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thousandths; 5 in the 6th place, 5 millionths; 5 in the 7th place, 5 ten millionths; 5 in the 8th place, 5 hundred millionths; 5 in the 9th place, 5 billionths; and all of them taken together are read thus: five hundred fifty-five million, five hundred fifty-five thousand, five hundred fifty-five billionths.

NOTATION AND NUMERATION OF DECIMAL FRACTIONS.

Since decimal fractions decrease in a tenfold proportion from the left towards the right, they must increase in the same proportion from the right towards the left; hence it follows that they are subject to the same law of notation, and consequently to the same modes of operation, that whole numbers are.

NUMERATION OF DECIMAL FRACTIONS. Art. 103. Numeration of decimal fractions is the method of reading them. It will be perceived that the name of the parts expressed by the numerator is the ordinal number of its denominator, and the denominator is always known to be a unit with as many ciphers annexed as there are figures in the numerator. Since the numerator expresses the number of parts, and the ordinal number of its denominator expresses the name of those parts, we have the following rule for reading decimal fractions.

RULE. Read the numerator, or number of parts, as if it were a whole number, and then read the ordinal number of its denominator, or name of those parts.

Pupils should be required to read the following decimal

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NOTATION OF DECIMAL FRACTIONS.

Art. 104. Notation of decimal fractions is the method of writing or expressing them by their proper figures.

By examining the table, it will be perceived that a figure in the first place at the right of the decimal point expresses tenths, in the second place, hundredths, in the third place,

thousandths, &c. Hence we have the following rule for writing decimal fractions.

RULE. Write a figure expressing the number of tenths, hundredths, thousandths, &c., of the given decimal, each in its proper place, observing to write a cipher in the place of each of the parts omitted.

Pupils should be required to write the following decimal numbers.

1. Five hundred and five thousandths.

2. Seven thousand five hundred and fifteen ten thousandths.

3. Twenty-five thousand and seventy-five hundred thousandths.

4. Forty-eight thousand and eighty-nine hundred thousandths.

5. One hundred twenty thousand two hundred and five millionths.

6. Six hundred seventy-six thousand and forty-five ten millionths.

7. Five hundred thousand and forty-five hundred millionths.

8. Nine hundred eighty-seven thousand and twenty-five billionths.

9. Eight hundred twenty thousand and twenty-five ten billionths.

10. Seven hundred and fifteen thousand and forty-five trillionths.

Art. 105. Annexing a cipher to a decimal fraction reduces it to the next lower order or denomination, but does not alter its size or value, since every significant figure continues to occupy the same place; thus, .5, .50, and .500, are all of the same size or value, each being equal to or of a unit.

Prefixing a cipher to a decimal fraction reduces it to the next lower order or denomination, and also diminishes its size or value in a tenfold proportion by removing the significant figures further from the decimal point; thus, .5, .05, and .005, differ in size or value, .5 being equal to or 1, .05 being equal or, and .005 being equal to Too or zoo.

Two or more decimal fractions, each containing the same number of figures, have a common denominator; thus, .005, .050, and .500, have 1000 for a common denominator.

Two or more decimal fractions, each containing a different

number of figures, may be changed to fractions having a common denominator, by annexing so many ciphers as are necessary to make the number of figures in each fraction the same; thus, .5, .05, and .500, may be changed to .500, .050, and .500, which have 1000 for a common denominator. A mixed number, that is, a whole number and decimal annexed, is equal to an improper fraction whose numerator is all the figures of the mixed number, taken as a whole number, and whose denominator is that of the decimal part of the mixed number. Thus, 45.75 is equal to 4575, which is obvious from the method of changing a mixed number to an improper fraction.

The denominations of federal money are purely decimal, dollars being units or whole numbers, dimes tenths of a dollar, cents hundredths of a dollar, and mills thousandths of a dollar; consequently, federal money and decimal fractions are subject to the same methods of operation.

ADDITION OF DECIMAL FRACTIONS.

Art. 106. Since decimal fractions increase in a tenfold proportion from right to left, like whole numbers, we have the following rule for addition of decimal fractions.

RULE. Write the numbers to be added, whether mixed or pure decimals, placing whole numbers under whole numbers, tenths under tenths, hundredths under hundredths, &c.

Find their sum as in addition of simple numbers; and, from the right, point off so many figures for decimals as are equal to the greatest number of decimal figures in any of the given numbers.

1.. What is the sum of .5, .7, and .8?

3. What is the sum of 5.5,

10.7, and 25.8?

2. What is the amount of .25, .75, and .95?

4. What is the amount of 25.25 and 75.75?

5. What is the sum of 85.385, 848.25, .3085, 28.75, 3.4867, and .835?

Ans. 967.0152.

6. What is the amount of 538.25, 375.025, 211.0025, and 34.75? Ans. 1159.0275. 7. What is the sum of .85745, .68379, .53048, and .00025 ? Ans. 2.07197. 8. What is the amount of 25.000007, 145.643, 175.89, and 17.00348? Ans. 363.536487.

9. What is the sum of $20.25, $12.5, $10.125, and $5.05? 10. What is the amount of $45.375, $75.125, $125.625, and $475.875?

SUBTRACTION OF DECIMAL FRACTIONS. Art. 107. Since decimal fractions increase in a tenfold proportion from right to left, we have the following

RULE. Write the less number under the greater, placing whole numbers under whole numbers, tenths under tenths, hundredths under hundredths, &c.

Then proceed as in subtraction of simple numbers, and from the right of the remainder point off so many figures for decimals as are equal to the greatest number of decimal figures in either of the given numbers.

1. From .5 take .25.

3. Subtract .5 from 10.

5. From 3.75 take 1.8975.

6. Subtract .3785 from 1.5.

2. From .75 take .5.
4. Subtract .75 from 25.

Ans. 1.8525.

Ans. 1.1215.

7. From $236 take .125 of a dollar. Ans. $235.875. 8. From seventy-five take twenty-five millionths.

Ans. 74.999975. 9. Subtract twenty thousandths from twenty thousand.

10. From one million take one millionth.

Ans. 19999.980.

Ans. 999999.999999.

11. Subtract $1.375 from 10 dollars and 10 cents. 12. From 100 dollars take 75 dollars and 25 cents.

MULTIPLICATION OF DECIMAL FRACTIONS.

Art. 108. When a whole number, or a fraction, is multiplied by a whole number, the product will be greater than the multiplicand in the same proportion that the multiplier is greater than a unit; thus, 5 x5=25; here the product is 5 times as great as the multiplicand; and .5 X 5 25 tenths, or 2.5; here, also, the product is 5 times as great as the multiplicand.

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When a whole number, or a fraction, is multiplied by a fraction, the product will be less than the multiplicand in the same proportion that the multiplier is less than a unit; thus, 12 X.560 tenths, or 6 units; here the product is only .5 tenths or 1 half of the multiplicand; and .5 × .5=.25 hundredths; here, also, the product is only .5 tenths or 1 half of the multiplicand.

From the above illustrations it is plain, that when a whole

number is multiplied by a decimal fraction, or when a decimal fraction is multiplied by a whole number, the product will be in the lowest denomination, or order of parts, named in the given decimal; thus, 175 X .005 thousandths.875 thousandths.

It is also obvious, that when a decimal fraction is multiplied by a decimal fraction, the product will be in the denomination produced by multiplying the lowest denomination, or order of parts, in the multiplicand by the lowest in the multiplier; thus, .375 thousandths X .45 hundredths .16875 hundred thousandths. Hence the following general rule for the multiplication of decimal fractions.

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RULE. Multiply as in whole numbers, then point off so many figures in the product for decimals, counting from the right, as there are decimal figures in both factors. If there are not as many figures in the product, prefix so many ciphers as are necessary to make up the required number.

1. What is the product of .15 multiplied by 5?

3. What is the product of .75 multiplied by 6?

5. What is the value of 10 yards of cloth, at .125 of a dollar a yard?

7. Multiply 200 by .75. 8. Multiply 5000.5 by .05. 9. Multiply 12.386 by .354. 10. Multiply .3785 by .003.

2. What is the product of 50 multiplied by .5?

4. What is the product of 12.5 multiplied by .12?

6. What is the value of .7 of an acre of land, at 30 dollars an acre?

Product, 150.

Product, 250.025.

Product, 4.384644.

Product, .0011355.

11. What is the value of 12.5 tons of coal, at $6.75 a ton? 12. A merchant purchased a ship, for which he paid 12500 dollars; what is .625 of the ship worth?

13. A gentleman purchased a lot of land for 750 dollars; what is .875 of it worth?

14. If an insurance office charge .0125 of the value of a house for insuring it against fire during one year, what will be the expense of insuring a house valued at $4675?

Decimal fractions are multiplied by 10, 100, 1000, &c., by removing the decimal point so many figures towards the right as there are ciphers in the multiplier.

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