23 4. A merchant owning of of of a ship, sold of his share. What part of the ship did he sell? 5. 3 men owned equally a saw-mill; one sold II. DECIMAL FRACTIONS. of 3 of A DECIMAL FRACTION is a fraction whose denominator is 10, or 100, or 1000, &c. The denominator of a decimal fraction is never written: the numerator is written with a point prefixed to it, and the denominator is understood to be a 1, with as many ciphers annexed as there are figures in the numerator. Thus, 3 is 31 is; 316 is 316; .3164 is 31646. 1000 10000 46 708 1642 96041 1. Write upon the slate, the decimals expressing the following fractions. 10 100 1000 10000 100000 When a whole number and a decimal are written together, the decimal point is placed between them. 24.6 is 24; 5.71 is 57; 48.364 is 48,364. 2. Write the following mixed numbers, expressing the fractions decimally. 38. 51622.8354. 2476366. 10 1000 10000 Thus, In whole numbers, any figure, wherever it may stand, expresses a quantity as great as it would express, if it were written one place further to the left. For instance, in the number 1111, the 1 hundred is of a thousand; the 1 ten is of a hundred, or 1 of a thousand; the 1 unit is of a ten, or of a thousand. In decimals, this system is continued below the place of units. For example, in the number 1.111, the 1 next to the right of the unit is 1-tenth, that is, of a unit; the 1 next to the right of the 1-tenth is of a tenth, or 1-hundredth of a unit; the one next to the right of the 1-hundredth, is of a hundredth, or 1-thousandth of a unit. one tenth. one unit. one hundredth. 1.11 1 Ciphers placed on the right hand of decimal figures, do not alter the value of the decimal; because, the figures 184 remain unchanged in their distance from the unit's place. For instance, .5, .50, and .500 are of equal value; being each equal to . But every cipher placed on the left of a decimal, renders it ten times smaller, by removing the figures one place further from the unit's place. Thus, if we prefix one cipher to .5, it becomes .05 [150]; if we prefix two ciphers, it becomes .005 [105]; and so on. 3. Write upon the slate, decimals expressing the fol lowing fractions. 100 1000 10000 100000 1000000* 6 TO READ DECIMAL FRACTIONS,Enumerate and read the figures, as if they were whole numbers, and conclude by pronouncing the name of the lowest denomination. 4. Copy upon the slate, and read the following decimals. .06 .065 .0007 24.02 ADDITION OF DECIMALS. 6. Add the following numbers into one sum. 63.75 and 524.0764 and .23 and 261.803. 63.75 524.0764 .23 261.803 849.8594 In arranging decimals for addition, we place tenths under tenths, hun dredths under hundredths, &c. We then begin with the lowest denomination, and proceed to add the columns as in whole numbers. 7. What is the sum of 2.164, 870.31, 756, 9.18, 157.0008, 26.104, and .3728? 8. What is the sum of 2706, 58.2, .2065, 6.441, 75, 14.2, and 990.752? In Federal Money, the dollar is the unit; that is, dollars are whole numbers; dimes are tenths, cents are hundredths, and mills are thousandths. See page 124. 9. Add together $24.6, $9.07, $5.009, and 5 cents. 10 Write the following sums of money in the form of decimals, and add them together. $46 and 9 cents, 14 cents, $7 and 8 mills, 6 dimes, 8 dimes and 7 mills. SUBTRACTION OF DECIMALS. 11. Subtract 52.6087 from 406.91. 406.91 354.3013 After placing tenths under tenths, &c., we subtract as in whole numbers. The blank places over the 7 and 8, are viewed as ciphers. 12. Subtract 943.076 from 8270.54. 15. Find the difference between .8 and .08, by subtracting the smaller decimal from the greater. 16. Find the difference between .45 and .31067. 17. What is the difference between 1 and .046? 18. Write 4 dollars and 8 mills in decimal form, and subtract therefrom, 6 dimes and 5 mills. 19. Subtract 7 cents and 3 mills from 10 dollars. MULTIPLICATION OF DECIMALS. Multiplying by any fraction, is taking a certain part of the multiplicand for the product; consequently, multiplying one fraction by another, must produce a fraction smaller than either of the factors. For example, multiplied by is, or, decimally, .9 multiplied by .8 is 용 .72. Hence you may observe, that the number of decimal figures in any product, must be equal to the number of decimal figures in both the factors. 8 20. Multiply 531 by .52. 65.7 by .43. 7.06 by .24. .439 by .38. .149 by .26. RULE FOR MULTIPLICATION OF DECIMALS. Mul tiply as in whole numbers; and in the product, point off as many figures for decimals, as there are decimal places in both factors. If the number of figures in the product he less than the number of decimal places in both factors prefix ciphers to supply the deficiency. 21. Multiply 1608 by .4,-that is, find .4 of 1608. 22. Multiply .45 of a dollar by 8. 23. How much is 36 times .495 of a dollar? 24. What cost 18 yards of cloth, at $4.072 per yd.? 25. What cost 28.7 yards of cloth, at $9 per yd.? 26. What cost 9.3 acres of land, at $8.41 per acre? 27. If 1 yard of silk cord cost 7 mills, [.007], what is the price of .9 of a yard? 28. What is 6 per cent. or .06 of 340.4? 29. Multiply 42.863 by 70.28. 30. Multiply 2046 by .932. 31. Multiply .7253 by .0423. 32. Multiply 6.5431 by .402. 33. What is the product of .04 multiplied by .07? 35. Multiply 7 and 5-hundredths by 6-thousandths. DIVISION OF DECIMALS. RULE FOR DIVISION OF DECIMALS. Divide as n whole numbers; and in the quotient, point off as many figures for decimals, as the decimal places in the dividend exceed those in the divisor; that is, make the decimal places in the divisor and quotient counted together, equal to the decimal places in the dividend. If there be not figures enough in the quotient to point off, prefix ciphers to supply the deficiency. When there are more decimal places in the divisor, than in the dividend, render the places equal, by annexing ciphers to the dividend, before dividing. After dividing all the figures in the dividend, if there be a remainder, ciphers may be annexed to it, and the division continued. The ciphers thus annexed, must be counted with the decimal places of the dividend. 36 Divide 64.395 by 40.5. Divide 5.8674 by 127 38. How many times .27)1.224 (4.533+ 108 144 135 90 81 90 81 is .27 contained in 1.224 ? The sign of addition, or more, here shows, that the true quotient is more than the preceding figures express. We might continually annex ciphers to this remainder, and carry on the division, but we should never arrive at a complete quotient. For the purposes of business, it is seldom necessary to extend the quotient below thousandths. 39. How many times is 1.23 contained in 3021.741 ? 40. How many times is 1243.4 contained in 5.37148? 41. How many times is .204 contained in 77112? 42. How many times is 4.2 contained in 194.334 ? 43. How many times is 30.02 contained in 94.657 ? 44. How many times is .44 contained in .1606? 45. What is the quotient of 42.65 divided by 36? 16. What is the quotient of .8 divided by 8? 9 |