10. Required the sum of twenty-nine and 3 tenths, four hundred and sixty-five, and two hundred and twenty-one thousandths. 11. What is the sum of one-tenth, one-hundredth, and onethousandth? 12. Find the sum of twenty-five hundredths, three hundred and sixty-five thousandths, six-tenths, and nine-millionths. 13. What is the sum of twenty-three millions and ten, one thousand, four hundred thousandths, twenty-seven, nineteenmillionths, seven, and five-tenths? 14. What is the sum of six-millionths, four ten-thousandths, 19 hundred-thousandths, sixteen-hundredths, and four-tenths? 15. 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. 16. A gentleman bought 6 houses, for which he paid, as follows: 1st, 2785.625 dollars; 2d, 3964.75 dollars; 3d, 5762.1875 dollars; 4th, 4960.50 dollars; 5th, 6912.375 dollars; 6th, 9156.3125 dollars: what did the six houses cost? 17. A farmer sold, at different times, the following quan tities of hay: 3.75 tons, 14.165 tons, 375.16247 tons, 54.8125 tons, 18.5 tons, 21.75 tons, and 25 tons: how much hay did he sell? 18. A vessel sailed, in 9 successive days, the following distances: 240.17 miles, 315.875 miles, 87.416 miles, 195.125 miles, 269.1875 miles, 291.06 miles, 197.0106 miles, 300.47925 miles, and 200 miles: what distance did the vessel sail? 19. Add 475.62; nine hundred and twelve thousandths; four hundred and sixty thousandths; thirty-seven thousand, eight hundred and ninety-nine; one hundred and ninety-nine millionths; 176942.125, and two hundred and ninety-six tenthousandths. в SUBTRACTION. 157. SUBTRACTION OF DECIMALS is the operation of finding the difference between two decimal numbers. 1. From 3.275 take .0879. ANALYSIS.-The subtraction is performed as in whole numbers, because the units of place in decimals have the same relative values as in whole numbers. OPERATION. 3.2750 .0879 3.1871 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. 154): Hence, Rule. I. Write the less number under the greater, so that figures of the same unit value shall fall in the same column. II. Subtract as in simple numbers, and place the decimal point, in the remainder, directly under that of the subtrahend. PROOF.-The same as in whole numbers. Examples. 1. From 3295 take .0879. 2. From 291.10001 take 41.375. 3. From 10.000001 take .111111. 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. How 157. What is Subtraction of Decimals? How do you set down the numbers for subtraction? How do you then subtract? many decimal places do you point off in the remainder? 10. From 10.0302 take 19 millionths. 11. From 2.01 take 6 ten-thousandths. 12. From thirty-five thousand 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. MULTIPLICATION. 158. MULTIPLICATION OF DECIMALS is the operation of taking one of two decimal numbers as many times as there are units in the other. 1. Multiply 3.05 by 4.102. ANALYSIS.-We may change the factors into common fractions, and then multiply them: the product of the numerators will be the product of the decimals. Since each denominator contains as many ciphers as there are places in the numerator (Art. 151); and since the product of the denominators will contain as many ciphers as both the factors, it follows that the product of the numerators must have as many places of figures as there are in both factors: Hence, the following Rule. OPERATION. 3.05 = 3150 100 305 4102 1000 1000 305 4102 X 1251110 1000 3.05 4.102 610 305 12.20 12.51110 100000 Multiply as in simple numbers, and point off in the product, from the right hand, as many figures for decimals as there are decimal places in both factors; and if there be not so many in the product, supply the deficiency by prefixing ciphers. Examples. 1. Multiply the number 3.049 by .012. 3. Multiply the number 496.0135 by 1.496. 4. Multiply one and one-millionth by one-thousandth. 5. Multiply one hundred and forty-seven millionths by onemillionth. 6. Multiply three hundred, and twenty-seven hundredths by 31. 7. Multiply 31.00467 by 10.03962. 8. What is the product of five-tenths by five-tenths? 9. What is the product of five-tenths by five-thousandths? 10. Multiply 596.04 by 0.00004. 11. Multiply 38049.079 by 0.00008. 12. What will 6.29 weeks' board come to, at 2.75 dollars per week? 13. What will 61 pounds of sugar come to, at 0.234 of a dollar per pound? 14. If 12.836 dollars are paid for one barrel of flour, what will 354 barrels cost? 15. Multiply 49000 by .0049. 16. Bought 1234 oranges for 4.6 cents apiece: how much did they cost? 17. What will 375.6 pounds of coffee cost, at .125 dollar per pound? 18. If I buy 36.251 pounds of indigo at 0.029 of a dollar per pound, what will it come to? 19. Multiply $89.3421001 by .0000028. 20. Multiply $341.45 by .007. 21. What is the product of the decimal .004 by the decimal .004? 22. Multiply .007853 by .035. 23. What is the product of $26.000375 multiplied by .00007? 158. What is Multiplication of Decimals? What is the rule for multiplication? Contractions in Multiplication. 159. Removing the decimal point one place to the right, increases the unit of each place ten times; two places, one hundred times, &c. Therefore, when a decimal number is to be multiplied by 10, 100, 1000, &c., the multiplication may be made by removing the decimal point as many places to the right as there are ciphers in the multiplier; and if there be not so many figures on the right of the decimal point, supply the deficiency by annexing ciphers. Examples. 24. Multiply the number 6.79 by 10; by 100. DIVISION. 160. DIVISION OF DECIMALS is the operation of finding how many times one decimal number is contained in another. 1. Let it be required to divide 1.38483 by 60.21. ANALYSIS.-The dividend must be equal to the product of the divisor and quotient (Art. 72); and hence, must contain as many decimal places as both of them therefore, There must be as many decimal places in the quotient as the number of places in the dividend exceeds the number in OPERATION. 60.21) 1.38483 ( 23 1.2042 18063 18063 Ans. .023 the divisor: Hence, the following 159. How do you multiply a decimal number by 10, 100, 1000, &c.? If there are not as many decimal figures as there are ciphers in the multiplier, what do you do? |