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: 3. If 4 of a yard of cloth cost $3, what is the price per yard ?

Ans. $51. 4. If $214 will buy 71% barrels of apples, how much are they per barrel ?

Ans. $28 5. If 44 gallons of molasses cost $2}, how much is it per quart?

Ans. 1511 fcts. 6. If i hhd. of wine cost $2501, how much is the wine per quart ?

Ans. $14:1=88 1911cts.

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DECIMAL FRACTIONS. $ 158. If the unit 1 be divided into 10 equal parts the parts are called tenths, because each part is one tenth of unity.

If the unit 1 be divided into one hundred equal parts, the parts are called hundredths, because each part is one hundredth of unity.

• If the unit 1 be divided into one thousand equal parts, the parts are called thousandths, because each

is one thousandth of unity : and we have similar expressions for the parts, when the unit is di. vided into ten thousand, one hundred thousand, &c. equal parts.

The division of the unit into tenths, hundredths, thousandths, &c. forms a system of numbers called Decimal Fractions.

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From which we see, that in each case the denominator gives denomination to the fraction; that is, determines whether the parts are tenths, hundredths, thousandths, &c.

§ 159. The denominatòr's of decimal fractions are seldom set down. The fractions are usually expressed by means of a point, or comma, placed at the left of the numerator. Thus, four tenths,

,4
forty-five hundredths, .
100 and twenty-five hundredths, 125
1047 ten thousandths,“-

,1047 The denominator of every decimal fraction, however, is always understood. It is a unit 1 with as many ciphers annexed as there are places of figures in the numerator.

The place next to the decimal point is called tenths; the next place to the right, the place of hun. dredths; the next, the place of thousandths; and so on for places further to the right according to the following Table.

DECIMAL NUMERATION TABLE,

Hundredths.

Thousandths.

Tons of thousandths.

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Hundreds of thousan.

Millionths.

Tens of millionths.

Read

4 tenths. ,6 4

64 hundredths. ,0 6 4

64 thousandths. ,6 5 4

6754 ten thousandths. 2 3 4

1234 hundred thousandths. 6 5 4 7654 millionths. 3 6 0 4 43604 ten millionths.

Ø 160. Decimal fractions are numerated from the left hand to the right, beginning with the tenths, hundredths, &c. as in the table.

The 64 hundredths is made up of 6 tenths and 4 hundredths; and the 64 thousandths is made up of 6 hundredths and 4 thousandths. Had it been required to express 4 tenths, 4 hundredths, &c. we should have written Four tenths,

,4. Four hundredths,

,0 4. Four thousandths,

,0 0 4. Four ten thousandths,

,000 4. Four hundred thousandths, ,000 0 4. Four millionths,

,00 000 4. Four ten millionths,

,00 0 0 0 0 4. Here we see, that the same figure expresses

dif. ferent values, according to the place which it occupies. But to of

16 is equal to Ito =,04. I's of Ito

17 =,004. ਠ of ਠ ਠ

ruit =,0004. To of Ιστοσ

Tothor =,00004. ty of Ισσσσσ

Totítod =,000004. To of τσσσσσσ

Toot77 =,0000004. $ 161. Therefore the value of the place diminishes in a tenfold proportion from the left hand to the right.

Hence, ten of the parts in any one of the places, are equal to one of the parts in the place next to the left; that is, ten thousandths make one hundredth, ten hundredths make one tenth, and ten tenths a unit 1.

This law of increase from the right hand towards the left, is the same as in whole numbers. Therefore, while numbers and decimal fractions may be united by placing the decimal point between them.

Thus, 36, 95 thirty-six, and 95 hundredths. 127,4

127 and four tenths.
163, 03 163 and three hundredths.

627, 0047 627 and 47 ten thousandths. The whole numbers increase in a tenfold propor. tion from the decimal point towards the left, and the decimals diminish in the same proportion from the decimal point towards the right.

A number composed partly of a whole number and partly of a decimal, is called a mixed number.

Write the following numbers.
Forty one, and three tenths.

41,3.
Sixteen, and three millionths. 16,000003.
Five, and nine hundredths.
Sixty-five, and fifteen thousandths.
Eighty, and three millionths.
Two, and three thousand millionths.
Four hundred and ninety-two thousandths.
Three thousand and twenty-one ten thousandths.
Forty seven, and twenty-one ten thousandths,
Thirty-nine, 3 tenths 4 thousandths 1 millionth.

$ 162. The denominations of Federal Money will correspond to the decimal division, if we regard 1 dollar as the unit. For, the dimes are tenths of the dollar, the cents are hundredths of the dollar, and the mills, being tenths of the cent, are thousandths of the dollar. Ec. 1. Express $47 and 25 cents decimally.

Ans. $47,25. 2. Express $39,39 cents and 7 mills decimally,

Ans. $39,397. 3. Express $12 and 3 mills decimally.

Ans. $12,003, 4. Express $147 and 4 cents decimally.

Ans. $147,04.

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$ 163. If ciphers be annexed to the numerator of a decimal fraction, the same number of ciphers must also be annexed to the denominator; for there must be as many ciphers in the denominator as there are places of figures in the numerator § 159. The numerator and denominator will therefore be multiplied by the same number, and the value of the fraction will not be changed $ 118. Hence,

Annexing ciphers to a decimal fraction does not alter its value.

Ez. 1. The fraction of is equal to 7%= that is, =,65,603,600.

§ 164. If ciphers be prefixed to the numerator of a decimal fraction, that is, placed at the left hand of the significant figures, the same number of ciphers must be annexed to the denominator. Now, the numerator will remain unchanged-while the denominator will be increased ten times for every cipher which is annexed, and the value of the fraction will be decreased in the same proportion $ 116. Hence,

Prefixing ciphers to a decimal fraction diminishes its value ten times for every cipher so prefixed.

Et. Take the fraction ,3. Prefixing a cipher, it becomes ,03, prefixing another it becomes ,003. But ,03 is one tenth of ,3, and ,003 is one hundredth of ,3.

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QUESTIONS. $ 158. When the unit 1 is divided into 10 equal parls, what is each part called? What is each part called when it is di. vided in 100 equal parts ? When into 1000, 10,000, &c. How are decimal fractions formed ? What gives denomination to the fraction ?

ģ 159. Are the denominators of decimal fractions generally set down ? How are the fractions expressed ? Is the der nator understood ? What is it? What is the place : decimal points called? The next ?-the third ? &c.

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