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 dollars 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 an 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-eights, &c. The figure above the line, is called the numerator, and that below it, the denominator, Thus 5 Numerator. The denominator (which is the divisor in division) shews how many parts the integer is divided into ; and the numerator (which is the remainder after division) shews 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 2, &c. PROBLEM I. To abbreviate or reduce fractions to their lowest terms. RULE. Divide the terms of the given f :tion b iny number which will divide them without a rainder, and the quotients again in the same manner; at so on, till it appears that there is no number greater than 1, which will divida them, and the fraction will be in its least terms. To find the value of a fraction in the known parts of the integer, as to coin, weight, measure, &c. RULE. Multiply the numerater by the common parts of the Integer, and divide by the denominator, &c. 2 What is the value of of a pound sterling? 3. Reduce of a shilling to its proper quantity. Ans. 44d 4. What is the value of 5. What is the value off of a shilling? Ans. 4 d of a pound troy? & How much is o. an hurated weight? Ans. 3grs. 7b 103oz 7 What is the value of of a mile ? 8. How much is 3 of an cwt. ? Ans. 6fur. 26po. 11ft. Ans. 3qrs. 3lb. 1oz. 124dr 9. Reduce of an Ell English to its proper quan ́ity. Ans. 2qrs. 3 na 10. How much is of a hhd. of wine? Ans. 54gal. 11. What is the value of of a day? Ans. 16h. 36min. 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 mentioned 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. 12 240 4 960 Denominator. 12 162 4 650 Num. Ans. &f=f&£ Ans. = 2. What part of an hundred weight is 3qrs. 14lb. ? 3qrs. 14lb. 98/h. Ans. 3. What part of a yard is 3qrs 3na.? Ans. 4. What part of a pound sterling is 13s. 4d.? Ans. 5. What part of a civil year is 3 weeks, 4 days? Ans. 6. What part of a mile is 6fur. 26po. 3yds. 2 ft. ? fur, po. yds. ft. feet. 6 26 3 2 4400 Num. a mile 5280 Denom. Ans. 448 28 7. Reduce 7oz. 4pwt. 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? Ans. Ans. 11. What part of a pound troy is 10oz. 10pwt. 10grs. DECIMAL FRACTIONS. A Decimal Fraction is that whose denominator is an unit, with a cypher, or cyphers annexed to it, Thus, fa fỗ02 Tâbò, &c. &c. 1009 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, o is written,5; 45,45; 725,725, &c. But if the numerator has not so many places as the denominator has cyphers, put so many cyphers before it, viz. at the left hand, as wifi make up the defect; so write To thus,,05; and thus, ,006, &c. NOTE. The point prefixed is called the separatrix. Decimats 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 pro portion, as in the following NUMERATION TABLE. Cyphers placed as 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 But cyphers 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 figure: 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. 4440 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 41,653 Sum, 103,312 which is 107 integers, and parts of an unit. Or, it is 108 units, and 3 tenth parts, 1 hundredth part, and 2 thousandth parts of an 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. |