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of the divisions How are we to find ihe true remainder?

74. 1. What is the rule for divicling by 25 ?

2. What is the rule for dividing by 12* ?

3. What is the rule for dividing by 33* ?

4. What is the rale for dividing by 125 ?

75. What are contractions in division ? What is a composite num-
ber?

76. What is the rule for division when the divisor is a composite
number ?

72 CONTRACTIONS.

77. Let it be required to divide 751 grapes into 16 equal
parts.

(4)751

4 x 4 = 16 -j 4)18T .... 3 first remainder.
40 .... 3x4 = 12
3

15 true rem. 4ns. -4S}|.
NOTE. The factors of the divisor 16, are 4 and 4.

ANALYSIS. If 751 grapes be divided by 4, there will be 187
bunches, each containing 4 grapes, and 8 grapes over. The unit
of 187 is one bunch ; that is, a unit 4 times <(s great as 1 grape.

If we divide 187 bunches by 4, we shall have 46 piles, each
containing 4 bunches, and 3 bunches over : here, again, the unit
of the quotient is 4 times as great as the unit of the dividend.

If, now we wish to find the number of grapes not included in
the 46 piles, we have 3 bunches with 4 grapes in a bunch, and
3 grapes besides : hence, 4 x 3 = 12 grapes ; and adding 3
grapes, we have a remainder, 15 grapes ; therefore, to find the
remainder, in units of the given dividend :

I. Multiply the last remainder by the last divisor but onr,
and add in the preceding remainder :

II. Multiply this result by the next preceding divisor,
and add in the remainder, and so on, till you reach the
unit of the dividend.

EXAMPLES,

1. Let it be required to divide 43720 by 45.
3)43720

5)14573 . l = lstrem. 1x5 + 3-8;
3)2914 . 3= 3d rem. 8x3 + 1 = 25

971 . 1 = 3d rein. 25 true r era.

Divide the following numbers by the factors, for the divisors :

2. 956789 by 7x8 = 56.

3. 4870029 by 8x9 = 72.

4. 674201 by*10x 11 = 110.

5. 4-15767 by 12x12 = 144.

6. 1913578 by 7x2x3 = 42.

7. 146187 by 3x5x7 = 105.

8. 26964 by 5x2 x 11 = 110.

9. 93696 by 3x7x11 = 231.

77. Give the rule for the remainder.

IN DIVISION. 73

CASE II.

78. When the Divisor is 10, 100, 1000, &c.

ANALYSIS. Since any number is made up of units, tens, hun-
dreds, &c. (Art. 28), the number of tens in any dividend will
denote how many times it contains 1 ten, and the units "will be the
remainder. The hundreds will denote how many times the divi-
dend contains 1 hundred, and the tens and units will be thi3 remain-
der ; and similarly, when the divisor is 1000, 10000, &c. ; hence,

RULE. Cut off from the right hand as many figures as
there are ciphers in the divisor the figures at the left ivill be
the quotient, and those at the right, the remainder.

EXAMPLES.

1. Divide 49763 by 10.

2. Divide 7641200 by 100.

3. Divide 496321 by 1000.

4. Divide 6i9T8 by 10000.

CASE III.

79. When there are ciphers on the right of the divisor.

I. Let it be required to divide 67389 by 700.

ANALYSIS. We may regard the OPERATION.

divisor as a composite number, of 7|00)673[89
which the factors are 7 and 100.
We first divide by 100 by striking

off the 89, and then find that 7 is 189 true remain,

contained in the remaining figures, " ^ns 96-

90 times, with a remainder of 1 ;

this remainder we multiply by 100, and then add 89, forming the
true remainder 189 : to the quotient 96, we annex 189 divided by
700, for the entire quotient : hence, the following

RULE I. Cut off" the ciphers by a line, and cut off" the
same number of figures from the right of the dividend.

II. Divide the remaining; figures of the dividend by the
remaining figures of the divisor, and annex to the remainder,
if there be one, the figures cut off from the dividend : this will
form the true remainder

EXAMPLES.
1. Divide 8749632 by 37000.

78. How do you divide when the divisor is 1 with ciphers annexed?
Give the reason of the rule.

79. How do you divide when there are ciphers on the right of the
divisor ? How do you form the true remainder ?

APPLICATIONS.

371000)87491632(236
74

Ans. 236JJJJJ.

17
Divide the following numbers :

2. 986327 by 210000.

3. 876000 by 6000.

4. 36599503 by 400700.

5. 5714364900 by 36500.

6. 18490700 by 73000.

7. 70807149 by 31500.

APPLICATIONS.

80. Abstractly, the object of division is to find from two
given numbers a third, which, multiplied by the first, will
produce the second. Practically, it has three objects :

1. Knowing the number of things and their entire cost, to
find the price of a single thing :

2. Knowing the entire cost of a number of things and the
price of a single thing, to find the number of things :

3. To divide any number of things into a given number of
equal parts.

For these cases, we have from the previous principles
(page 57), the following

RULES.

I. Divide the entire cost by the number of the things :
the quotient will be the price of a single thing.

II. Divide the entire cost by the price of a single thing :
the quotient will be the number of things.

III. Divide the whole number of things by the number of
parts into which they are to be divided : the quotient will
be the number in each part.

QUESTIONS INVOLVING THE PREVIOUS RULES.

1. Mr. Jones died, leaving an estate worth 4500 dollars, to
be divided equally between 3 daughters and 2 sons : what
was the share of each ?

80. What is the object of division, abstractly? How many objects has
it, practically ? Name the three objects. Give the rules for the three cases.

APPLICATIONS. 75

2. What number must be multiplied by 124 to produce
40796?

3. The sum of 19125 dollars is to be distributed equally
among a certain number of men, each to receive 425 dollars :
how many men are to receive the money ?

4. A merchant has 5100 pounds of tea, and wishes to pack
it in 60 chests : how much must he put in each chest ?

5. The product of two numbers is 51679680, and one of
the factors is 615 : what is the other factor ?

6. Bought 156 barrels of flour for 1092 dollars, and sold
the same for 9 dollars per barrel : how much did I gain ?

7. Mr. James has 14 calves worth 4 dollars each, 40 sheep
worth 3 dollars each ; he gives them all for a horse worth
150 dollars : what does he make or lose by the bargain ?

8. Mr. Wilson sells 4 tons of hay at 12 dollars per ton,
80 bushels of wheat at 1 dollar per bushel, and takes in
payment a horse worth 65 dollars, a wagon worth 40 dollars,
and the rest in cash : how much money did he receive ?

9. How many pounds of coffee, worth 12 cents a pound,
must be given for 368 pounds of sugar, worth 9 cents a
pound ?

10. The distance around the earth is computed to be about
25000 miles : how long would it take a man to travel that
distance, supposing him to travel at the rate of 35 miles a
day?

11. If 600 barrels of flour cost 4800 dollars, what will
21 7 2 barrels cost?

12. If the remainder is 17, the quotient 610, and the divi-
dend 45767, what is the divisor?

13. The salary of the President of the United States is
25000 dollars a year : how much can he spend daily and
save of his salary 4925 dollars at the end of the year ?

14. A farmer purchased a farm for which he paid 18050
dollars. He sold 50 acres for 60 dollars an acre, and the re-
mainder stood him in 50 dollars an acre : how much land
did he purchase ?

15. There are 31173 verses in the Bible: how many
verses must be read each day, that it may be read through
in a year ?

16. A farmer wishes to exchange 250 bushels of oats at
42 cents a bushel, for flour at 7 dollars per barrel : how many

76 APPLICATIONS.

It. The owner of an estate sold 240 acres of land and had
312 acres left : how many acres had he at first ?

18. Mr. James bought of Mr. Johnson two farms, one con-
taining 250 acres, for which he paid 85 dollars per acre ; the
second containing 175 acres, for which he paid 70 dollars an
acre ; he then sold them both for 75 dollars an acre : did he
make or lose, and how much ?

19. A farmer has 279 dollars with which he wishes to buy
cows at 25 dollars, sheep at 4 dollars, and pigs at 2 dollars
apiece, of each an equal number : how many can he buy of
each sort ?

20. The sum of two numbers is 3475, and the smaller is
1162 : what is the greater ?

21. The difference between two numbers is 1475, and the
greater number is 5760 : what is the smaller ?

22. If the product of two numbers is 346712, and one of
the factors is 76 : what is the other factor?

23. If the quotient is 482, and the dividend 135442 : what
is the divisor ?

24. A gentleman bought a house for two thousand twenty-
five dollars, and furnished it for seven hundred and six dol-
lars ; he paid at one time one thousand and ten dollars, and
at another time twelve hundred and seven dollars : how much
remained unpaid ?

25. At a certain election the whole number of votes cast
for two opposing candidates was 12672: the successful can-
ceive ?

26. Mr. Place purchased 15 cows : he sold 9 of them for
35 dollars apiece, and the remainder for 32 dollars apiece,
when he found that he had lost 123 dollars : how much did
he pay apiece for the cows ?

27. Mr. Gill, a drover, purchased 36 head of cattle at 64
dollars a head, and 88 sheep at 5 dollars a head ; he sold the
vance : how much did he receive for both lots ?

28. Mr. Nelson supplied his farm with 4 yoke of oxen at
93 dollars a yoke ; 4 plows at 11 dollars apiece ; 8 horses at
97 dollars each ; and agrees to pay for them in wheat at
1 dollar and a half per bushel ; how many bushels must he
give ?

APPLICATIONS. 77

29. If a man's salary is 800 dollars a year and his expenses
425 dollars, how many years will elapse before he will be
worth 10000 dollars, if he is worth 2500 dollars at the pre-
sent time ?

30. How long can 125 men subsist on an amount of food
that will last 1 man 4500 days ?

31. A speculator bought 512 barrels of flour for 3584 dol-
lars and sold the same for 4608 dollars : how much did he
gain per barrel ?

32. A merchant bought a hogshead of molasses containing
96 gallons at 35 cents per gallon ; but 26 gallons leaked out,
and he sold the remainder at 50 cents per gallon : did he
gain or lose, and how much ?

33. Two persons counting their money, together they had
342 dollars ; but one had 28 dollars more than the other :

34. Mrs. Louisa Wilsie has 3 houses valued at 12530 dol-
lars, 11324 dollars, and 9875 dollars : also a farm worth 6720
dollars. She had a daughter and 2 sons. To the daughter
she gives one-third the value of the houses and one-fourth the
value of the farm, and then divides the remainder equally
among the boys : how much did each receive ?

35. A person having a salary of 1500 dollars, saves at the
end of the year 405 dollars : what were his average daily
expenses, allowing 365 days to the year ?

36. Mr. Bailey has 7 calves worth 4 dollars apiece,
9 sheep worth 3 dollars apiece, and a fine horse worth 175
dollars. He exchanges them for a yoke of oxen worth 125
dollars and a colt worth 65 dollars, and takes the balance in
hogs at 8 dollars apiece : how many does he take ?

37. Mr. Snooks, the tailor, bought of Mr. Squire, the mer-
chant, 4 pieces of cloth ; the first and second pieces each
measured 45 yards, the third 47 yards, and the fourth 53
yards ; for the whole he paid 760 dollars : what did he pay
for 35 yards ?

38. Mr. Jones has a farm of 250 acres, worth 125 dollars
per acre, and offers to exchange with Mr. Gushing, whose
farm contains 185 acres, provided Mr. Gushing will pay him
20150 dollars difference: what was Mr. Cushing's farm
valued at per acre ?

78 APPLICATIONS.

39. The volcano in the island of Bourbon, in 1796, threw
out 45000000 cubic feet of lava : how long would it take 25
carts to carry it off, if each cart carried 12 loads a day, and
40 cubic feet at each load ?

40. The income of the Bishop of Durham, in England, is
292 dollars a day ; how many clergymen would this support
in a salary of 730 dollars per annum ?

41. The diameter of the earth is 7912 miles, and the diame-
ter of the sun 112 times as great : what is the diameter of the
sun?

42. By the census of 1850, the whole population of the
United States was 23191876 ; the number of births for the
previous year was 629444 and the number of deaths 324394 :
supposing the births to be the only source of increase, what
was the population at the beginning of the previous year ?

43. Mr. Sparks bought a third part of neighbor Spend-
thrift's farm for 2750 dollars. Mr. Spendthrift then sold half
the remainder at an advance of 250 dollars, and then Mr.
Sparks bought what was left at a further advance of 250
dollars : how much money did Mr. Sparks pay Mr. Spend-
thrift, and what did he get for his whole farm ?

44. George Wilson bought 24 barrels of pork at 14 dollars
a barrel ; one-fourth of it proved damaged, and he sold it at
half price, and the remainder he sold at an advance of 3 dol-
lars a barrel : did he make or lose by the operation, and how
much ?

45. A miller bought 320 bushels of wheat for 576 dollars,
and sold 256 bushels for 480 dollars : what did the remain-
der cost him per bushel ?

46. A merchant bought 117 yards of cloth for 702 dollars,
and sold 76 yards of it at the same price for which he bought
it ; what did the cloth sold amount to ?

47. If 46 acres of land produce 2484 bushels of corn ; how
many bushels will 1 20 acres produce ?

48. Mr. J. Williams goes into business with a capital of
25000 dollars ; in the first year he gains 2000 ; in the second
year 3500 dollars ; in the third year 4000 dollars ; he then
invests the whole in a cargo of tea and doubles his money ;
he then took out his original capital and divided the residue
equally among his 5 "children : what was the portion of
each ?

UNITED STATES MONEY. 79

UNITED STATES MONEY.

81. Numbers are collections of units of the same kind.
In forming these collections, we first collect the lowest or pri-
mary units, until we reach a certain number ; we then
change the unit and make a second collection, and after
reaching a certain number we again change the unit, and so on.

In abstract numbers, we first collect the units 1 till we
reach ten ; we then change the unit, to 1 ten, and collect till
we reach 10 ; we then change the unit to 100, and so on.

A SCALE expresses the relations between the orders of units,
in any number. There are two kinds of scales, uniform and
varying. In the abstract numbers, the scale is uniform, the
units of the scale being 10, at every step.

82. United States money is the currency established by Con-
gress, A.D. 1786. The names or denominations of its units are,
Double Eagles, Eagles, Dollars, Dimes, Cents, and Mills.

The coins of the United States are of gold, silver, and cop-
per, and are of the following denominations :

1. Gold : Double-eagle, eagle, half-eagle, three-dollars,
quarter-eagle, dollar.

2. Silver: Dollar, half-dollar, quarter-dollar, dime, half-
dime, and three-cent piece.

3. Copper : Cent, half-cent.

TABLE.

10 Mills make 1 Cent, Marked ct.

10 Cents -

- 1 Dime,

- - d.

10 Dimes -

- 1 Dollar,

- - \$.

10 Dollars -

- 1 Eagle,

- - E.

Mills.

Cents.

Dimes.

Dollars.

Eagles.

10

= 1

100

= 10

= 1

1000

= 100

= 10

= 1

10000

= 1000

= 100

= 10

= 1

81. "What are numbers? How are numbers formed? How are sim-
ple numbers formed ? What is the scale ? What is the primary unit
in simple numbers ?

80 UNITED STATES MONEY.

83. It is seen, from the above table, that in United States
money, the primary unit is 1 mill ; the units of the scale, in
passing from mills to cents, are 10. The second unit is 1
cent, and the units of the scale, in passing to dimes, are 10.
The third unit is 1 dime, and the units of the scale in passing
to dollars, are 10. The fourth unit is 1 dollar, and the units
of the scale in passing to eagles, are 10. This scale is the
same as in simple numbers ; therefore,

The units of United States money may be added, sub-
tracted, multiplied, and divided, by the same rules that
have already been given for simple numbers.

NUMERATION TABLE.

5 7, is read 5 cents and 7 mills, or 57 mills.
1 6 4, - - 16 cents and 4 mills, or 164 mills.
6 2. 1 2 0, - - 62 dollars 12 cents and no mills.
27.623,- - 27 dollars 62 cents and 3 mills.
4 0. 4 1, - - 40 dollars 4 cents and 1 mill.

The period, or separatrix, is generally used to separate the
cents from the dollars. Thus \$67.256 is read 67 dollars 25
cents and 6 mills. Cents occupy the two first places on the
right of the period, and mills the third.

United States money is read in dollars, cents and mills.

82. What is United States money? What are the names of its
units ? What are the coins of the United States ? Which gold ?
Which silver ? Which copper ?

83. In United States money what is the primary unit? What is the
Hcale in passing from one denomination to another? I low does this
compare with the scale in simple numbers ? What then follows V
What is used to separate dollars from cents ? How is United States

84. What is reduction ? How many kinds of reduction are there ?
Name them. How may cents be changed into mills? How may dol-
lars be changed into cents ? How into mills ?

UNITED STATES MONEY. 81

REDUCTION OF UNITED STATES MONEY.

84. Reduction of United States Money is changing the
unit from one denomination to that of another, without altering
the value of the number. It is divided into two parts :

1st. To reduce from a greater unit to a less, as from dol-
lars to cents.

2d. To reduce from a less unit to a greater, as from mills
to dollars.

85. To reduce from a greater unit to a less.
From the table it appears,

1st. That cents may be changed into mills by annexing
one cipher.

2d. That dollars may be changed into cents by annexing
two ciphers, and into mills by annexing three ciphers.

3d. That eagles may be changed into dollars by annexing
one cipher.

The reason of these rules is evident, since 10 mills make a
cent, 100 cents a dollar, and 1000 mills a dollar and 10
dollars 1 eagle.

EXAMPLES.

1. Reduce 25 eagles, 14 dollars, 85 cents and 6 mills to
the denomination of mills.

OPERATION.

25 eagles =250 dollars,

"264 dollars =2 64 00 cents,

26485 cents=264850 mills,

Ans. 264856 mills.

2. In 3 dollars 60 cents and 5 mills, how many mills ?
3 dollars =300 cents,

60 cents,

160 = 3600 mills, to which add the 5 mills.
6

82 REDUCTION OF

3. In 37 dollars 31 cents 8 mills, how many mills ?

4. In 375 dollars 99 cents 9 mills, how many mills ?

5. How many mills in 67 cents ?

6. How many mills in \$54 ?

7. How many cents in \$125 ?

8. In \$400, how many cents ? How many mills ?

9. In \$375, how many cents ? How many mills ?

10. How many mills in \$4 ? In \$6 ? In \$10.14 cents.

11. How many mills in \$40.36 cents 8 mills ?

12. How many mills in \$71.45 cents 3 mills ?

86. To reduce from a less unit to a greater.
1. How many dollars, cents and mills in 26417 mills?

ANALYSIS. We first divide the mills by 10, OPERATION.

giving 2641 cents and 7 mills over; we then 10)264117

divide the cents by 100, giving 26 dollars, and 100)26141

41 cents over : hence the answer is 26 dollars *!> . -. *,
41 cents and 7 mills : therefore,

I. To reduce mills to cents : cut off the right hand figure.

II. To reduce cents to dollars : cut off the two right hand
figures: and,

III. To reduce mills to dollars : cut off the three right
hand figures.

EXAMPLES.

1. How many dollars cents and mills are there in 67897
mills ?

2. Set down 104 dollars 69 cents and 8 mills.

3. Set down 4096 dollars 4 cents and 2 mills.

4. Set down 100 dollars 1 cent and 1 mill.

5. Write down 4 dollars and 6 mills.

6. Write down 109 dollars and 1 mill.

7. Write down 65 cents and 2 mills.

8. Write down 2 mills.

9. Reduce 1607 mills, to dollars cents and mills.
10. Reduce 170464 mills, to dollars cents and mills.
IK Reduce 8674416 mills, to dollars cents and mills.

12. Reduce 94780900 mills, to dollars cents and mills.

13. Reduce 74164210 mills, to dollars cents and mills.

8G. How do you change mills into cents ? How do you change cento
Into dollars ? How do you change mills to dollars ?

UNITED STATES MONEY. 83

87. One number is said to be an aliquot part of another,
when it is contained in that other an exact number of times.
Thus ; 50 cents, 25 cents, &c., are aliquot parts of a dollar :
so also 2 months, 3 months,. 4 months and 6 months are ali-
quot parts of a year. The parts of a dollar are sometimes
expressed fractionally, as in the following

TABLE OF ALIQUOT PARTS.

\$1 =100 cents.

| of a dollar = 50 cents.

| of a dollar = 33 J cents.

J of a dollar = 25 cents,

of a dollar = 20 cents.

I of a dollar^ 121 cents.

fa of a dollar = 10 cents.

^ of a dollar = 6J cents,

z^j- of a dollar = 5 cents,

of a cent = 5 mills.

1. Charles gives 9| cents for a top, and 3J cents for 6
quills : how much do they all cost him ?

2. John gives \$1.37 for a pair of shoes, 25 cents for a
penknife, and 12 J cents for a pencil : how much does he pay
for all ?

OPERATION.

ANALYSIS. We observe that half a cent is equal \$1.375
to 5 mills. We then place the mills, cents and dol- '25

lars in separate columns. We then add as in simple I9f\

numbers. i - J

\$1.750

OPERATION.

3. James gives 50 cents for a dozen oranges, \$0.50
12| cents for a dozen apples: and 30 cents for .125
a pound of raisins : how much for all ? .30

\$0.925 '

88. Hence, for the addition of United States money, we
have the following

RULE. I. Set down the numbers so that units of the
same value shall stand in the same column.

87. What is an aliquot part ? How many cents in a dollar ? In half
a dollar ? In a third of a dollar ? In a fourth of a dollar ?

84 APPLICATIONS IN

II. Add up the several columns as in simple numbers,
and place the separating point in the sum directly under
that in the columns.

PROOF. The same as in simple numbers.

EXAMPLES.

1. Add \$61.214. \$10.049, \$6.041, \$0.271, together.

(1.) (2.) (3.)

\$ cts. m. \$ cts. m. \$ cts. m.

67.214 59.316 81.053

10.049 87.425 67.412

6.041 48.872 95.376

0.271 56.708 87.064

\$83.575 \$330.905

APPLICATIONS.

1. A grocer purchased a box of candles for 6 dollars
89 cents : a box of cheese for 25 dollars 4 cents and 3 mills ;
a keg of raisins for 1 dollar 12| cents, (or 12 cents and 5
mills ;) and a cask of wine for 40 dollars 37 cents 8 mills :
what did the whole cost him ?

2. A farmer purchased a cow for which he paid 30 dollars
and 4 mills ; a horse for which he paid 104 dollars 60 cents
and 1 mill ; a wagon for which he paid 85 dollars and
9 mills : how much did the whole cost ?

3. Mr. Jones sold farmer Sykes 6 chests of tea for \$75.641 ;
9 yards of broadcloth for \$27.41 ; a plow for \$9.75 ; and a
harness for \$19.674 : what was the amount of the bill ?

4. A grocer sold Mrs. Williams 18 hams for \$26.497 ; a bag
of coffee for \$17.419 ; a chest of tea for \$27.047 ; and a
firkin of butter for \$28.147 : what was the amount of her
bill?

5. A father bought a suit of clothes for each of his four
boys ; the suit of the eldest cost \$15.167 ; of the second,
\$13.407 ; of the third, 12.75 ; and of the youngest, \$11.047 :
how much did he pay in all ?

88. How do you set down the numbers for addition ? How do you
add up the columns ? How do you place the separating point ? How

UNITED STATES MONEY. 85

6. A father has six children ; to the first two he gives
each \$375.416 ; to each of the second two, \$287.55 ; to each
of the remaining two, \$259.004 : how much did he give to
them all?

7. A man is indebted to A, \$630.49 ; to B, \$25 ; to C,
87 J cents ; to D, 4 mills : how much does he owe ?

8. Bought 1 gallon of molasses at 28 cents per gallon ; a
half pound of tea for 78 cents ; a piece of flannel for 12 dol-
lars 6 cents and 3 mills ; a plow for 8 dollars 1 cent and

1 mill ; and a pair of shoes for 1 dollar and 20 cents : what
did the whole cost ?

9. Bought 6 pounds of coffee for 1 dollar 12J cents ; a
wash-tub for 75 cents 6 mills ; a tray for 26 cents 9 mills ; a
broom for 27 cents ; a box of soap for 2 dollars 65 cents
7 mills ; a cheese for 2 dollars 87^ cents : what is the whole
amount ?

10. What is the entire cost of the following articles, viz. :

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