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bought for $175?

25. If 2 gallons of molasses cost 65 cents, what will 3
hogsheads cost ?

26. What is the cost of -6 bushels of coal at the rate of
1 Us. Qd. a chaldron?

27. What quantity of corn can I buy for 90 guineas, at the
rate of 6 shillings a bushel ?

28. A merchant failing in trade owes $3500, and his
effects are sold for $2100 : how much does B. receive, to
whom he owes $420 ?

29. If 3 yards of broadcloth cost as much as 4 yards of
cassimere, how much cassimere can be bought for 18 yards
of broadcloth ?

30. If 7 hats cost as much as 25 pair of gloves, worth 84
cents a pair, how many hats can be purchased for $216 ?

31. How many barrels of apples can be bought for $114.33,
if 7 barrels cost $21.63?

32. If 27 pounds of butter will buy 45 pounds of sugar,
how much butter will buy 36 pounds of sugar ?

33. If 42J tons of coal cost $206.21, what will be the cost
of 2J tons ?

34. If 40 gallons run into a cistern, holding 700 gallons, in
an hour, and 15 run out, in what time will it be filled ?

35. A piece of land of a certain length and 12 J rods in
width, contains 1 J acres, how much would there be in a piece
of the same length 26 f rods wide ?

36. If 13 men can be boarded 1 week for $39,585, what
will it cost to board 3 men and 6 women the same time, the
women being boarded at half price ?

37. What will 75 bushels of wheat cost, if 4 bushels 3
pecks cost $10.687?

38. What will be the cost, in United States money, of 324
yards 3qrs. of cloth, at 5s. d. New York currency, for 2
yards ?

39. At $1.12J a square foot, what will it cost to pave a
floor 18 feet long and 12ft. (tin. wide ?



222 CAUSE AND EFFECT.



CAUSE AND EFFECT.

231. Whatever produces effects, as men at work, animals
eating, time, goods purchased or sold, money lent, and the
like, may be regarded as causes.

Causes are of two kinds, simple and compound.

A simple cause has but a single element, as men at work, a
portion of time, goods purchased or sold, and the like.

A compound cause is made up of two or more simple ele-
ments, such as men at work taken in connection with time, and
the like.

232. The results of causes, as work done, provisions con-
sumed, money paid, cost of goods, and the like, may be re-
garded as effects. A simple effect is one which has but a
single element ; a compound effect is one which arises from
the multiplication of two or more elements.

233. Causes which are of the same kind, that is, which can
be reduced to the same unit, may be compared with each
other ; and effects which are of the same kind may likewise
be compared with each other. From the nature of causes and
effects, we know that

1st Cause : 2d Cause : : 1st Effect : 2d Effect ;
and, 1st Effect : 2d Effect : : 1st Cause : 2d Cause.

234. Simple causes and simple effects give rise to simple
ratios. Compound causes or compound effects give rise to
compound ratios.



331. What arc causes? How many kinds of causes are there?
What is a simple cause ? What is a compound cause ?
1 232. What are effects? What is a simple effect? What is a com-
pound effect?

233. What causes are of the same kind ? What causes may be com-
pared with each other ? What do we infer from the nature of causes
and effects ?

234. What gives rise to simple ratios ?



DOUBLE RULE OF THREE. 223

DOUBLE RULE OF THREE.

236. The Double Rule of Three is an application of the
principles of compound proportion. It embraces all that class
of questions in which the causes are compound, or in which
the effects are compound ; arid is divided into two parts :

1st When the compound causes produce the same effects ;
2<2. When the compound causes produce different effects.

237. When the compound causes produce the same effects.
1. If 6 men can dig a ditch in 40 days, what time will 30

men require to dig the same ?



ANALYSIS. The first cause



STATEMENT.



men. men.



is compounded of 6 men, and ' . on

40 days, the time required to : OIJ

do the work, and n equal to days. days.

what 1 man would do in 40 : x

G x 40=240 days. 240 : 30 xx

The second cause is com-
pounded of 30 men and the
number of days necessary to #0

do th'} same work, viz : x



ditch, ditch.
: 1 : i



But since the effects are the x ~ 8 davs -

same, viz : the work done, the causes must be equal ; hence, the
products of the elements of the causes are equal. Therefore, in the
solution of all like examples,

Write the cause containing the unknown element on the left
of the vertical line for a divisor, and the other cause on the
right for a dividend.

NOTE. This class of questions has generally been arranged
under the head of " Rule of Three Inverse."

EXAMPLES.

1. A certain work can be done in 12 days, by working 4
hours a day : how many days would it require the same
number of men to do the same work, if they worked 6 hours
a day?

336. What is the double Rule of Three ? What class of questions
does it embrace ? Into how many parts is it divided ? What are they ?

337. What is the rule when the effects are equal ? Under what rule
has this class of cases been arranged ?



224: DOUBLE RULE OF THREE.

2. A pasture of a certain extent supplies 30 horses for 18
days : how long will the same pasture supply 20 horses ?

3. If a certain quantity of food will subsist a family of 12
persons 48 days, how long will the same food subsist a family
of 8 persons ?

4. If 30 barrels of flour will subsist 100 men for 40 days,
how long will it subsist 25 men ?

5. If 90 bushels of oats will feed 40 horses for six days,
how many horses would consume the same in 1 2 days ?

6. If a man perform a journey of 22 J days, when the days
are 12 hours long, how many days will it take him to per-
form the same journey when the days are 15 hours long?

7. If a person drinks 20 bottles of wine per month when it
costs 2s. per bottle, how much must he drink without increas-
ing the expense when it costs 2s. 6e?. per bottle ?

8. If 9 men in 18 days will cut 150 acres of grass, how
many men will cut the same in 27 days ?

9. If a garrison of 536 men have provisions for 326 days,
how long will those provisions last if the garrison be increased
to 1304 men ?

10. A pasture of a certain extent having supplied a body
of horse, consisting of 3000, with forage for 18 days : how
many days would the same pasture have supplied a body of
2000 horse ?

11. What length must be cut off from a board that is 9
inches wide, to make a square foot, that is, as much as is
contained in 12 inches in length and 12 in breadth ?

12. If a certain sum of money will buy 40 bushels of oats
at 45 cents a bushel, how many bushels of barley will the
same money buy at 72 cents a bushel ?

13. If 30 barrels of flour will support 100 men for 40
days, how long would it subsist 400 men ?

14. The governor of a besieged place has provisions for 54
days, at the rate of 2/6. of bread per day, but is desirous of
prolonging the siege to 80 days in expectation of succor : what
must be the ration of bread ?



DOUBLE RULE OF THREE.



225



238. When the Compound Causes produce different
Effects.

In this class of questions, either a cause, or a single ele-
ment of a cause may. be required ; or an effect, or a single
element of an effect may be required.

1. If a family of 6 persons expend $300 in 8 months, how
much will serve a family of 15 persons for 20 months ?



ANALYSIS. In this example the second
effect is required ; and the statement may be
read thus : If 6 persons in 8 months expend
$300, 15 persons in 20 months will expend
how many (or x) dollars ?



OPERATION

15 5

( *0 & 25

X



#=1875 Ans.



STATEMENT.

1st Cause : 2d Cause : : 1st Effect : 2d Effect



15)

20 j



Or, 6x8 :. 15x20



$300
300



2. If 16 men, in 12 days, build 18 feet of wall, how many
men must be employed to build 72 feet in 8 days ?

ANALYSIS. In this example an element of
the second cause is required, viz : the number
of men. The question may be read thus :
If 16 men, in 12 days, build 18 feet of wall,
how many (or x) men, in 8 days, will build
72 feet of wall ?



. ,

* $
$



x



OPERATION.
^ ,4
" * 9
Jf
12



=96 men.



STATEMENT.

1} 1S ^

1 Q ^79

. io . \ A.



in ,

12 j
Or, 16 x 12 :

3. If 32 men build a wall 36 feet long, 8 feet high, and
4 feet thick, in 4 days, working 12 hours a day how long
a wall, that is 6 feet high, and 3 feet thick can 48 men build
in 36 days, working 9 hours a day ?



238. When the compound causes produce different effects, what will
always be required ?
15



226 DOUBLE BULE OF THKEE. '

OPERATION.



) 48 36) x

Y : 36 : : 8> : 6
) 9' 4) 3



ANALYSIS. In this example an element of the
second effect is required, viz : the length of the
wall, and the question may be read thus : If
32 men, in 4 days, working 12 hours a day,
can build a wall 36 feet long, 8 feet high, and
4 feet thick, 48 men in 36 days, working 9
hours a day, can build a wall how many (or x)
feet long, 6 feet high, and 3 feet thick ?

#1=648 feet.

STATEMENT.

32
4
12

Or, 32x4x12 : 48x36x9 : : 36x8x4 : #x6x3.
239. Hence, we have the following

RULE. I. Arrange the terms in the statement so that the
causes shall compose one couplet, and the effects the other,
putting x in the place of the required element :

II. Then if x fall in one of the extremes, make the
product of the means a dividend, and the product of the
extremes a divisor; but if x fall in one of the means, make
the product of the extremes a dividend, and the product of
the means a divisor.

EXAMPLES.

1. If I pay $24 for the transportation of 96 barrels of flour
200 miles, what must I pay for the transportation of 480 bar-
rels 75 miles ?

2. If 12 ounces of wool be sufficient to make 1| yards of
cloth 6 quarters wide, what number of pounds will be required
to make 450 yards of flannel 4 quarters wide ?

3. What will be the wages of 9 men for 11 days, if the
wages of 6 men for 14 days be $84 ?

4. How long would 406 bushels of oats last 7 horses, if 154
bushels serve 14 horses 44 days ?

. If a man travel 217 miles in 7 days, travelling 6 hours
7 tfay, how far would he travel in 9 days if he travelled 11
fiours a day ?

939. What is the rule for finding tho unknown part ?



DOUBLE SULE OF THREE. 227

6. If 27 men can mow 20 acres of grass in 5$- days, work-
ing 3f hours a day, how many acres can 10 men mow in 4|
days, by working 8 J hours a day ?

7. How long will it take 5 men to earn $11250, if 25 men
can earn $6250 in 2 years ?

8. If 15 weavers, by working 10 hours a day for 10 days,
can make 250 yards of cloth, how many must work 9 hours
a day for 15 days to make 60 7 J yards?

9. A regiment of 100 men drank 20 dollars' worth of wine
at 30 cents a bottle : how many men, drinking at the same
rate, will require 1 2 dollars' worth at 25 cents a bottle ?

10. If a footman travel 341 miles in 7^ days, travelling
12 J hours each day, in how many days, travelling 10^ hours
a day, will he travel 155 miles?

11. If 25 persons consume 300 bushels of corn in 1 year,
how much will 139 persons consume in 8 months, at the
same rate ?

12. How much hay will 32 horses eat in 120 days, if 96
horses eat 3J tons in 7| weeks ?

13. If $2. 45 will pay for painting a surface 21 feet long
and 13 J feet wide, what length of surface that is lOf feet
wide, can be painted for $31.72 ?

14. How many pounds of thread will it require to make
60 yards of 3 quarters wide, if 7 pounds make 14 yards
6 quarters wide ?

15. If 500 copies of a book, containing 210 pages, require
12 reams of paper, how much paper will be required to print
1200 copies of a book of 280 pages?

16. If a cistern 17J feet long, 10 feet wide, and 13 feet
deep, hold 546 barrels of water, how many barrels will a
cistern 12 feet long, 10 feet wide, and 7 feet deep, contain ?

17. A contractor agreed to build 24 miles of railroad in 8
months, and for this purpose employed 150 men. At the
end of 5 months but 10 miles of the road were built : how
many more men must be employed to finish the road in the
time agreed upon ?

18. If 336 men, in 5 days of 10 hours each, can dig a trench
of 5 degrees of hardness, 70 yards long 3 wide and 2 deep :
what length of trench of 6 degrees of hardness, 5 yards wide
and 3 yards deep, may be dug by 240 men in 9 days of 12
hours each ?



228 PARTNERSHIP.



PARTNERSHIP.

240. PARTNERSHIP is the joining together of two or more
persons in trade, with an agreement to share the profits or
losses.

PARTNERS are those who are united together in carrying
on business.

CAPITAL, is the amount of money or property employed :
DIVIDEND is the gain or profit :
Loss is the opposite of profit :

241. The Capital or Stock is the cause of the entire profit :
Each man's capital is the cause of his profit :

The entire profit or loss is the effect of the whole capital :
Each man's profit or loss is the effect of his capital : hence,

Wliole Stock : Each man's Stock
: : Whole profit or loss : Each man's profit or loss.

EXAMPLES.

1. A and B buy certain goods amounting to 160 dollars, of
which A pays 90 dollars and B, 70 ; they gain 32 dollars by
the purchase : what is each one's share ?

OPERATION.

160 : 90 : : 32 : A's share ; or,



160 : : 70 : 32 : B's share ; or,




240. What is a partnership ? What are partners ? What is capital
or stock ? What is dividend ? What is loss ?

241. What is the cause of the profit? What is the cause of each
man's profit? What is the effect of the whole capital ? What is the
effect of each man's capital ? What proportion exists between causes
and their effects ? What is the rule ?



COMPOUND PARTNERSHIP. 229

Hence, the following

RULE. As the whole stock is to each man's share, so is the
whole gain or loss to each man's share of the guin or loss.

EXAMPLES.

1. A and B have a joint stock of $2100, of which A owns
$1800 and B $300 ; they gain in a year $1000 : what is
each one's share of the profits ?

2. A, B and C fit out a ship for Liverpool. A contributes
$3200, B $5000, and C $4500 ; the profits of the voyage
amount to $1905 : what is the portion of each ?

3. Mr. Wilson agrees to put in 5 dollars as often as Mr.
Jones puts in 7 ; 'after raising their capital in this way, they
trade for 1 year and find their profits to be $3600 : what is
the share of each ?

4. A. B and C make up a capital of $20,000 ; B and C
each contribute twice as much as A ; but A is to receive one-
third of the profits for extra services ; at the end of the year
they have gained $4000 : what is each to receive ?

5. A, B and C agree to build a railroad and contribute
$18000 of capital, of which B pays 2 dollars, and C, 3 dollars
as often as A pays 1 dollar ; they lose $2400 by the opera-
tion : what is the loss of each ?

COMPOUND PARTNERSHIP.
242. When the causes of profit or loss are compound.

"When the partners employ their capital for different periods
of time, each cause of profit or loss is compound, being made
up of the two elements of capital and t^me. The product of
these elements, in each particular case, will be the cause of
each man's gain or loss ; and their sum will be the cause of
the entire gain or loss : hence, to find each share,

Multiply each man 1 stock by the time he continued it in
trade ; then say, as the sum of the products is to each product,
so is the whole gain or loss to each man's share of the gain or



243. "When is the cause of profit or loss compound ? What arc the
elements of the compound caus ? What is the rule in this case?



230 COMPOUND PARTNERSHIP.



EXAMPLES.

1. A and B entered into partnership. A put in $840 for 4
months, and B, $650 for 6 months ; they gained $363 : what
is each one's share ?

OPERATION.

A, $840x4-3360

B. 650 x 63900

J 3360 : : QPQ f $168 A's.
J3900 :: 363: j $195 B's.

2. A puts in trade $550 for 7 months and B puts in $1625
for 8 months ; they make a profit of $337 : what is the
share of each ?

3. A and B hires a pasture, for which they agreed to pay
$92.50. A pastures 12 horses for 9 weeks and B 11 horses
for 7 weeks : what portion must each pay ?

4. Four traders form a company. A puts in $400 for ft
months ; B $600 for 7 months ; C $960 for 8 months ; D
$1200 for 9 months. In the course of trade they lost $750 ;
how much falls to the share of each ?

5. A, B and C contribute to a capital of $15000 in the
following manner : every time A puts in 3 dollars B puts in
$5 and C, $7. A's capital remains in trade 1 year ; B's If-
years ; and C's 2f years ; at the end of the time there is a
profit of $15000 : what is the share of each ?

6. A commenced business January 1st, with a capital of
$3400. April 1st, he took B into partnership, with a capital
of $2600 ; at the expiration of the year they had gained.
$750 : what is each one's share of the gain ?

7. James Fuller, John Brown and William Dexter formed
a partnership, under the firm of Fuller, Brown & Co., with a
capital of $20000 ; of which Fuller furnished $6000, Brown
$5000, and Dexter $9000. At the expiration of 4 months,
Fuller furnished $20^)0 more ; at the expiration of 6 months,
Brown furnished $2500 more ; and at the end of a year Dex-
ter withdrew $2000. At the expiration of one year and a
half, they found their profits amounted to $5400 : what was
each partner's share ?



PERCENTAGE.



231



PERCENTAGE.

243. PERCENTAGE is an allowance made by the hundred.
The base of percentage, is the number on which the per-
centage is reckoned.

PER CENT means by the hundred : thus, 1 per cent means

1 for every hundred ; 2 per cent, 2 for every hundred ; 3 per
cent, 3 for every hundred, &c. The allowances, 1 per cent,

2 per cent, 3 per cent, &c., are called rates, and may be
expressed decimally, as in the following

TABLE.



1 per cent is


-01


7 per cent is


.07


3 per cent is


.03


3 per cent is


.08


4 per cent is


.04"


15 per cent is


.15


5 per cent is


.05


68 per cent is


.68


6 percent is


.06


99 per cent is


.99



100 per cent is 1.
150 per cent is 1.50
130 per cent is 1.30
200 per cent is 2.
. \ per cent is .005
3| per cent is .035
5| per cent is 0575



ALSO,

for, 1-0$ is equal to 1 .
for, |g is equal to 1.50
for, |$# is equal to 1.30
for, f $ is equal to 2.00
for, T-^-^2 is equal to .005
for, 3J = .03+.005 = .035
for, 5j=.05+.075 = .OT5



EXAMPLES.

Write, decimally, 8J per cent ; 9 per cent ; 6| per cent ;
65J per cent ; 205 per cent ; 327 per cent.

244. To find the percentage of any number.

1. What is the percentage of $320, the rate being 5 per
cent?



343. What is per centage? What is the base? What does per cent
mean ? What do you understand by 3 per cent ? What is the rate, or
rate per cent ?

244. How do yon find the percentage of any number ?



232 PERCENTAGE.

ANALYSIS. The rate being 5 per cent, is ex- OPERATION.
pressed decimally by .05. We are then to take 320

.05 of the base (which is $320) ; this we do by
multiplying $320 by .05.

Hence, to find the percentage of a number, $16. 00 Ans.

Multiply the number by the rate oppressed decimally, and
the product will be the percentage.

EXAMPLES.

1. What is the percentage of $657, the rate being 4J per
cent?

OPERATION

NOTE. When the rate cannot be .657

reduced to an exact decimal, it is most Q^I

convenient to multiply by the fraction,

and then by that part of the rate which 219 = | per cent,

is expressed in exact decimals. 2628 = 4 per cent.

$28.47 = 41 per cent.
Find the percentage of the following numbers :



1. 2J per cent of 650 dollars.

2. 3 per cent of 650 yards.

3. 4 per cent of Slbcwl.

4. 6J per cent of $37.50.

5. 5| per cent of 2704 miles.

6. \ per cent of 1000 oxen.
7 2| per cent of $376.

8. 2^ per cent of 860 sheep.

9. 5 per cent of $327.33.



10. 66| per cent of 420 cows.

11. 105 per cent of 850 tons.

12. 116 per cent of 875/6.

13. 241 per cent of $875.12.

14. 37J per cent of $200.

15. 33^ per cent of $687.24.

16. 87J per cent of $400.

17. 62J per cent of $600.

18. 308 per cent of $225.40.



19. A has $852 deposited in the bank, and wishes to draw
out 5 per cent of it : how much must he draw for ?

20. A merchant has 1200 barrels of flour : he shipped
64 per cent of it and sold the remainder : how much did he
sell?

21. A merchant bought 1200 hogsheads of molasses. On
getting it into his store, he found it short 3| per cent : how
many hogsheads were wanting ?

1 22. What is the difference between 5| per cent of $800
and 6J per cent of $1050?



PERCENTAGE. 233

23. Two men had each $240. One of them spends 14
per cent, and the other 18| per cent : how many dollars more
did one spend than the other ?

24. A man has a capital of $12500 : he puts 15 per cent
of it in State Stocks : 33 J per cent in Railroad Stocks, and
25 per cent in bonds and mortgages : what per cent has he
left, and what is its value ?

25. A farmer raises 850 bushels of wheat : he agrees to
sell 18 per cent of it at $1.25 a bushel ; 50 per cent of it at
$1.50 a bushel, and the remainder at $1.75 a bushel : how
much does he receive in all ?

245. To find the per cent which one number is of another.
1. What per cent of $16 is $4 ?

ANALYSIS. The question is, what part of OPERATION.

$16 is $4, when expressed in hundreths: JL- 1 .25.

The standard is $16 (Art. 228) : hence, the or 25 p er cent,
part is -j*g:^ .25; therefore, the per cent is
25 : hence, to find what per cent one number is of another,

Divide by the standard or base, and the quotient, reduced
to decimals, will express the rate per cent.

NOTE. The standard or base, is generally preceded by the word
of.

EXAMPLES.

1. What per cent of 20 dollars is 5 dollars?

2. Forty dollars is what per cent of eighty dollars ?

3. What per cent of 200 dollars is 80 dollars ?

4. What per cent of 1250 dollars is 250 dollars ?

5. What per cent of 650 dollars is 250 dollars ?

6. Ninety bushels of wheat is what per cent of ISOO&usJi.?

7. Nine yards of cloth is what per cent of 870 yards ?

8. Forty-eight head of cattle are what per cent of a drove
of 1600 ?

9. A man has $550, and purchases goods to the amount
of $82.75 : what per cent of his money does he expend?

245. How do you find the per cent which one number is of another ?



234 PERCENTAGE.

10. A merchant goes to New York with $1500 ; he first
lays out 20 "per cent, after which he expends $660 : what
per cent was his last purchase of the money that remained
after his first ?

11. Out of a cask containing 300 gallons, 60 gallons are
drawn : what per cent is this ?

12. If I pay $698.23 for 3 hogsheads of molasses and sell
them for $837.996, how much do I gain per cent on the
money laid out ?

13. A man purchased a farm of 75 acres at $42.40 an
acre. He afterwards sold the same farm for $3577.50 : what
was his gain per cent on the purchase money ?

STOCK, COMMISSION AND BROKERAGE.

246. A CORPORATION is a collection of persons authorized
by law to do business together. The law which defines their
rights and powers is called a Charter.

CAPITAL or STOCK is the money paid in to carry on the
business of the Corporation, and the individuals so contributing
are called Stockholders. This capital is divided into equal
parts called Shares, and the written evidences of ownership
are called Certificates.

247. When the United States Government, or any of the
States, borrows money, an acknowledgment is given to the
lender, in the form of a bond, bearing a fixed interest. Such
bonds are called United States Stock, or State Stock.

The par value of stock is the number of dollars named in
each share. The market value is what the stock brings per
share when sold for cash.

If the market value is above the par value, the stock is
said to be at a premium, or above par ; but if the market
value is below the par value, it is said to be at a discount, or
below par.

346. What is a corporation ? What is a charter? What is capital
or stock ? What are shares ?

347. What are United States Stocks? What are State Stocks?
What is the par value of a stock ? What is the market value ? If the
market is above the par value, what is said of the stock ? If it is below,
what is said of the stock ? What is the market value when above par ?
What when below ?



COMMISSION AND BROKERAGE. 235

Let l=par value of 1 dollar :

l+premium= market value of 1 dollar, 'when above

par :
1 discount =: market value of 1 dollar when below par.

248. Commission is an allowance made to an agent for
buying or selling, or taking charge of property, and is gen-
erally reckoned at a certain rate per cent.


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Online LibraryCharles DaviesSchool arithmetic. Analytical and practical → online text (page 16 of 24)