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that $1000 gives $7.50

300 2.25
70

.525
6

-.045

For A's real estate=$10.32
In like manner we find A's tax for personal estate to be $8.6175
His 3 polls, at $1.25 each,

3.75

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NOTE.-A Tax is a sum of money required of individuals by government, for the use of the nation: or by a town, county, society, or corporation, for the purpose of defraying expenses. A tax is sometimes imposed by government, in small sums, upon every male citizen above a certain age; and this being the same to all, not varying with the amount of an individual's property, is called a tax of so much a head, or so much a poll: the word poll meaning head. Or, the more concise expression is poll-tax. In some states no poll.tax is allowed. Other taxes are usually rated or assessed on property. Property is of two kinds, real and personal: real property, or real estate, consists of possessions which are fixed and immoveable; as lands and buildings. Personal property compre. hends all other possessions, which of course are moveable.

2. A town is valued at $298,769.50, and is taxed $6,177.89. There are 675 polls taxed $.30 each: what is the tax on a dollar ?

Ans. $.02 on $1. 3. Suppose a tax of $755 be laid on a town, and the inventory of all the estates in the town amounts to $9345: what must B pay, whose estate is valued at $149 ?

Ans. $12.037+ 4. A certain town is taxed $2140; the whole property of the town is valued at $500000; there are 200 polls, which are taxed $.70 each; A's property is valued at $1400, and he pays but one poll: what will be the tax on $1, and what will be A's tax?

Ans. $.004 on a dollar, and A's tax $6.30.

REMARK.-The scholar will bear in mind, that the table in the preceding page was computed at 7} mills on the dollar; therefore it will not answer for any other rate.

5. A certain town, valued at $64530, raises a tax of $2259.90; there are 540 polls, which are taxed $.60 each: what is the tax on a dollar, and what will be A's tax, whose real estate is valued. at $1340, his personal property at $874, and who pays for 2 polls ?

Ans. 67.62.

ing

Questions. What is a lax? What is the meaning of the word poll ?-real estate ?--personal ? What is (first) taken out of the whole tax |--remainder how assessed ? How assessed on real estate? How on personal estate ?

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VULGAR FRACTIONS. * FRACTIONS are either proper, improper, compound, or mixed.

A proper fraction, is a fraction whose numerator is less than its denominator; as, ļ, 11

3 13, &c. An improper fraction, is a fraction whose numerator is larger than its denominator; as, 5, 3, 17, &c.

A compound fraction, is a fraction expressed in a compound form, being a fraction of a fraction, or two or more fractions connected together, as of of of 5 of ki, which are read thus one-half of three-fourths of two-sevenths of five-elevenths of nineteen-twentieths, &c.

Any whole number may be made an improper fraction, by drawing a line under it, and putting a unit, or 1, for a denominator, as, 6 may be expressed fractionwise, thus, j, and 12 thus, 42, &c.

A prime number is that which can only be measured (that is, divided) by itself or a unit, as 5, 7, &c.

That number which is produced by multiplying several numbers together, is called a composite number; thus, 360 is a composite number produced by 3x4x6x5=360, &c.

1. If an apple be cut into four parts, by what fraction is 1 part expressed ?—2 parts ?—3 parts ?–4 parts ?-how many parts make a whole apple?

2. What part of an orange is a į part of 3 oranges ?
3. Reduce to a decimal.

Ans. .375.
4. Reduce Iqr. 2na. to a decimal of a yard.
5. What is the value of .9375 of an acre ? Ans. 3R. 30P.
6. What is the largest common measure of jo? Ans. 8.
7. Reduce 1779 to its lowest terms?

Ans. 1 8. Reduce 996 to its lowest terms?

Ans. The former rule given to reduce a fraction to its lowest termis, was to divide both terms of the fraction by its largest common measure; but it can be done thus,-Divide both of the terms of the fraction by any number that will divide them without a remainder, and the quotient again in the same manner, and so on as long as they are susceptible of division by a common measure. 1. Reduce 288 to its lowest terms.

Thus, 8)488=26=*= the answer.
2. Reduce to its lowest terms.
3. Reduce to its lowest term.
4. Reduce

12

to its lowest term.

Ans.

Questions.--How many kinds of vulgar fractions are there? What are they called? What is a proper fraction ?-Improper ?-Compound ?-Mixed ? How is a whole made an improper fraction? What is called a composite number?

CASE 2. To reduce a mixed number to its equivalent improper fraction.

RULE.

-Multiply the whole number by the denominator of the fraction, and add the numerator of the fraction to the product fur the numerator, under which write the given denominator.

EXAMPLES.

1. Reduce 59 to its equivalent improper fraction.

59 7

4 the fraction sought." 2. Reduce 1274, to its equivalent improper fraction.

Ans. 213. 3. Reduce 653 to its equivalent improper fraction.

Ans. 410. 4. Reduce 1511 to its equivalent improper fraction.

Ans. 12 CASE 3. To reduce a whole number to an equivalent fraction, having a given denominator.

RULE.

Multiply the whole number by the given denominator; place the product over the said denominator, and it will form the fraction required.

EXAMPLES.

1. Reduce 6 to a fraction, whose denominator shall be 8.

Thus, 64.8=48 and 48 the ans.

Proof. 48=48--8=6. 2. Reduce 15 to a fraction, whose denominator shall be 12.

Ans. 1

CASE 4.7 To reduce an improper fraction to its equivalent whole or mixed number.

* All fractions represent a division of a numerator by the denominator, and are taken together as proper and adequate expressions of the quotient. Thus, & denotes 3 to be divided by 4, and =.75.

+ This Case is evidently the reverse of Case 2.

2. Reduce of foff of 1 to a single fraction.

Yultiply all the numerators continually together for a new numerator, and all the denominators for a new denominator, and they will form the simple fraction required.

If part of the compound fraction be a whole or mixed nur zaber, it must be reduced to an improper fraction, by

If the denominator of any number of a compound fracthese equal numerators and denominators may be expunged, tion be equal to the numerator of another number thereof, will produce the fractions required in lower terms. and the other numbers continually multiplied (as by the rule) |

Reduce of of off to a single fraction. Or, by, expunging the equal numerators and denominators, it will give as before.

REDICTION OF FTLSAR FRACTIOXS.

RTLE. Divide the numerator by the denominator, the quotient will be the whole number, and the remainder, if any, will be tłe numerator to the giren denominator.

EXAMPLES. 1. Reduce '1' to its equiralent, whole, or mixed number.

8)293

364 Ans. 2. Reduce 'H' to its equivalent, whole, or mixed num

Ans. 1274. 3. Reduce to its equivalent whole number. Ans. 9.

CASE 5.
To reduce a compound fraction to an equivalent simple

RULE.

EXAMPLES.
IX 2X 3x4

==}. the ans.
2x 3x4x 5

Thus, 1x2x3x4

Case 2d or 3d

=} Ans.

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Ans. :

REDUCTION OF VULGAR FRACTIONS.

113 3. Reduce $ of } of 4 of 124 to a single fraction.

Ans. 1=111 4. Reduce 1 of 37 of f to a single fraction. Ans.

CASE 6.

To reduce fractions of different denominations to equivalent fractions baving a common denominator.

RULE

Multiply each numerator into all the denominators, except its own, for a new numerator, and all the denominators into each other continually, for a common denominator.

EXAMPLES.

1. Reduce 1, 4, and , to equivalent fractions, having a common denominator.

1x5x840, the new numerator for 4
2x4x8=64,

do.
5x 4 x 5=100,

do. 4x5x8=160, the common denominator. Therefore the new equivalent fractions are 140, Motos

and 185, the answer.

2. Reduce , }, }, 6, and }, to fractions having a common denominator. Ans. 179, 194, 195, list:

3. Reduce }, šof 5, 74, and is, to a common denominator.

Ans. 238, 184, 1872, 1842 4. Reduce 1'}, of 11, 12, and , to a common denomi. nator.

Ans. 1990, 1896, 1996, 1996

CASE 7. To reduce any given fractions to others, which shall have the least common denominator.

4.32

RULE.

Find the least common multiple (already taught) of all the denominators of the given fractions, and it will be the common denominator required.

Divide the common denominator by the denominator of each fraction, and multiply the quotient by the numerator, and the product will be the numerator of th fraction required.

EXAMPLES.

1. Reduce }, 4, and }, to fractions having the least common denominator possible.

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