3. Eighteen and five hundredths. 4. Twenty-nine and three thousandths 7. Seventy-five and nine tenths. 8. Two thousand and two thousandths. 9. Eighteen and eighteen thousandths. 10. Five hundred five and one thousand six millionths. 11. Three hundred and forty-two ten millionths. 12. Twenty-five hundred and thirty-seven billionths. ART. 182. It will be seen that decimals increase from right to left, and decrease from left to right, in the same ratio as simple numbers; hence they may be added, subtracted, multiplied, and divided, in the same manner ADDITION OF DECIMALS. ART. 183. Ex. 1. Add together 5.018, 171.16, 88.133, Ans. 1392.09356. 1113.6, .00456, and 14.178. OPERATION. 5.0 18 171.16 8 8.1 3 3 1113.6 .00456 14.178 139 2.0 9 356 We write the numbers so that figures of the same decimal place shall stand in the same column, and then, beginning at the right hand, add them as whole numbers, and place the decimal point in the result directly under those above. RULE. Write the numbers so that figures of the same decimal place shall stand in the same column. Add as in whole numbers, and point off, in the sum, from the right hand as many places for decimals as equal the greatest number of decimal places in any of the numbers added. Proof. The proof is the same as in addition of simple numbers. EXAMPLES FOR PRACTICE. 2. Add together 171.61111, 16.7101, .00007, 71.0006, and 1.167895. Ans. 260.489775. 3. Add together .16711, 1.766, 76111.1, 167.1, .000007, and 1476.1. Ans. 77756.233117. QUESTIONS. Art. 182. How do decimals increase and decrease? How may they be added, subtracted, multiplied, and divided? — Art. 183. How are decimals arranged for addition? What is the rule for addition of decimals? What is the proof? 4. Add together 151.01, 611111.01, 16.5, 6.7, 46.1, and .67896. Ans. 611331.99896. 5. Add fifty-six thousand and fourteen thousandths, nineteen and nineteen hundredths, fifty-seven and forty-eight ten thousandths, twenty-three thousand five and four tenths, and fourteen millionths. Ans. 79081.608814. 6. What is the sum of forty-nine and one hundred and five ten thousandths, eighty-nine and one hundred seven thousandths, one hundred twenty-seven millionths, forty-eight ten thousandths? Ans. 138.122427. 7. What is the sum of three and eighteen ten thousandths, one thousand five and twenty-three thousand forty-three millionths, eighty-seven and one hundred seven thousandths, fortynine ten thousandths, forty-seven thousand and three hundred nine hundred thousandths? Ans. 48095.139833. SUBTRACTION OF DECIMALS. ART. 184. Ex. 1. From 74.806 take 49.054. OPERATION. 7 4.8 06 49.0 5 4 25.7 52 Ans. 25.752. Having written the less number under the greater, so that figures of the same decimal place stand in the same column, we subtract as in whole numbers, and place the decimal point in the result, as in addition of decimals. RULE. Write the less number under the greater, so that figures of the same decimal place shall stand in the same column. Subtract as in whole numbers, and point off the remainder as in addition of decimals. Proof. The proof is the same as in subtraction of simple numbers. QUESTIONS. Art. 184. What is the rule for subtraction of decimals? What is the proof? Ans. 99.999. Ans. 72.927. 10. From 100 take .001. 12. From three hundred sixty-five take forty-seven ten thousandths. Ans. 364.9953. 13. From three hundred fifty-seven thousand take twentyeight and four thousand nine ten millionths. 14. From .875 take .4. 15. From .3125 take .125. 17. From 3.7 take 1.8. 20. From .666 take .041. Ans. 356971.9995991. MULTIPLICATION OF DECIMALS. ART. 185. Ex. 1. Multiply 18.72 by 7.1. OPERATION. 18.72 7.1 1872 13104 132.912 =132812 = Ans. .475 Ans. .1875 Ans. .51. Ans. 1.9. Ans. 5.4375. Ans. 7.875. Ans. .625. Ans. 132.912. We multiply as in whole numbers, and point off on the right of the product as many figures for decimals as there are decimal figures in the multiplicand and multiplier counted together. The reason for pointing off decimals in the product as above will be seen, if we convert the multiplicand and multiplier into common fractions, and multiply them together. Thus, 18.72 1873 1872; and 7.1 = 71% = . Then 1872 × 70 =132812 132.912, Ans., the same as in the operation. Ex. 2. Multiply 5.12 by .012. OPERATION. 5.12 .012 1024 512 .0 6144 Ans. = 1000 Since the number of figures in the product is not equal to the number of decimals in the multiplicand and multiplier, we supply the deficiency by placing a cipher on the left hand. The reason of this process will appear, if we perform the question thus: 5.12 = 51% = 12, and .012= 1880. Then 512 6144 .06144, Ans., the same as before. Art. 185. In multiplication of decimals how do you point off the product? Will you give the reason for it? When the number of figures in the product is not equal to the number of decimals in the multiplicand and multiplier, what must be done? RULE.-Multiply as in whole numbers, and point off as many figures for decimals, in the product, as there are decimals in the multiplicand and multiplier. If there be not so many figures in the product as there are decimal places in the multiplicand and multiplier, supply the deficiency by prefixing ciphers. NOTE. When a decimal number is to be multiplied by 10, 100, 1000, &c., remove the decimal point as many places to the right as there are ciphers in the multiplier; and if there be not figures enough in the number, annex ciphers. Thus, 1.25 X 10: = 12.5; and 1.7 X 100 = 170. Proof. The proof is the same as in multiplication of simple numbers. EXAMPLES FOR PRACTICE. 3. Multiply 18.07 by .007. Ans. .12649. Ans. 18.58922. Ans. .00000114. Ans. 1137. Ans. 2.20947. Ans. .00046967. Ans. 22.09. 10. Multiply eighty-seven thousandths by fifteen millionths. Ans. .000001305. 11. Multiply one hundred seven thousand, and fifteen ten thousandths by one hundred seven ter thousandths. Ans. 1144.90001605. 12. Multiply ninety-seven ten thousandths by four hundred, and sixty-seven hundredths. Ans. 3.886499. 13. Multiply ninety-six thousandths by ninety-six hundred thousandths. Ans. .00009216. Ans. 1. 14. Multiply one million by one millionth. Ans. .14. 16. Multiply one hundred one thousandths by ten thousand one hundred one hundred thousandths. Ans. .01020201. 17. Multiply one thousand fifty, and seven ten thousandths by three hundred five hundred thousandths. Ans. 3.202502135. 18. Multiply two million by seven tenths. Ans. 1400000. QUESTIONS. What is the rule for multiplication of decimals? What is the proof? How do you multiply a decimal by 10, 100, 1000, &c. ? 19. Multiply four hundred, and four thousandths by thirty and three hundredths. Ans. 12012.12012. 20. What cost 46lb. of tea at $1.125 per pound? Ans. $51.75. 21. What cost 17.125 tons of hay at $18.875 per ton? Ans. $323.234375. 22. What cost 18lb. of sugar at $0.125 per pound? Ans. $2.25. 23. What cost 375.25bu. of salt at $0.62 per bushel ? Ans. $232.655. DIVISION OF DECIMALS. ART. 186. Ex. 1. Divide 45.625 by 12.5. Ans. 3.65. OPERATION. We divide as in whole numbers, and 12.5) 4 5.6 2 5 (3.65 since the divisor and quotient are the 375 812 750 625 625 two factors, which, being multiplied together, produce the dividend, we point off two decimal figures in the quotient, to make the number in the two factors equal to the product or dividend. = The reason for pointing off will also be seen by performing the question with the decimals in the form of common fractions. Thus, 45.625 = = 125. Then 45625 125 = 3653.65, Ans., as = 45,8820% 45625 and 12.5 = 12% 1000 = = 45625 = 456250 = 185 before. Ex. 2. Divide .175 by 2.5. OPERATION. 2.5).175 (.07 175 Ans. .07. We divide as in whole numbers, and since we have but one figure in the quotient, we place a cipher before it, which removes it to the place of hundredths, and thus makes the decimal places in the divisor and quotient 'equal to those of the dividend. The reason for prefixing the cipher will appear more obvious by solving the question with the decimals in the form of common fractions. Art. 186. In division of decimals how do you point off the quotient? What is the reason for it? If the decimal places of the divisor and quotient are not equal to the dividend, what must be done? |