 times the cost of 1 ton ; 4 times 15 is 60, 1 v l Z 10

and 1 fifth of 60 is 12.



4



NOTE. Both operations may be combined -"* 2

in one by the use of the vertical line and can-
cellation : hence,

| 12 Ans.

Divide the whole number by the denominator of the fraction
and multiply the quotient by the numerator ;

Or : Multiply the whole number by the numerator of the
fraction and divide the product by the denominator.



EXAMPLES.



1. Multiply 24 by ?.

2. Multiply 42 by



3. Multiply 105 by

4. Multiply 64 by



5. What is the cost of of a yard of cloth at 8 dollars a
yard ?

6. If an acre of land is valued at 75 dollars, what is -^ of
it worth ?

7. If a house is worth 320 dollars, what is T 9 ^- of it worth ?

8. If a man travel 46 miles in a day, how far does he
travel in of a day ?

9. At 18 dollars a ton, what is the cost of ^ of a ton of
hay?

10. If a man earn 480 dollars in a year, how much does
he earn in -J-J of a year ?

182. To multiply one fraction by another.

1 . If a bushel of corn cost f of a dollar, what will -f of a
bushel cost ?

OPERATION.

ANALYSIS. 5-sixths of a bushel will cost Jx|$. ^. _
times as much as 1 bushel, or 5 times 4 !

1 sixth of a bushel : i of is &, (Art. 180), g

and 5 times -fa is $=$ : hence,



8 5 =



181. How do you multiply a whole number by a fraction ?



COMMON FRACTIONS. 167

Multiply the numerators together for a new numerator and
the denominators together for a new denominator.

NOTES. 1. When the multiplier is less than 1, we do not take
the whole of the multiplicand, but only such a part of it as the
multiplier is of 1.

2. When the multiplier is a proper fraction, multiplication does
not imply increase, as in the multiplication of Avhole numbers.
The product is the same part of the multiplicand which the multi-
plier is of 1.



EXAMPLES.



1. Multiply I by

2. Multiply A by



3. Find the pro't of

4. Find the pro't of f ft, f \.



|, J,



5. If silk is worth ft of a dollar a yard, what is f of a yard
worth ?

6. If I own ^ of a farm and sell | of my share, what part
of the whole farm do I sell ?

7. At of a dollar a pound, what will ft of a pound of
tea cost ?

8. If a knife cost * of a dollar and a slate -f as much, what
does the slate cost ?

OPERATION.

9. Multiply 5 J by -J- of |. 5^=^ ; i O f f =-

21 v 8 - 7

NOTE. Before multiplying, * 3ir I ;
reduce both fractions to the form * i

of simple fractions.



9 | 1=1 Ans.



GENERAL EXAMPLES.



1 . Mult, l of I of 4- by

2. Mult i by $ of If.

3. Mult. J of 3 by i of



4. Mult. 5 of | of f by 4J.

5. Mult. 14 of'-f of 9 by Gf

6. Mult, f of 6 of -| by f of 4.

183. When the multiplicand is a whole and the multi-
plier a mixed number.



183. How do you multiply one fraction by another? When the
multiplier is less than 1, what part of the multiplicand is taken ? If the
fraction is proper, does multiplication imply increase ? What part is the
product of the multiplicand ?



168 DIVISION OF

7. What is the product of 48 by 8 ?

NOTE. First multiply 48 by , which gives 48 x = 8
8 ; then by 8, which gives 384, and the sum, 392 40 v Q OQJ.
is the product : hence,

392

Multiply first by the fraction, and then by the whole
number, and add the products.



8. Mult. 67 by 9,

9. Mult. 12 by



10. Mult. 108 by 1

11. Mult. 5f by 3|.



12. What is the product of 6|, 2 and J of 12 ?

13. What will 24 yards of cloth cost at 3| dollars a yard ?

14. What will 6 bushels of wheat cost at 3j dollars a
bushel ?

15. A horse eats ^\ of - of 12 tons of hay in three months ;
how much did he consume ?

16. Jf of of a dollar buy a bushel of corn, what will
^ of T 6 T of a bushel cost ?

17. What is the cost of 5| gallons of molasses at 96 J cents
a gallon ?

18. What will 7| dozen caudles cost at T 3 T of a dollar per
dozen ?

19. What must be paid for 175 barrels of flour at 7| dol-
lars a barrel ?

20. If | of -f- of 2 yards of cloth can be bought for one dol-
lar, how much can be bought for | of 13| dollars ?

21. What is the cost of 15| cords of wooc^at 3|- dollars a
cord?

DIVISION OF FRACTIONS.

184. Division of Fractions is the operation of finding a
number which multiplied by the divisor will produce the divi-
dend, when one or both of the parts are fractional.

185. To divide a fraction by a ivhole number.

1. If 4 bushels of apples cost -jj- of a dollar, what will
1 bushel cost ?

183. How may you, multiply when the multiplicand is a icJiolc and the
multiplier a mixed number?

184. What is division of fractions?

185. How do you divide a fraction by a whole number ?



COMMON FRACTIONS.



169



ANALYSIS. Since 4 bushels cost f of a dollar,
J. bushel will cost \ of f of a dollar. Dividing
the numerator of the fraction f by 4, we have
(Art. 159).



OPERATION.



Multiplying the denominator by 4 will pro- A -^-4 -^j~~-
duce the same result (Art. 160) : hence,

Divide the numerator or multiply the denominator by the
divisor.



NOTE. By the use of the vertical line and the
principles of cancellation (Art. 148), all operations
in divisions of fractions may be greatly abridged.





9 | 2=f



EXAM

1. Divide ff- by 6.
3. Divide ^f- by 9.
3. Divide ^ by 15.
4. Divide -fff by 75.


PLES.

5. Divide || by 6.
6. Divide by 12.
7. Divide if by 20.
8. Divide iff by 27.



9. If 6 horses eat T ^j of a ton of hay in 1 month, how much
will one horse eat ?

10. If 9 yards of ribbon cost f of a dollar, what will 1 yard
cost?

11. If 1 yard of cloth cost 4 dollars, how much can be
bought for f of a dollar ?

12. If 5 pounds of coffee cost if of a dollar, what will
1 pound cost ?

13. At $6 a barrel, what part of a barrel of flour can be
bought for -f of a dollar ?

14. If 10 bushels of barley cost 3J dollars, what will
1 bushel cost ?



NOTE. We reduce the mixed number to
an improper fraction and divide as in the
case of a simple fraction.



OPERATION.



J/-f-10 = i Ans.

15. If 21 pounds of raisms cost 4| dollars, what will 1
pound cost ?

16. If 12 men consume 6f pounds of meat in a day ; how
much does 1 man consume ?



170 DIVISION OF

186. To divide a whole number by a fraction.

I. At f of a dollar apiece, how many hats can be bought
for 6 dollars ?

ANALYSIS. Since of a dollar will OPERATION.

buy one hat, 6 dollars will buy as many 6-=-4-= 6x5-f-4 = 7i.
hats as is contained times in 6 ; and
as there are 5 times as many fifths as
whole things in any number, in 6 there
are 30 fifths, and 4 fifths is contained in 2 ;

30 fifths 7i times : hence, _

Invert the terms of the divisor and multiply the whole num-
ber by the new fraction.

EXAMPLES.

1. Divide 14 by J. 3. Divide 63 by f .



2. Divide 212 by



4. Divide 420 by



5. At -^ of a dollar a yard, how many yards of cloth can
be bought for 9 dollars ?

6. If a man travel ^ of a mile in 1 hour, how long will it
take him to travel 10 miles ?

7. If y of a ton of hay is worth 9 dollars, what is a ton
worth ?

187. To divide one fraction by another.

1. At f of a dollar a gallon, how much molasses can be
bought for | of a dollar ?

ANALYSIS. Since of a dollar OPERATION.

will buy 1 gallon, I of a dollar will J-T-?-= I x 4^^4

- - - - " ^ "



buy as many gallons as \ is contained g

times in \ : one is contained in I, I o

times : but & is contained 5 times as

many times as 1, or *- times ; but 2 161

fifths is contained half as many times

as i, or f $ times, equal to 2- 1 3 -j times : hence,



I. Invert the terms of the divisor.

II. Multiply the numerators together for the numerator
of the quotient, and the denominators together for the de-
nominator of the quotient.

186. How do you divide a whole number by a fraction ?



COMMON FRACTIONS. 171

NOTES. 1. If the vertical line is used, the denominator of the
dividend and the numerator of the divisor fall 011 the left, and the
other terms on the right.

2. Cancel all common factors.

3. If the dividend and divisor have a common denominator,
they will cancel, and the quotient of their numerators will be the
answer.

4. When the dividend or divisor contains a whole or mixed
number, or compound fractions, reduce them to tiie form of simple
fractions before dividing.



EXAMPLES.



1. Divide -ft by ft.

2. Divide -ft by T f .

3. Divide 3 by |f



4. Divide } of f by T ^ of 1J.

5. Divide f of 21 by f of 3|.

6. Divide 6| by 2J.



7. At l of a dollar a pound, how much butter can be
bought for | of a dollar ?

8. If 1 man consume 1^ pounds of meat in a day, how
many men would 8J- pounds supply ?

9. If 6 pounds of tea cost 4J dollars, what does it cost a
pound ?

10. At it of a dollar a basket, how many baskets of peaches
can be bought for 11^ dollars ?

11. If of a ton of coal cost 6| dollars, what will 1 ton
cost, at the same rate ?

12. How much cheese can be bought for -J of a dollar at
of a dollar a pound ?

13. A man divided 2f dollars among his children, giving
them y 7 ^ of a dollar a piece ; how many children had he ?

14. How many times will J-J- of a gallon of beer fill a vessel
holding i of f gallons ?

15. How many tunes is of ^ of 27 contained in - of i
of42?

16. If 5-J- bushels of potatoes cost 2f dollars, how much do
they cost a bushel ?

17. If John can walk 21 miles in -^ of a day, how far can
he walk in 1 day ?

18. If a turkey cost If dollars, how many can be bought
for 12f dollars ?

19. At f of | of a dollar a yard, how many yards of rib-
bon can be bought for -|i of a dollar ?

187. How do you divide one fraction by another 9



172 REDUCTION OF

REDUCTION OP COMPLEX FRACTIONS.

188. Complex Fractions are only other forms of expression

for the division of fractions : thus ; 1 is the same as % divided

by -?j ; and may be written, % x f =f =2^-.

181). To reduce a complex fraction to the form of a sim-
ple fraction.

1. Reduce _ to its simplest form.

*i

OPERATION.
4

?j^ - !=^xA =T s^ Ans.-, hence,

4 2 TJ- 09

3

RULE. Divide the numerator of the complex fraction by its
denominator,

Or : Multiply the numerator of the upper fraction into the
denominator of the loiuer,for a numerator ; and the denomi-
nator of the upper fraction into* the numerator of the lower, for
a denominator. 9

NOTES. 1. When either of the terms of a complex fraction is a
mixed number, or compound fraction, it must first be reduced to
the form of a simple fraction.

2. When the vertical line is used, the numerator of the upper and
the denominator of the lower numbers fall on the right of the verti-
cal line, and the other terms on the left.

EXAMPLES.

Reduce the following complex fractions to their simplest form :



1. Reduce jL

2. Reduce ^1



3. Reduce



4. Reduce f of i.



5 Reduce




6. Reduce f.
8f

1. Reduce



8. Reduce



__

* of 15

214f

25H



9. Reduce '^,



10 . Reduce



of 48



DENOMINATE FRACTIONS.



173



DENOMINATE FRACTIONS.

190. A DENOMINATE Fraction is one in which the unit of
the fraction is a denominate number. Thus, f of a yard is a
denominate fraction.

191. REDUCTION of denominate fractions is the operation
of changing a fraction from one denominate unit to another
without altering its value.

There are four cases :

1st. To change from a greater unit to a less, as from yards
to inches :

2d. To change from a less unit to a greater :

3d. To find the value of a fraction in integers of lower
denominations :

4th. To find the value of integers in a fraction of a larger
unit.

These cases will be arranged in sets of two and two.



192. To change from a
greater unit to & less.

1. In $ of a yard, how
many inches ?

OPERATION.

f x 3 x 12=if=20 inches.

ANALYSIS. Since in 1 yard
there are 3 feet, in f yards there
are $ times 3 feet=-^- feet. And
since in 1 foot there are 12
inches, in ^ feet there are 1 9 -
times 12 inches = I a = 20 inch's :
hence,

RULE. Multiply the frac-
tion and the products which
arise by the units of the scale,
in succession, until you reach
the unit required.



193. To change from a
less unit to a greater.

1. In 20 inches, how many
yards ?

OPERATION.

20 xAxi=H=* J ards <
ANALYSIS. Since 12 inches
make 1 foot, in 20 inches there
are as many feet as 12 inches is
contained times in 20 inches
= H feet; and as 3 feet make
1 yard, in ^ feet there are as
many yards as 3 feet is contained
times in ^ fect=|=f yards:
hence,

RULE. Divide the fraction
and the quotients which arise,
by the units of the scale, in suc-
cession, until you reach the
unit required.



188. What are complex fractions?

189. How do you reduce complex to simple fractions ?



174 DENOMINATE FRACTIONS.

NOTE. In every operation of reduction, in which there are
common factors, be sure and cancel them before making the final
multiplication.

EXAMPLES.

1. Reduce -g-f-g- of a hogshead to the fraction of a quart.

2. Reduce -^ of a bushel to the fraction of a pint.

3. Reduce -g^ir of a pound Troy to the fraction of a grain.

4. What part of a foot is -J-&TF of a furlong ?

5. What part of a minute is -^Vo- f a day ?

6. Reduce ^Vjizr f a cwt. to the fraction of an ounce.

7. Reduce f of a gallon to the fraction of a hogshead.

8. What part of a is of a shilling ?

9. What part of a hogshead is -g- of a quart ?

10. What part of a mile is -fr of a foot ?

11. Reduce 4-^0 of to the fraction of a farthing.

12. Reduce yV of an Ell Eng. to the fraction of a nail.

13. Reduce |- of a nail to the fraction of a yard ?

14. Reduce J of % of a foot to the fraction of a mile.

15. Reduce 5 ^ 7 6 of a ton to the fraction of a pound.

16. Reduce J[ of 3| pwt. to the fraction of a pound Troy.
It. What part of a mile is j of a rod ?

18. What part of an ounce is -fo of a scruple ?

19. -^f-g- of a day is what portion of 10 minutes?

20. What part of J- of a foot is yf-g- of a furlong ?

21. Reduce -g^g- of a hogshead of ale to the fraction of a
pint.

190. What is a denominate fraction ?

191. What is reduction of denominate fractions? How many casca
are there V Name them.

192. How do you change from a greater unit to a less ?

193. How do you change from a less unit to a greater ?



DENOMINATE FRACTIONS.



175



194. To find the value of
a fraction in integers of loiver
denominations.

1. What is the value of f
of a pound Troy ?

ANALYSIS. of a pound re-
duced to the fraction of an ounce
is |xl2=^. of an ounce, (Art.
177.), which is equal to 9-
ounces : f of an ounce reduced
to the fraction of a pennyweight
is | x 20=^ of a pwt., or 12pwt.

OPERATION.

burner. 4

12 oz. pwt.

Denom. 5)48(9... 12
45
3
20

5)60
60

RULE. I. Multiply the
numerator of the fraction by
the number which will re-
duce it to the next lower de-
nomination and divide the
product by the denominator.

II. If there is a remain-
der, reduce it in the same
manner, and so on, till
the lowest denomination is
obtained.



195. To find ike value of
integers in a fraction of a
higher denomination.

2. Reduce 9oz. 12pwts. to
the fraction of a pound Troy.

ANALYSIS. In 1 pound there
are 240 pennyweights: 1 pen-
ny weight is ^ of a pound ; and
9 ounces 12pwts. = l&Zpwts. is
of a pound= of a pound.



OPERATION.

1 lb. oz. pwts.
12 9.. 12
12 20
20 Num - l92_
240 Denom. 40" ~



RULE. I. Reduce the given
ntegers to the lowest de-
nomination named, and the
result will be the numerator
jf the required fraction.

II. Eeduce 1 unit of the
required denomination, to the
denomination of the numera-
or, and the result will be
he denominator of the re-
quired fraction.



EXAMPLES.

3. What is the value of - of a tun of wine ?

4. What part of a tun of wine is 3hhd. Slgal. 2gt. ?

194. How do you find the value of a fraction in integers of lower de-
nominations ?

195. How do yon find the value of integers in a fraction of a higher
denomination ?



176 ADDITION AND SUBTRACTION OF



5. What is the value of y 9 ^ of a yard ?

6. What is the value of -| of a month ?

7. What is the value of f of a chaldron ?

8. What is the value of % of a mile ?

9. What is the value of -fe of a ton ?

10. What is the value of $ of 3 days ?


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