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# School arithmetic. Analytical and practical online

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Font size EXAMPLES.

(1.) (2.)

T. cwt. qr. Ib. A. It. P.

7)1 19 2 12 9)113 3 25

Quotient. " 5216 122 25

(3.) (4.)

L. mi. fur. rd. bu. pk. qL

8)47 1 7 8 11)25 3 1

Quotient.

132 DIVISION OF

Divide the following :

5. l*lcwt. Qqr. 2/6. 6oz. by 7.

6. 49*/d. 3?r. 3/m. by 9.

7. 131A 1,8. by 12.

8. 1138 12s. 4a. by 53.

9. TOT. 17cwtf. 7/6. by 79.
10. 276u. Spyfc. 7^. by 84.

11. Bought 65 yards of cloth for which I paid 72 14s.
. : what did it cost per yard?

12. If 15 loads of hay contain 35 T. 5cwt., what is the

13. If a man, lifting 8 times as much as a boy, can raise
201/6. 12oz., how much can the boy lift?

14. If a vessel sail 25 42' 40" in 10 days, how far will
she sail in one day ?

15. Divide Vhhd. ZZgal. 2qt. by 12.

16. What is the quotient of 656w. Ipk. 3qt. divided by 12?

17. In 4 equal packages of medicine there are 13B> 7 3
23 13 4gr. ; how much is there in each package ?

18. In 25hhd. of molasses, the leakage has reduced the
whole amount to 1534gra/. \qt. \pt. : if the same quantity
has leaked out of each hogshead, how much will each hogs-

19. In 9 fields there are 113A 37?. 25P. of land : if the
fields contain an equal amount, how much is there in each
field?

20. If in 30 days a man travels 746mi. 5/wr., travelling
the same distance each day, what is the length of each day's
journey ?

21. Suppose a man had 98/6. 2oz. Wpwt. 6gr. of silver ;
how much must he give to each of 7 men if he divides it
equally among them?

22. When J75#a/. 2qt. of beer are drank in 52 weeks,
how much is consumed in one week ?

23 A rich man divided 1686w. Ipk. Qqt. of corn among
35 poor men : how much did each receive ?

24. In sixty-three barrels of. sugar there are 7T. 16cwtf.
3qr. 12/6. : how much is there in each barrel ?

25. A farmer has a granary containing 232 bushels 3
ks 7 quarts of wheat, and he wishes to put it in 105 bags :

ow much must each bag contain ?

26. If 90 hogsheads of sugar weigh 56 T. Hcwt. Zqr. 15/6,
what u the weight of 1 hogshead ?

DENOMINATE NUMBERS. 133

27. One hundred and seventy-six men consumed in a week
IScwt 2qr. 15/6. 6oz. of bread : how much did each man
consume ?

28. If the earth revolves on its axis 15 in 1 hour, how far
does it revolve in 1 minute ?

29. If 59 casks contain 44Md. ttgal. 2qt. Ipt. of wine,
what are the contents of one cask ?

30. Suppose a man has 246ml Qfur. 36rd. to travel in 12
days : how far must he travel each day?

31. If I pay 12 14s. 5d 3/ar. for 35 bushels of wheat,
what is the price per bushel ?

32. A printer uses one sheet of paper for every 16 pages of
an octavo book : how much paper will be necessary to print
500 copies of a book containing 336 pages, allowing 2 quires
of waste paper in each ream ?*

33. A man lends his neighbor 135 6s. 8d., and takes in
part payment 4 cows at 5 8s. apiece, also a horse worth
50 : how much remained due ?

34. Out of a pipe of wine, a merchant draws 12 bottles,
each containing 1 pint 3 gills ; he then fills six 5-gallon demi-
johns ; then he draws off 3 dozen bottles, each containing
1 quart 2 gills : how much remained in the cask ?

35. A farmer has 6 T. Scivt. 2qr. 14/6. of hay to be re-
moved in 6 equal loads : how much must be carried at each

36. A person at his death left landed estate to the amount
of 2000, and personal property to the amount of 2803 17s.
4c?. He directed that his widow should receive one-eighth of
the whole, and that the residue should be equally divided
among his four children : what was the widow and each
child's portion ?

37. If a steamboat go 224 miles in a day, how long will
it take to go to China, the distance being about 12000 miles?

38. How long would it take a balloon to go from the earth
to the moon, allowing the distance to be about 240000 miles,
the balloon ascending 34 miles per hour ?

* In packing and selling paper, the two outside quires of every ream
are regarded as waste, and each of the remaining quires contains 34
perfect sheets: hence, in this example, the waste "paper is considered
as belonging only to the entire reams.

134 LONGITUDE AND TIME.

LONGITUDE AND TIME.

124. The circumference of the earth, like that of other
circles, is divided into 360, which are called degrees of lon-
gitude.

125. The sun apparently goes round the earth once in 24
hours. This time is called a day.

Hence, in 24 hours, the sun apparently passes over 360 of
longitude ; and in 1 hour over 360 -=-24 = 15.

126. Since the sun, in passing over 15 of longitude, re-
quires 1 hour or GO' of time, 1 will require 60'-=- 15 = 4=
minutes of time ; and V of longitude will be equal to one
sixteenth of 4' which is 4" : hence,

15 of longitude require 1 hour
1 of longitude requires 4 minutes.
1' of longitude requires 4 seconds.

Hence, we see that,

1. If the degrees of longitude be multiplied by 4, the pro-
duct will be the corresponding time in minutes.

2. If the minutes in longitude be multiplied by 4, the pro-
duct will be the corresponding time in seconds.

127. When the sun is on the meridian of any place, it is
12 o'clock, or noon, at that place.

Now, as the sun apparently goes from east to west, at the
instant of noon, it will be past noon for all places at the east,
and before noon for all places at the west.

If then, we find the difference of time between two places,
and know the exact time at one of them, the corresponding
time at the other will be found by adding their difference, if
that the other be east, or by subtracting it if west.

124. How is the circumference of the earth supposed to be divided ?

125. How does the sun appear to move ? What is a day ? How far
does the sun appear to move in 1 hour ?

126. How do you reduce degrees of longitude to time ? How do you
reduce minutes of longitude to time ?

127. What is the hour when the sun is on the meridian ? When the
sun is on the meridian of any place, how will the time be for all places
cast? How for all places west? If you have the difference of time,
how do you find the time V

LONGITUDE AND TIME. 135

1. The longitude of New York is 74 1' west, and that of
Philadelphia 75 10' west : what is the difference of longi-
tude and what their difference of time ?

2. At 12 M. at Philadelphia, what is the time at New
York?

3. At 12 M. at New York, what is the time at Philadelphia ?

4. The longitude of Cincinnati, Ohio, is 84 24' west :
what is the difference of time between New York and Cin-
cinnati ?

5. What is the time at Cincinnati, when it is 12 o'clock at
New York?

6. The longitude of New Orleans is 89 2' west : what
time is at New Orleans, when it is 12 M. at New York ?

7. The meridian from which the longitudes are reckoned
passes through the Greenwich Observatory, London : hence,
the longitude of that place is : what is the difference of
time between Greenwich and New York ?

8. What is the time at Greenwich, when it is 12 M. at
New York?

9. The longitude of St. Louis is 90 15' west : what is the
time at St. Louis, when it is 3/i. 25m. P.M. at New York ?

10. The longitude of Boston is 71 4' west, and that of
New Orleans 89 2' west : what is the time at New Orleans
when it is 7 o'clock 12??i A.M. at Boston ?

11. The longitude of Chicago, Illinois, is 87 30' west :
what is the time at Chicago, when it is 12 M. at New York?

PROPERTIES OF NUMBERS.

COMPOSITE AND PRIME NUMBERS.

128. An Integer, or whole number, is a unit or a collection
of units.

129. One number is said to be divisible by another, when
the quotient arising from the division is a whole number. The
division is then said to be exact.

NOTE. Since every* number is divisible by itself and 1, the
term divisible will be applied to such numbers only, as have other
divisors.

128. What is an Integer ?

136 PROPERTIES OF NUMBERS.

130. Every divisible number is called a composite number,
(Art. 54), and any divisor is called & factor: thus, 6 is a com-
posite number, and the factors are 2 and 3.

131. Every number which is not divisible is called a prime
number : thus, 1, 2, 3, 5, 7, 11, &c. are prime numbers.

132. Every prime number is divisible by itself and 1 ;
but since these divisors are common to all numbers, they are
not called factors.

133. Every factor of a number is either prime or compo-
site : and since any composite factor may be again divided, it
follows that,

Any number is equal to the product of all its prime factors.

For example, 12=: 6 x 2 ; but 6 is a composite number, of
which the factors are 2 and 3 ; hence,

12=2 x 3 x 2 ; also, 20=10 x 2=5 x 2 x 2.
Hence, to find the prime factors of any number,

Divide the number by any prime number that will exactly
divide it : then divide the quotient by any prime number that
will exactly divide it, and so on, till a quotient is found which
is a prime number ; the several divisors and the last quotient
will be the prime factors of the given number. '

NOTE. It is most convenient, in practice, to use the least prime
number, which is a divisor.

1. What are the prime factors of 42 ?

OPERATION.

ANALYSIS. Two being the least divisor 2)42

that is a prime number, we divide by it, giv- o\ 91

ing the quotient 21, which we again divide o)4L

by 3, giving 7: hence, 2, 3 and 7 are the 7

prime factors. 2x3x7 = 42.

129. When is one number divisible by another ? By what is every
number divisible ? Is 1 called a divisor ?

130. What is a composite number ? What is a factor ?

131. What is a prime number ?

132. By what divisors is every prime number divided ?

133. To what product is every number equal? Give the rule for
finding the prime factors of a number. What number is it most conve-
nient to use as a divisor ?

PRIME FACTORS. 137

What arc the prime factors of the following numbers ?

1. Of the number 9 ?

2. Of the number 15?

3. Of the number 24 ?

4. Of the number 16?

5. Of the number 18 ?

6. Of the number 32 ?

7. Of the number 48 ?

8. Of the number 56?

9. Of the number 63 ?
10. Of the number 76?

NOTE. The prime factors, when the number is small, may
generally be seen by inspection. The teacher can easily multiply
the examples.

134. When there are several numbers whose prime factors
are to be found,

Find the prime factors of each and then select those factors
which are common to all the numbers.

11. What are the prime factors common to 6, 9 and 24 ?

12. What are the prime factors common to 21, 63 and 84?

13. What are the prime factors common to 21, 63 and 105 ?

14. What are the common factors of 28, 42 and 70 ?

15. What are the prime factors of 84, 126 and 210 1

16. What are the prime factors of 210, 315 and 525 ?

135. DIVISIBILITY OF NUMBERS.

1. 2 is the only even number which is prime.

2. 2 divides every even number and no odd number.

3. 3 divides any number when the sum of its figures is di-
visible by 3.

4. 4 divides any number when the number expressed by
the two right hand figures is divisible by 4.

5. 5 divides every number which ends in or 5.

6. 6 divides any even number which is divisible by 3.

7. 10 divides any number ending in 0.

GREATEST COMMON DIVISOR.

130. The greatest common divisor of two or more num-
bers, is the greatest number which will divide each of them,
separately, without a remainder. Thus, 6 is the greatest
common divisor of 12 and 18.

134. How do you find the prime factors of two or more numbers ?

138 COMMON DIVISOR.

NOTE. Since 1 divides every number, it is not reckoned among
the common divisors.

137. If two numbers have no common divisor, they are
called prime with respect to each other.

138. Since a factor of a number always divides it, it fol-
lows that the greatest common divisor of two or more num-
bers, is simply the greatest factor common to these numbers.

Hence, to find the greatest common divisor of two or
more numbers,

I. Resolve each number into its prime factors.

II. The product of the factors common to each result will
be the greatest common divisor.

EXAMPLES.

1. What is the greatest common divisor of 24 and 30 ?

ANALYSIS. There are four prime OPERATION.

factors in 24, and 3 in 30 : the factors 24 = 2x2x2x3
2 and 3 are common : hence, 6 is the 30 = 2 X 3 X 5

greatest common divisor. 2 X ^(> com. divisor.

2. What is the greatest common divisor of 9 and 18 ?

, 3. What is the greatest common divisor of 6, 12, and 30 ?

4. What is the greatest common divisor of 15, 25 and 30 ?

5. What is the greatest common divisor of 12, 18 and 72 ?

6. What is the greatest common divisor of 25, 35 and 70 ?

7. What is the greatest common divisor of 28, 42 and 70 ?

8. What is the greatest common divisor of 84, 126 and
210?

139. When the numbers are large, another method of find-
ing their greatest common divisor is used, which depends ou
the following principles :

135. What even number is prime ? What numbers will 2 divide ?
What numbers will 3 divide ? What numbers will 4 divide ? 5 ? 6 ?
10?

136. What is the greatest common divisor of two or more numbers ?

137. When are two numbers said to be prime with respect to each
other?

138. What is the greatest factor of two numbers ? How do you find
the greatest common divisor of two or more numbers ?

PROPERTIES OF NUMBERS. 139

1. Any number which willdividetwo numbers separately, will
divide their sum ; else, we should have a

whole number equal to a proper fraction. 24+27=51

2. Any number which will divide two numbers separately,
ivill divide their difference; and any

number which will divide their differ- 51 27 = 24
ence and one of the numbers, will divide
the other ; else, we should have a whole number equal to a
proper fraction.

1.

*/

What is the greatest common divisor of 27 and 51 ?

Divide 51 by 27 ; the quotient is 1 and the remainder 24 ; then

divide the preceding divisor 27 by the re- OPERATION.

mainder 24 : the quotient is 1 and the re 27)51(1

mainder 3 : then divide the preceding 27

divisor 24 by the remainder 3 ; the quo-

tient is 8 and the remainder 0. 24 ) 27 ( 1

Now, since 3 divides the difference 3,

and also 24, it will divide 27, by principle 3)24(8

2d ; and since 3 divides the remainder 24, 04

and 27, it will also divide 51 : hence it is

a common divisor of 27 and 51 ; and since it is the greatest com-
mon factor, it is their greatest common divisor. Since the above
reasoning is as applicable to any other two numbers as to 27 and
51, we have the following rule :

Divide the greater number by the less, and then divide the
preceding divisor by the remainder, and so on, till nothing re-
mains : the last divisor will be the greatest common divisor.

EXAMPLES.

1. What is the greatest common divisor of 216 and 408 ?

2. Find the greatest common divisor of 408 and 740.

3. Find the greatest common divisor of 315 and 810.

4. Find the greatest common divisor of 4410 and 5670.

5. Find the greatest common divisor of 3471 and 1869.

6. Find the greatest common divisor of 1584 and 2772.

NOTE. If it be required to find the greatest common divisor of
more than two numbers, first find the greatest common divisor of

139. When the numbers are large, on what principles docs the oper-
ation of finding the greatest common divisor depend ? What is the
rule for finding it ?

140 COMMON MULTIPLE*

two of them, then of that common divisor and one of the remain
ing numbers, and so on for all the numbers ; the last common
divisor will be the greatest common divisor of all the numbers.

7. What is the greatest common divisor of 492, 744 and
1044?

8. What is the greatest common divisor of 944, 1488, and
2088?

9. What is the greatest common divisor of 216, 408 and
740?

10. What is the greatest common divisor of 945 1560 and
22683 ?

LEAST COMMON MULTIPLE.

140. The common multiple, of two or more numbers, is any
number which will exactly divide.

The least common multiple of two or more numbers, is the
least number which they will separately divide without a re-
mainder.

NOTES. 1. If a dividend is exactly divisible by a divisor, it can
be resolved into two factors, one of which is the divisor and the
other the quotient.

2. If the divisor be resolved into its prime factors, the cor-
responding factor of the dividend may be resolved into the same
factors : hence, the dividend will contain every prime factor of the
divisor.

3. The question of finding the least common multiple of several
numbers, is therefore reduced to finding a number which shall con-
tain all their prime factors and none others.

1. Let it be required to find the least common multiple of
6, 8 and 12.

ANALYSIS. We see, from inspec- OPERATION.

tion, that the prime factors of 6 are 2x3 2x2x2 2x2x3

2 and 3 : of 8 ; 2, 2 and 2 : and 6 8 12

of 12 ; 2, 2 and 3.

Every number that is a prime factor must appear in the least com-
mon multiple, and none others: hence, it will contain all the prime

140. What is the least common multiple of two or more numbers ?
State the principles involved in finding it. Give the rule for finding it.
What is the multiple when the numbers have no common prime fac-
tors ?

COMMON MULTIPLE. 141

factors of any one of the numbers, as 8, and such other prime fac-
tors of the others, 6 and 12, as are not found among the prime fac-
tors of 8 ; that is, the factor 3 : hence,

2 x 2 x 2 x 3 = 24, the least common multiple.
To find the least common multiple of several numbers.

I. Place the numbers on the same line, and divide by any-
prime number that will exactly divide two or more of them,
and set down in a line below the quotients and the undivided
numbers.

II. Then divide as before until there is no prime number
greater than 1 that will exactly divide any two of the numbers.

III. Then multiply together the divisors and the numbers of
the lower line, and their product will be the least common
multiple.

NOTE. 1. The object of dividing by any prime number that will
divide two or more of the numbers, is to find common factors. x

2. If the numbers have no common prime factor, their product
will be their least common multiple.

EXAMPLES.

OPERATION.

1. Find the least common mul-
tiple of 3, 4 and 8. 2)3 4 8

Ans. 2x2x3x1x2 = 24. 2)3

2. Find the least common mul-
tiple of 3, 8 and 9. 3)3 8 9

Ans. 3x1x8x3=72. 1 8 3

3. Find the least common multiple of 6, 7, 8 and 10.

4. Find tKe least common multiple of 21 and 49.

5. Find the least common multiple of 2, 7, 5, 6, and 8.

6. Find the least common multiple of 4, 14, 28 and 98

7. Find the least common multiple of 13 and 6.

8. Find the least common multiple of 12, 4 and 7.

9. Find the least common multiple of 6, 9, 4, 14 and 16.

10. Find the least common multiple of 13, 12 and 4.

11. Find the least common multiple of 11, 17, 19, 21, and

14:2 CANCELLATION.

CANCELLATION.

141. CANCELLATION is a method of shortening Arithmeti-
cal operations by omitting or cancelling common factors.

1. Divide 24 by 12. First, 24 = 3 x 8 ; and 12 = 3 x 4.

ANALYSIS. Twenty-four divided by 12 is OPERATION.

equal to 3 x 8 divided by 3 x 4 ; by cancelling 24 \$ x 8

or striking out the 3's, we have 8 divided by ~~nr ~* ~r = 2
4, which is equal to 2.

142. The operations in cancellation depend on two princi-
ples :

1. The cancelling of a factor, in any number, is equivalent
to dividing the number by that factor.

2. If the dividend and divisor be both divided by the same
number, the quotient will not be changed.

PRINCIPLES AND EXAMPLES.

1. Divide 63 by 21.

ANALYSIS. Resolve tlie dividend and divi- OPERATION.
sor into factors, and then cancel those which 63 _ * x 9

are common.

"

2. In 7 times 56, how many times 8 ?

ANALYSIS. Resolve 56 into the OPERATION.

two factors 7 and 8, and then cancel 56x7_\$x7x7
the 8. -g- -J-

3. In 9 times 84, how many times 12 ?

4. In 14 times 63, how many times 7 ?

5. In 24 times 9, how many times 8 ?

6. In 36 times 15, how many times 45 ?

ANALYSIS. We see that 9 is a factor of 36
and 45. Divide by this factor, and write the OPERATK N.
quotient 4 over 36, and the quotient 5 below 4 3

45. Again, 5 is a factor of 15 and 5. Divide \$6 x I'SJ
15 by 5, and write the quotient 3 over 15. _ =1

Dividing 5 by 5, reduces the divisor to 1, which 40

need not be set down : hence, the true quotient \$

4x3=12.

141. What is cancellation ?

143. On what do the operations of rnneellitlon depend ?

CANCELLATION. 143

143. Therefore, to perform the operations of cancellation :

1. Resolve the dividend and divisor into such factors as
shall give all the factors common to both.

II. Cancel the common factors and then divide the product
of the remaining factors of the dividend by the product of the
remaining factors of the divisor.

NOTES. 1. Since every factor is cancelled by division, the quo-
tient 1 always takes the place of the cancelled factor, but is omit-
ted when it is a multiplier of other factors.

2. If one of the numbers contains a factor equal to the product
of two or more factors of the other, they may all be cancelled.

3. If the product of two or more factors of the dividend is equal
to the product of two or more factors of the divisor, such products
may ba cancelled.

4- It is generally more convenient to set the dividend on the
right of a vertical line and the divisor on the left.

EXAMPLES.

1. What number is equal to 36 multiplied by 13 and the
product divided by 4 times 9 ?

ANALYSIS. We may place the numbers whose OPERATION.

product forms the dividend on the right of a verti- ^ #0

cal line, and those which form the divisor on the A 10
left. We see that 4x9=36 ; we then cancel 4, 9,

and 36. Ans. 13.

2. What is the result of 20 x 4 x 12, divided by
10x16x3?

OPERATION.

ANALYSIS. First, cancel the factor 10, in 10
and 20, and write the quotients 1 and 2 above
the numbers. We then see that 16 x 3 48, and
that 4x12=48; cancel 16 and 3 in the divisor,
and 4 and 12 in the dividend ; hence, the quo-
tient is 2. Am

3. Divide the product of 126 x 16 x 3, by 7 x 12.

ANALYSIS We see that 7 is a factor OPERATION.

of 126 giving a quotient of 18. We 1

cancsl 7, and place 18 at the right of
126. We then cancel 6, in 12 and 18, \ *
and write the quotients 2 and 3. We ^

then cancal the factor 2, in 2 and 16, X

and set down the quotients 1 and 8. Ans. 3x8x3 = '
The product of 1x1 is the divisor,
and the product of 3 x 8 x 3 = 72, the dividend.

14:4: CANCELLATION.

4. What is the quotient of 3x8x9x7x15, divided by
63x24x3x5?

ANALYSIS. The 63 is cancelled by 7 x 9 ; 24
by 3 x 8 ; 3 aiid 5, by 15 ; hence, the quotient is 1.

OPERATION.
\$

H

5. Divide the product of 6x1x9x11, by 2x3x7x3
X21.

6. Divide the product of 4 X 14 x 16 x 24, by 7 x 8x32
Xl2.

7. Divide the product of 5 x 11 x 9 x 7 x 15 x 6, by 30 x 3
x21 x3x5.

8. Divide the product of 6 x 9 x 8 x 11 x 12 x 5, by 27 x 2
x 32 x 3.

9. Divide the product of 1 x 6 x 9 x 14 x 15 x 7 x 8, by 36
x 126x56x20.

10. Divide the product of 18 x 36 x 72 x 144, by 6 x 6 x 8
x 9x12x8.

11. Divide the product of 4 x 6 x 3 x 5, by 5 x 9 x 12 x 16.

12. Multiply 288 by 16, and divide the product by 8 x 9
x2x2.

13. In a certain operation the numbers 24, 28, 32, 49, 81,
are to be multiplied together and the product divided by
8x4x7x9x6: what is the result ?

14. Multiply 240 by 18 and divide the product by 6
times 90.

15. Divide 16 x 20 x 8 x 3, by 30 x 8 x 6.

16. How many pounds of butter worth 15 cents a pound,
may be bought for 25 pounds of tea at 48 cents a pound ?

1 7. How much calico at 25 cents a yard must be given
for 100 yards of Irish sheeting at 87 cents a yard ?

18. How many yards of cloth at 46 cents a yard must be
given for 23 bushels of rye at 92 cents a bushel ?

143. Give the rule for the operation of cancellation.

CANCELLATION. 145

19. How many bushels of oats at 42 cents a bushel must
be given for 3 boxes of raisins each containing 26 pounds, at
14 cents a pound ?

20. A man buys 2 pieces qf cotton cloth, each containing
33 yards at 11 cents a yard, and pays for it in butter at 18
cents a pound : how many pounds of butter did IIQ give ?

21. If sugar can be bought for 7 cents a pound, how many
bushels of oats at 42 cents a bushel must I give for 56 pounds ?

22. If wool is worth 36 cents a pound, how many pounds
must be given for 27 yards of broadcloth worth 4 dollars a
yard?

23. If cotton cloth is worth 9 cents a yard, how much
must be given for 3 tons of hay worth 15 dollars a ton ?

24. How much molasses at 42 cents a gallon must be given
for 216 pounds of sugar at 7 cents a pound?

Online LibraryCharles DaviesSchool arithmetic. Analytical and practical → online text (page 10 of 24)