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UNITED STATES MONEY.

204. The denominations of United States Money 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.

EXAMPLES.

1. Express $39 and 39 cents and 7 mills, decimally.
2. Express $12 and 3 mills, decimally.

3. Express $147 and 4 cents, decimally.
4. Express $148 4 mills, decimally.
5. Express $4 6 mills, decimally.

6. Express $9 6 cents 9 mills, decimally.
7. Express $10 13 cents 2 mills, decimally.

ANNEXING AND PREFIXING CIPHERS.

205. Annexing a cipher is placing it on the right of a number.

If a cipher is annexed to a decimal it makes one more decimal place, and therefore, a cipher must also be annexed to the denominator (Art. 202).

The numerator and denominator will therefore have been multiplied by the same number, and consequently the value of the fraction will not be changed (Art. 161): hence,

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

We may take as an example, .3=.

If we annex a cipher. to the numerator, we must, at the same time, annex one to the denominator, which gives,

204. If the denominations of Federal Money be expressed decimally what is the unit? What part of a dollar is 1 dime? What part of a dime is a cent? What part of a cent is a mill? What part of a dollar is 1 cent? 1 mill?

205. When is a cipher annexed to a number? Does the annexing of ciphers to a decimal alter its value? Why not? What do three tenths become by annexing a cipher? What by annexing two ciphers? Three ciphers? What do 8 tenths become by annexing a cipher? By annexing two ciphers? By annexing three ciphers?

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If a decimal point be placed on the right of an integral number, and ciphers be then annexed, the value will not be changed thus, 5 5.0 5.00 5.000, &c.

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206. Prefixing a cipher is placing it on the left of a number.

If ciphers are prefixed to the numerator of a decimal fraction, 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 annexed; and hence, the value of the fraction will be diminished ten times for every cipher prefixed to the numerator (Art. 160).

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

Take, for example, the fraction .2.

.2 becomes 100%

= .02

by prefixing one cipher,

.2 becomes = .002 by prefixing two ciphers,

002 1000

.2 becomes 2.0002 by prefixing three ciphers: in which the fraction is diminished ten times for every cipher prefixed.

ADDITION OF DECIMALS.

207. It must be remembered, that only units of the same kind can be added together. Therefore, in setting down decimal numbers for addition, figures expressing the same unit must be placed in the same column.

206. When is a cipher prefixed to a number? When prefixed to a decimal, does it increase the numerator? Does it increase the denominator? What effect then has it on the value of the fraction? What do.2 become by prefixing a cipher? By prefixing two ciphers? By prefixing three? What do .07 become by prefixing a cipher? By prefixing two? By prefixing three? By prefixing four?

207. What parts of unity may be added together? How do you set down the numbers for addition? How will the decimal points fall? How do you then add? How many decimal places do you point off in the sum ?

The addition of decimals is then made in the same manner as that of whole numbers.

1. Find the sum of 37.04, 704.3, and .0376.

Place the decimal points in the same column: this brings units of the same value in the same column: then add as in whole numbers: hence,

OPERATION.

37.04

704.3

.0376

741.3776

RULE.-I. Set down the numbers to be added so that figures of the same unit value shall stand in the same

column.

II. Add as in simple numbers, and point off in the sum from the right hand, as many places for decimals as are equal to the greatest number of places in any of the numbers added. PROOF.-The same as in simple numbers.

EXAMPLES.

1. Add 4.035, 763.196, 445.3741, and 91.3754 together. 2. Add 365.103113, .76012, 1.34976, .3549, and 61.11 together.

3. 67.407+97.004+4+.6+.06+.3.

4. .0007+1.0436+.4+.05+.047.

5. .0049+47.0426+37.0410+360.0039.

6. What is the sum of 27, 14, 49, 126, 999, .469, and .2614?

7. Add 15, 100, 67, 1, 5, 33, .467, and 24.6 together. 8. What is the sum of 99, 99, 31, .25, 60.102, .29, 100.347?

and

9. Add together .7509, .0074, 69.8408, and .6109. 10. Required the sum of twenty-nine and 3 tenths, four hundred and sixty-five, and two hundred and twenty-one thousandths.

11. Required the sum of two hundred dollars one dime three cents and 9 mills, four hundred and forty dollars nine mills, and one dollar one dime and one mill.

12. What is the sum of one-tenth, one hundredth, and one thousandth?

13. What is the sum of 4, and 6 ten-thousandths?

14. Required, in dollars and decimals, the sum of one dollar one dime one cent one mill, six dollars three mills, four dollars eight cents, nine dollars six mills, one hundred dollars six dimes, nine dimes one mill, and eight dollars six cents.

15. What is the sum of 4 dollars 6 cents, 9 dollars 3 mills, 14 dollars 3 dimes 9 cents 1 mill, 104 dollars 9 dimes 9 cents 9 mills, 999 dollars 9 dimes 1 mill, 4 mills, 6 mills, and 1 mill?

16. If you sell one piece of cloth for $4,25, another for $5,075, and another for $7,0025, how much do you get for all?

17. What is the amount of $151,7, $70,602, $4,06, and $807,2659 ?

18. A man received at one time $13,25; at another $8,4; at another $23,051; at another $6; and at another $0,75 : how much did he receive in all?

19. Find the sum of twenty-five hundredths, three hundred and sixty-five thousandths, six tenths, and nine millionths.

20. What is the sum of twenty-three millions and ten, one thousand, four hundred thousandths, twenty-seven, nineteen millionths, seven and five tenths?

21. What is the sum of six millionths, four ten-thousandths, 19 hundred thousandths, sixteen hundredths, and four tenths?

22. If a piece of cloth cost four dollars and six mills, eight pounds of coffee twenty-six cents, and a piece of muslin three dollars seven dimes and twelve mills, what will be the cost of them all?

23. If a yoke of oxen cost one hundred dollars nine dimes and nine mills, a pair of horses two hundred and fifty dollars five dimes and fifteen mills, and a sleigh sixty-five dollars eleven dimes and thirty-nine mills, what will be their entire cost?

24. Find the sum of the following numbers: Sixty-nine thousand and sixty-nine thousandths, forty-seven hundred and forty-seven thousandths, eighty-five and eighty-five hundredths, six hundred and forty-nine and six hundred and forty-nine ten-thousandths?

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SUBTRACTION OF DECIMALS

208. Subtraction of Decimal Fractions is the operation of finding the difference between two decimal numbers.

1. From 3.275 to take .0879.

NOTE-In this example a cipher is annexed to the minuend to make the number of decimal places equal to the number in the subtrahend. This does not alter the value of the minuend (Art. 205): hence,

OPERATION.

3.2750

.0879

3.1871

RULE.-I. Write the less number under the greater, so that figures of the same unit value shall stand in the same column. II. Subtract as in simple numbers, and point off the deci mal places in the remainder, as in addition.

PROOF. Same as in simple numbers.

EXAMPLES.

1. From 3295 take .0879.

2. From 291.10001 take 41.375.
3. From 10.000001 take 111111.
4. From 396 take 8 ten-thousandths.
5. From 1 take one thousandth.

6. From 6378 take one-tenth.

7. From 365.0075 take 3 millionths.

8. From 21.004 take 97 ten-thousandths.

9. From 260.4709 take 47 ten-millionths.

10. From 10.0302 take 19 millionths.

11. From 2.01 take 6 ten-thousandths.

12. From thirty-five thousands take thirty-five thousandths.

13. From 4262.0246 take 23.41653.

14. From 346.523120 take 219.691245943.

15. From 64.075 take .195326.

16. What is the difference between 107 and .0007?

17. What is the difference between 1.5 and .3785?

18. From 96.71 take 96.709.

208. What is subtraction of decimal fractions? How do you set down the numbers for subtraction? How do you then subtract? How many decimal places do you point off in the remainder ?

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