2. Multiply the remainder, if any, by the next lower denomination, and divide by the denominator, as before; and the several quotients will be the answer. EXAMPLES. How much is lb. avoirdupoise? How much is lb. ? How much is lb.? How much is 3lb.? glb.? How much is lb. ? How much is of a shilling? How much is How much is of a shilling? How much is of a shil of a shilling? How much is ? ling? How much is 1. What is the value of of a pound sterling? 3. What is the value of 4. What is the value of of a pound sterling? Ans. 18s. 4d. of a shilling? of a shilling? Ans. 10 pence 14qrs. of a pound Troy? Ans. 9oz. 5. What is the value of 3 Ans. 12oz. 12 dr. 7. Reducc of a hundred weight to its proper quantity? Ans. 3qr. 3lb. 1oz. 123dr. 8. What is the value of § of an Ell English? 9. What is the value of § of a yard? Ans. 2qr. 3дna. Ans. 3qr. 1 na. 10. How much is 15 of a hogshead of wine? Ans. 35gals. 11. How much is § of a mile? Ans. 6fur. 26rd. 3yds. 2ft. 12. How much is of a day? Ans. 12h. 55min. 231 sec. 13. How much is 32 of an acre? Ans. 3 roods. 25 rods. Questions I. What are Fractions? From what do all fractions arise? Of how many kinds are fractions? II. How is a Vulgar fraction represented? III. What is the number below the line called? and why? IV. What is the number above the line called? and why? In Division, what is the denominator? and what is the numerator? V. What is a simple or proper fraction? What is an improper fraction? What is a mixed nnmber? How may a whole number be expressed as a fraction? Prob. I. How do you reduce fractions to their lowest terms? What are the terms of a fraction? II. How do you change a whole or mixed number to an improper fraction? III. How do you change an improper fraction to a whole or mixed number? IV. How do you multiply a whole number by a fraction? V. How do you multiply a fraction by a whole number? VI. How do you divide a whole number by a fraction? VII. How do you reduce a given quantity to the fractiou of a greater denomination of the same kind? VIII. How do you find the value of a fraction in whole numbers of less denominations? DECIMAL FRACTIONS. 1. A Decimal* Fraction is that whose denominator is always 1 with a cipher, or a number of ciphers annexed to it. Thus, 5 6 56 Yo, To, 1000' &c. &c. 2. The integer is always divided into 10, 100, 1000, &c. equal parts. Therefore the denominator is always 10, 100, 1000, &c. 3. The true value of a decimal fraction is expressed by writing the numerator only with a point before it. Thus, is written,,5; 100,,25; 100,645. 4. If the numerator has not so many places of figures as the denominator has ciphers, we must put as many ciphers on the left hand as will make up the defect. Thus, 6 written,06 and 1000 is written ,006, &c. 5. The point prefixed is called a separatrix. * So called from the Latin word decem, which signifies ten. is 6. Each figure takes its value by its distance from the unit's place; the first figure on the right hand of units, or the separatrix, signifies so many tenths; the second so many hundredths; the third so many thousandths, &c.; thus decreasing in a tenfold proportion from the left towards the right hand. 7. Ciphers placed at the right hand of a decimal fraction do not alter its value, since every significant figure continues to possess the same place. Thus, ,5,50,500, &c. are all of the same value, and each equal to or. 5 10 8. Every cipher placed at the left hand of a decimal fraction decreases its value tenfold, by removing each significant figure farther from the place of units. Thus,,5 in the first place is 5 tenths; ,05 in the second place is 5 hundredths; ,005 in the third place is 5 thousandths, &c. the following TABLE. See =6 tenths. "65 thousandths. ,1234 "1234 ten thousandths. ,45012-45012 hun. thousandths. 8,000005 "8 and 5 millionths. 4,5,65 "45 and 65 hundredths. 365,123456" 365 & 123456 millionths. 9. The magnitude of a decimal fraction depends mostly on the first, or left hand figure, which, if it be less than ,9, we may extend to an infinite number of figures, and it will not be equal to ,9. Thus, ,899999 is not equal to ,9. 10. Decimals are read in the same manner as whole num bers, giving the name of the lowest denomination, or right hand figure to the whole. Thus,52 is read 52 hundredths; ,425 is read 425 thousandths, &c. 11.. When whole numbers and decimals are expressed in the same number, it is called a mixed number. Write, in decimals, the following mixed numbers. 1. Twenty-five and six tenths=25,6. 2. Sixteen, and seventy-five hundredths. 3. Forty-one, and one hundred and forty-five thou sandths. 4. 362, and nine millionths. 5. Nineteen, and 34756 hundred thousandths. 6. 794, and twenty-five ten thousandths. 84 101; 5618; 411500; 95000; 145000000, &c. ADDITION OF DECIMALS. RULE. 1. Place the numbers, whether mixed or pure decimals, under each other, according to the value of their places. 2. Add them together as whole numbers, and place the separatrix exactly under the separating point above. EXAMPLES. 1. What is the sum of 28,753; 365,41; 18,75; 145,6. 28,753 365,41 18,75 145,6 558,513 We place tenths under tenths, hundredths under hundredths, &c., accord ing to the value of their places, and add the columns as whole numbers; and the amount is 558 units, and 513 thousandths of a unit. Note. We may here observe, that the denominations of Federal Money correspond exactly with decimals, the dollars being units, dimes being tenths, cents hundredths, and mills thousandths of dollars, &c. 2. Add together the following sums of dollars and decimals of a dollar, viz: 13,755, 2,50, 25,3, and 41,144. Ans. $82,699-82dols. 69cts. 9m. 3. Find the amount of 79,45dols., 36,045dols., 128,5 dols., 95,006dols., 135,25dols., and 14,146dols. Ans. 488dols. 39cts. 7m. 4. Add together the following mixed numbers, viz: 5,98471+18,568+2,005+9,15+35,1009+,35762. Ans. 71,16623. 5. What is the whole sum of 5,91 acres, 3,5 acres, 8,596 acres, ,795 acres, and 14 acres? Ans. 32,801 acres. 6. Required the sum of 25,164lbs., 9,56lbs., 87,31lbs., 256,25lbs., 9,18lbs., and 125,9lbs. Ans. 513,364lbs. 7. Add together 276, 54,321,,65, 112, 12,5 and ,0463. Ans. 455,5173. 25 ounces, 795 ounces, 3,850 8. What is the sum of 5 ounces, 151 ounces and 25100 ounces? 6 Ans. 56,985. 9. Find the amount of forty-five hundredths, two hundred and fifty-six thousandths, sixty-five hundredths, ten, and six hundred and forty-four thousandths. SUBTRACTION OF DECIMALS. RULE. Ans. 12. Place the numbers according to their value, then subtract as in whole numbers, and point off the decimals as in addition of decimals. Ans. 9,999999. 10. From 10, subtract one millionth part of a unit. MULTIPLICATION OF DECIMALS. RULE. Multiply as in whole numbers, and point off as many figures in the product for decimals, as there are decimal |