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EXAMPLES IN NUMERATION. 1. Read the decimal .685. SOLUTION.—This expresses 6 tenths, 8 hundredths, and 5 thousandths; or since 6 tenths equals 600 thousandths, and 8 hundredths equals 80 thousandths, and all united equal 685 thousandths, it may also be read 685 thousandths; hence the following rules :

Rule 1.Begin at tenths, and read the terms in order towards the right, giving each its proper denomination.

Rule II.Read the decimal as a whole number, and give it the denomination of the last term at the right.

NOTE.—In the second method we may determine the denominator by numerating from the decimal point, and the numerator by numerating towards the decimal point.

Read the following decimals: 2. .73. 6. 7.039.

10. 146.0302056. 3. .241. 7. 8.1367. 11. 376.10070354. 4. 3.703. 8. 7.03081 12. 487.000081035. 5. 2.004 9. 9.10076 13. 586.0004003256.

EXAMPLES IN NOTATION. 1. Express 45 thousandths in the form of a decimal. SOLUTION 1st.–45 thousandths equals 40 thou- OPERATION. sandths plus 5 thousandths, or 4 hundredths and 5 45 thousandths. thousandths; hence we write the 5 in the third or

: =.045. Ans.

= 045 thousandths place, the 4 in the second or hundredths place, and fill the vacant tenths place with a cipher, and we have .045.

SOLUTION 2D.—We write the 45 and then, since the last figure must stand in the third or thousandths place, the denomination being thousandths, write a cipher before the 2 and place the decimal point before it, and we have .045.

Rule 1. Place the decimal point, and then write each term so that it may express its proper denomination, using ciphers when necessary.

Rule II.- Write the numerator, and then place the decimal point so that the right hand term shall be of the same denomination as the decimal.

Express the following in decimal form:

2. Four hundred and seventy-1 4, 9 tenths, 8 thousandths, and 7 five thousandths.

millionths. 3. Seven thousand four hundred 5 Five thousand and one miland sixty-five ten-thousandths. lionths.

6. Six hundred, and seven hun- | thousand, 4 hundred and forty-eight dredths.

and seven-ninths millionths. 7. Fifty thousand seven hundred 12. Nine hundred and twentyand six millionths.

six million, 4 thousand and 7 hun8, 9 thousandths, 6 hundred- dred millionths. thousandths, and two hundred-mil- 13. Four thousand and thirtylionths.

nine tenths. 9 One hundred and one thou- 14. Fifty-six million, and fiftysand one hundred and one ten-mil- six millionths. lionths.

15. Four thousand and two and 10. Two hundred and forty one-fifth hundredths. thousand, four hundred and six 16. 6 ten-thousandths, 5 milthousandths.

| lionths, and 337 billionths. 11. Six hundred and fifty-seven

Express the following fractions and mixed numbers decimally :

15. 43, 743, 45,6954 | 19. 4331, voitoo.
16. 10100) 104600
17. 24 400, 4100.

21. 4734465, 9451033600. 18. ' 633 423

22. 10000 104001400

6436 437 5 0 0 63
1000 T: 100000000

4

18. 98, 100:

4

2

1000

REDUCTION OF DECIMALS. 269. The Reduction of Decimals consists of three cases, as follows:

1st. To reduce decimals to common fractions.
2d. To reduce common fractions to decimals.
3d. To reduce decimals to a common denominator.

CASE I. 270. To reduce a decimal to a common fraction. 1. Reduce . 75 and also .16to a common fraction. SOLUTION.—.75 expressed in the form of a OPERATION. common fraction equals 18, which reduced to 75= 75=3. Ans. its lowest terms, becomes . SOLUTION.—.163 is 16 hundredths, which, OPERATION.

16? by writing the denominator, becomes 10

. 16 530 which equals , or 35%, which, reduced to = 300=%, Ans. its lowest terms, equals . Hence the following

Rule.- Write the denominator under the decimal, omitting the decimal point, and reduce the common fraction to its lowest terms.

Reduce the following to common fractions:
2. .125.

Ans. t. 12. .831. Ans. . 3. .3125.

Ans. e. 13. .938. Ans. 16 4. .73125. Ans. 167. 14. .081. Ans. The 5. 7.375. Ans. 73. 15. .063.

Ans. 15. 6. 5.008.

Ans. 5125: 16. 2.067. Ans. 216 7. 7.5555 Ans. 75. 17. 3.43%. Ans. 3116 8. 8.25625. Ans. 840. 18. 4.003. Ans. 4130 9. 7.46875. Ans. 745. 19. 5.006. Ans. 511o. 10. 9.65625. Ans. 931. 20. 6.104. Ans. 6785 11. 14.75325. Ans. 143.13. 21. 7.060%. Ans. 7 str.

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CASE II. 271. To reduce a common fraction to a decimal. 1. Reduce to a decimal. SOLUTION.- equals f of 7. 7 equals 70 tenths ; OPERATION. 3 of 70 tenths is 8 tenths and 6. tenths remaining :

{=} of 7= 6 tenths equal 60 hundredths ; } of 60 hundredths is 7 hundredths and 4 hundredths remaining : 4

8; 7.000 hundredths equal 40 thousandths; f of 40 thou

.875 sandths is 5 thousandths. Therefore ; equals .875.

Rule.-1. Annex ciphers to the numerator and divide by the denominator.

II. Point off as many decimal places in the quotient as there are ciphers used.

NOTES.-1. In many cases the division will not terminate; the common fraction cannot then be exactly expressed by a decimal. Such decimals are called interminate or infinite decimals.

2. The symbol + annexed to a decimal, indicates that it contains other decimal terms. The symbol — annexed to a decimal indicates that the last decimal term is increased by 1. This is often done when the next term is greater than 5.

Reduce the following common fractions to decimals : 2. 15. Ans. .9375., 8. 74. Ans. .7628866–. 3. 5.

Ans. .46875. 9. 291. Ans. .78515625. 4. 1. Ans. .65625. 10. 125. Ans. .1220703125. 5. 1 Ans. .796875.

Ans. .06%. Ans. .3793103+.

Ans. .00831. 7. 41. Ans. .8723404+. | 13. to. Ans. .0063.

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14. 100

Ans. .060. 19. 5.0064. 15. 7.04. Ans. 7.016. 20. 5.307.

Ans. 7.016. 20. 5.307. 16. 1.2075. Ans. 1.2015. 21. 7.30123. 17. 3075. Ans. .17766277. 22. 6.30. 18. 3.001ấo. Ans. 3.0000625. / 23. 7.30028.

Ans. 5.00015625.

Ans. 5.314. Ans. 7.303875. Ans. 6.315625. Ans. 7.302875.

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CASE JII. 272. To reduce decimals to a common denominator.

1. Reduce .4, .25, and .875 to a common denominator

SOLUTION.–For the decimals to have a common denominator, they must occupy the same number of

OPERATION. decimal places ; .875 occupies three places, express

.875 ing thousandths; hence each of the other decimals

.250 must occupy three places that they may express

.400 thousandths. .25 equals .250 and .4 equals .400.

Rule.- Annex ciphers to the simple decimals and expand the complex ones so as to make each decimal occupy the same number of decimal places.

NOTE.-Decimals will be reduced to their least common denominator when they are reduced to the same number of places as the decimal which occupies the greatest number of places. Reduce the following to their least common denominator : 2. .25, .025, .37.

Ans. .250, .025, .370. 3. .523, 4.36, and 5.0315. Ans. .5230, 4.3600, 5.0315. 4. 3, .4036, and 5.025. Ans. .3750, .4036, 5.0640. 5. .375, 26, and 12. Ans. .37500, .64000, .53125. 6. .8135, 1967, 5.03}, and 17.

Ans. .813500, .506250, 5.034000, .265625. 7. .45302, 2, .0154, and 2.003.10.

Ans. .45302000, .49500000, .01525000, 2.00003125. 8. 101.012, 4236, 167, and 1.

Ans. 101.017500, 42.187500, .005625, .800000. 9. 7511950, , 101.0175, and .005625.

Ans. 75119.037500, .875000, 101.017500, .005625. 10. .00097656, 15, .125.

Ans. .00097656, .46875000, .12500000. 11. .30149617, 4.008, 5.7832, .29167.

Ans. .300070, 4.008000, 5.783125, .291670.

ADDITION OF DECIMALS. 273. Addition of Decimals is the process of finding the sum of two or more decimals.

1. What is the sum of 45.37, 56.508, 75.45, and 86.497 ? SOLUTION.-We write the numbers so that terms of OPERATION. the same order shall stand in the same column, and

45.37 begin at the right to add. 7 thousandths plus 8 thou

56.508 sandths are 15 thousandths, which equals 1 hun

75.45 dredth and 5 thousandths; we write the 5 thousandths,

86.497 and add the 1 hundredth to the next column:1 and 9 are 10 and 5 are 15 and 7 are 22 hundredths, which

263.825 equals 2 tenths and 2 hundredths; we write the 2 hundredths and add the tenths to the next column, etc.

Rule.--I. Write the number so that terms of the same order stand in the same colunin.

II. Add as in whole numbers, and place the decimal point between the units and tenths of the sum.

NOTE.—When there are complex decimals, reduce all the decimals to a common denominator before adding.

2. Add 12.34, 432.015, 302.23, .00025. Ans. 746.58525. 3. Add 137.4263, 3426.01, 412.003, 3.0005.

Ans. 3978.4398. 4. Add 6340.205, .000632, 4.73, .00325, .99935.

Ans. 6345.938232. 5. Add 4.25, 3, 463.2504, 5.0.16, .4036. Ans. 473.343. 6. Add .000432, 400.25, 72.0026, 17, 4.32502.

Ans. 477.113102. 7. Add 500.0006, 80, 5.033, -7654, .001.

Ans. 506.30725. 8. Add .4532, 7.004, 1005.70011, , .000}.

Ans. 1013.78646. 9. Add 6011: 17, 50.305, 6850.275, 1\, .000045

Ans. 6961.227016. 10. Add .432758, .21, .29999997, .00000003.

Ans. .982758. 11. Add .22}+.3333}+.4444444.

Ans. 1. 12. Find the sum of 2 decimal units of the 2d order, 21 of the 3d order, 4} of the 4th, 31 of the 5th, 516 of the 6th, and 93 of the 7th order.

Ans. .02295725.

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