14. The national debt of England amounts to about 279 millions of pounds sterling; how long would it take to count this debt in dollars (4s. 6d. sterling) reckoning without intermission twelve hours a day at the rate of 50 dols. a minute, and 365 days to the year? Ans. 94 years, 134 days, 5 hours, 20 min. FRACTIONS. FRACTIONS, or broken numbers, are expressions for any assignable part of a unit or whole number, and (in general) are of two kinds, viz. VULGAR AND DECIMAL. A Vulgar Fraction, is represented by two numbers placed one above another, with a line drawn between them, thus, , &c. signifies three fourths, five eighths, &c. 3 4959 The figure above the line, is called the numerator, and that below it, the denominator; The denominator (which is the divisor in division) shows how many parts the integer is divided into; and the numerator (which is the remainder after division) shows how many of those parts are meant by the fraction. A fraction is said to be in its least or lowest terms, when it is expressed by the least numbers possible, as when reduced to its lowest terms will be, and is equal to 3, &c. PROBLEM I. To abbreviate or reduce fractions to their lowest terms. RULE. Divide the terms of the given fraction by any number which will divide them without a remainder, and the quotients again in the same manner; and so on, till it appears that there is no number greater than 1, which will divide them, and the fraction will be in its 11. What is the value of of a day? Ans. 16 h. 36 min. 55 sec. PROBLEM III. To reduce any given quantity to the fraction of any greater denomination of the same kind. RULE. Reduce the given quantity to the lowest term inentioned for a numerator; then reduce the integral part to the same term, for a denominator; which will be the fraction required. EXAMPLES. 1. Reduce 13s. 6d. 2qrs. to the fraction of a pound. 20 integral part 12 240 4 960 Denominator. 13 6 2 given sum. 12 £. 2. What part of a hundred weight is 3 qrs. 14 lb. ? 3 qrs. 14 .98 lb. Ans. Ans. 1 4. What part of a pound sterling is 13s. 4d.? Ans. 5. What part of a civil year is 3 weeks, 4 days? 3. What part of a yard is 3 qrs. 3 na.? Ans. 6. What part of a mile is 6 fur. 26 po. 3 yds, 2 ft.? fur. po. yds. ft. feet. 6 26 3 2-4400 Num. Ans. 7. Reduce 7 oz. 4 pwt. to the fraction of a pound troy. Ans. 8. What part of an acre is 2 roods, 20 poles? Ans. 9. Reduce 54 gallons to the fraction of a hogshead of Ans. Ans. wine. 10. What part of a hogshead is 9 gallons? 11. What part of a pound troy is 10 oz. 10 pwt. 10 grs. A Decimal Fraction is that whose denominator is a unit, with a cipher, or ciphers annexed to it, Thus, f: TOO, TOO! &c. &c. 11. What is the value of of a day? Ans. 16 h. 36 min. 55 sec. PROBLEM III. To reduce any given quantity to the fraction of any greater denomination of the same kind. RULE. Reduce the given quantity to the lowest term inentioned for a numerator; then reduce the integral part to the same term, for a denominator; which will be the fraction required. EXAMPLES. 1. Reduce 13s. 6d. 2qrs. to the fraction of a pound. 20 integral part 13 6 2 given sum. 12 12 2. What part of a hundred weight is 3 qrs. 14 lb. ? 3 qrs. 14 lb. 98 lb. Ans. 98 3 na. ? 112 Ans. 15 3. What part of a yard is 3 qrs. Ans. 25 Ans. 6. What part of a mile is 6 fur. 26 po. 3 yds, 2 ft. ? fur. po. yds. ft. feet. 6 26 3 2-4400 Num. 7. Reduce 7 oz. 4 pwt. to the fraction of a pound troy. Ans. 8. What part of an acre is 2 roods, 20 poles? Ans. 9. Reduce 54 gallons to the fraction of a hogshead of wine. 10. What part of a hogshead is 9 gallons? 11. What part of a pound troy is 10 oz. 10 DECIMAL FRACTIONS. Ans. Ans. pwt. 10 grs. Ans. 45 A Decimal Fraction is that whose denominator is a unit, with a cipher, or ciphers annexed to it, Thus, fʊ: Tỗ, TOOT, &c. &c. The integer is always divided either into 10, 100, 1000, &c. equal parts; consequently the denominator of the fraction will always be either 10, 100, 1000, or 10000, &c. which being understood, need not be expressed; for the true value of the fraction may be expressed by writing the numerator only with a point before it on the left hand thus, is written,5; 25,45; 723,725, &c. 1000 But if the numerator has not so many places as the denominator has ciphers, put so many ciphers before it, viz. at the left hand, as will make up the defect; so write T thus, ,05; and thus, ,006, &c. 00 NOTE. The point prefixed is called the separatrix. Decimals are counted from the left towards the right hand, and each figure takes its value by its distance from the unit's place; if it be in the first place after units, (or separating point) it signifies tenths; if in the second, hundredths, &c. decreasing in each place in a tenfold proportion, as in the following 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: so,5,50 and ,500 are all the same value, and equal to or 1. But ciphers placed at the left hand of decimals, decrease their value in a tenfold proportion, by removing them further from the decimal point. Thus, ,5,05,005, &c. are five tenth parts, five hundredth parts, five thousandth parts, &c. respectively. It is therefore evident that the magnitude of a decimal fraction, compared with another, does not depend upon the number of its figures, but upon the value of its first left hand gue: for instance, a fraction beginning with any figure less than,9 such as ,899229, &c. if extended to an infinite number of figures, will not equal,9. ADDITION OF DECIMALS. RULE.-1. Place the numbers, whether mixed or pure decimals, under each other, according to the value of their places. 2. Find their sum as in whole numbers, and point off so many places for the decimals, as are equal to the greatest number of decimal parts in any of the given numbers. EXAMPLES 1. Find the sum of 41,653+36,05+24,009+1,6 100 Sum, 103,312, which is 103 integers, and parts of a unit. Or, it is 103 units, and 3 tenth parts, 1 hundredth part, and 2 thousandth parts of a unit, or 1. Hence we may observe, that decimals, and FEDERAL MONEY, are subject to one and the same law of notation, and consequently of operation. For since dollar is the money unit; and a dime being the tenth, a cent the hundredth, and a mill the thousandth part of a dollar, or unit, it is evident that any number of dollars, dimes, cents and mills, is simply the expression of dollars, and decimal parts of a dollar: Thus, 11 dollars, 6 dimes, 5 cents, 11,65 or 11 dol. &c. - 2. Add the following mixed numbers together. (4) (2) (3) Yards. |