Hence, for addition of decimals we have the following RULE. I. Set down the numbers to be added so that tenths shall fall under tenths, hundredths under hundredths, &c. This will bring all the decimal points directly under each other. II. Then add as in simple numbers and point off in the from the right hand, so many places for decimals as are equal to the greatest number of places in any of the given numbers. sum, Q. 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? EXAMPLES. 1. Add 4,035, 763,196, 445,3741 and 91,3754 together. Ans. 1303,9805. 2. Add 365,103113, 76012, 1,34976, ,3549 and 61,11 together. Ans. 428,677893. 3. 67,40797,004+4+,6+,06+,3=169,371. 4. ,0007+1,0436+,4+,05+,047=1,5413. 5. ,0049+47,0426+37,0410+360,0039=444,0924. 6. Required the sum of twenty-nine and 3 tenths, four hundred and sixty-five, and two hundred and twenty-one thousandths. Ans. 494,521. 7. Required the sum of two hundred dollars one dime three cents and nine mills, four hundred and forty dollars nine mills, and one dollar one dime and one mill. Ans. $641,249, or 641 dollars 2 dimes 4 cents 9 mills. 8. What is the sum of one tenth, one hundredth, and one thousandth. Ans. 111. 9. What is the sum of 4, and 6 ten thousandths. Ans. 4,0006. 10. 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. Ans. $129,761. 11. 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. Ans. $1132,365. SUBTRACTION OF DECIMAL FRACTIONS. § 126. Subtraction of Decimal Fractions teaches how to find the difference between two decimal numbers. EXAMPLES. 1. From 3,275 take,0879. 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 (see § 123). Hence, we have the following RULE. OPERATION. 3,2750 ,0879 3,1871 I. Set down the less number under the greater, so that figures occupying places of the same value shall fall directly under each other. II. Then subtract as in simple numbers, and point off in the remainder as many places for decimals as are equal to the greatest number of places in either of the given numbers. Q. What does subtraction teach? 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? 2. From 3295 take,0879. 3. From 291,10001 take 41,375. 4. From 10,000001 take,111111. 5. From three hundred and ninety-six, sandths. 6. From 1 take one thousandth. 7. From 6378 take one tenth. 8. From 365,0075 take 3 millionths. Ans. 3294,9121. Ans. ,999. Ans. 365,007497. 9. From 21,004 take 97 ten thousandths. 10. From 260,4709 take 47 ten millionths. Ans. 20,9943, Ans. 260,4708953. 11. From 10,0302 take 19 millionths. Ans. 10,030181. 12. From 2,01 take 6 ten thousandths. Ans. 2,0094. MULTIPLICATION OF DECIMAL FRACTIONS. EXAMPLES. 100 § 127. 1. Multiply,37 by,8. If we multiply the fraction 37 by 18%, we find the product to be 290 1000 and generally, the number of ciphers in the denominator of the product, will be equal to the number of decimal places in the two factors. 2. Multiply,3 by ,02. OPERATION. ,3,02=3×20=1000=,006 answer. To express the 6 thousandths decimally we have to prefix two ciphers to the 6, and this makes as many decimal places in the product as there are in both multi plicand and multiplier. Therefore, to multiply one decimal by another, we have the following RULE. Multiply as in simple numbers, and point off in the product, from the right hand, as many figures for decimals as are equal to the number of decimal places in the multiplicand and multiplier; and if there be not so many in the product, supply the deficiency by prefixing ciphers. Q. After multiplying, how many decimal places will you point off in the product? When there are not so many in the product, what do you do? Give the rule for the multiplication of decimals. 4. Multiply one and one millionth by one thousandth. Ans. ,001000001. 5. Multiply one hundred and forty-seven millionths, by one millionth. Ans. ,000000000147. 6. Multiply three thousand, and twenty-seven hundredths by 31. 7. Multiply 31,00467 by 10,03962. Ans. 9308,37. Ans. 311,2751050254. 8. What is the product of five-tenths by five-tenths. Ans. 25. 9. What is the product of five-tenths by five thousandths. Ans. ,0025. Ans. ,0238416. 10. Multiply 596,04 by 0,00004. 11. Multiply 38049,079 by 0,00008. Ans. 3,04432632. § 128. NOTE. 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 hand 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 Q. 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? DIVISION OF DECIMAL FRACTIONS. § 129. Division of Decimal Fractions is similar to that of simple numbers. We have just seen, that, if one decimal fraction be multiplied by another, the product will contain as many places of decimals as there were in both the factors Now, if this product be divided by one of the factors the quotient will be the other factor (see § 35.). Hence, in division, the dividend must contain just as many decimal places as the divisor and quotient together. The quotient, therefore, will contain as many places as the dividend, less those of the divisor. Hence, for the division of decimals we have the following RULE. Divide as in simple numbers, and point off in the quotient, from the right hand, so many places for decimals as the decimal places in the dividend exceed those in the divisor; and if there are not so many, supply the deficiency by prefixing ciphers. Q. If one decimal fraction be multiplied by another, how many decimal places will there be in the product? How does the number of decimal places in the dividend compare with those in the divisor and quotient? How do you determine the number of decimal places in the quotient? If the divisor contains four places and the dividend six, how many in the quotient? If the divisor contains three places and the dividend five, how many in the quotient? Give the rule for the division of decimals. 6. What is the quotient of 37,57602, divided by 3? By,3? By ,03? By ,003? By ,0003? |