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 I.-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. EXAMPLES IN NOTATION. 1. Express 45 thousandths in the form of a decimal. OPERATION. 45 thousandths =.045. Ans. SOLUTION 1ST.-45 thousandths equals 40 thousandths plus 5 thousandths, or 4 hundredths and 5 thousandths; hence we write the 5 in the third or 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 4 and place the decimal point before it, and we have .045. Rule I.-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- | 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. 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 .16 to a common fraction. SOLUTION.-.75 expressed in the form of a common fraction equals, which reduced to its lowest terms, becomes . SOLUTION.-.163 is 163 hundredths, which, by writing the denominator, becomes 163 OPERATION. .75=70%, Ans. OPERATION. 50 163 3 163 100 100 100 300, Ans. Rule. Write the denominator under the decimal, omitting the decimal point, and reduce the common fraction to 271. To reduce a common fraction to a decimal. 1. Reduce to a decimal. SOLUTION. equals of 7. 7 equals 70 tenths; of 70 tenths is 8 tenths and 6 tenths remaining: 6 tenths equal 60 hundredths; of 60 hundredths is 7 hundredths and 4 hundredths remaining: 4 hundredths equal 40 thousandths; of 40 thousandths is 5 thousandths. Therefore equals .875. OPERATION. }=} of 7= 8,7.000 Rule.-I. 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: 14. 8. 15.7.0. 16. 1.207. Ans. .0603. 19. 5.00. Ans. 5.00015625. Ans. 7.016. 20. 5.307. Ans. 1.2015. 21. 7.30123. 307 Ans. .177662. 22. 6.30. 17. 1978. Ans. 5.314. Ans. 7.303875. Ans. 6.315625. 18. 3.00. Ans. 3.0000625. | 23. 7.30023. Ans. 7.302875. CASE III. 272. To reduce decimals to a common denominator. OPERATION. 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 decimal places; .875 occupies three places, expressing thousandths; hence each of the other decimals must occupy three places that they may express thousandths. .25 equals .250 and .4 equals .400. .875 .250 .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. 3. .523, 4.36, and 5.0315. 4. 3, .4036, and 5.018. 5. .375, 18, and 1. 81 Ans. .250, .025, .370. Ans. .5230, 4.3600, 5.0315. Ans. .3750, .4036, 5.0640. Ans. .37500, .64000, .53125. 6. .8135, 5.033, and 14. 이 99 Ans. .813500, .506250, 5.034000, .265625. 7. .45302, 2, .015, and 2.002 0. Ans. .45302000, .49500000, .01525000, 2.00003125. 8. 101.01, 426, 1600, and . Ans. 101.017500, 42.187500, .005625, .800000. 9. 75119, 7, 101.0175, and .005625. Ans. 75119.037500, .875000, 101.017500, .005625. 10. .00097656, 15, .125. Ans. .00097656, .46875000, .12500000. 11. .301881, 4.008, 5.7812, .29167. 143000, 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 the same order shall stand in the same column, and begin at the right to add. 7 thousandths plus 8 thousandths are 15 thousandths, which equals 1 hundredth and 5 thousandths; we write the 5 thousandths, and add the 1 hundredth to the next column: 1 and 9 are 10 and 5 are 15 and 7 are 22 hundredths, which equals 2 tenths and 2 hundredths; we write the 2 hundredths and add the tenths to the next column, etc. OPERATION. 45.37 56.508 75.45 86.497 263.825 Rule.-I. Write the number so that terms of the same order stand in the same column. 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. Ans. 473.343. 5. Add 4.25, 3, 463.2504, 5.016, .4036. Ans. 477.113102. 7. Add 500.0006, 6, 5.03, .7654, .001. 5 Ans. 506.30725. 8. Add .4532, 7.00, 1005.7001, 2, .000. Ans. 1013.78646. 9. Add 60-1981, 50.305, 6850.275, 1, .0000. Ans. 6961.227016. 10. Add .432758, .21, .29999997, .00000003. 11. Add .223+.33333+.4444444. 12. Find the sum of 2 decimal units of the the 3d order, 4 of the 4th, 31 of the 5th, 5 93 of the 7th order. Ans. .982758. 2d order, 2 of of the 6th, and Ans. .02295725. |