6. How much is of an hundred weight? Ans. Sqrs. 7lb. 10 oz. 7. What is the value of of a mile? Ans. 6fur. 26pc. 11ft. 8 How much is of an cwt.? Ans. 3qrs. 3lb. 1oz. 124dr. 9. Reduce of an Ell English to its proper quantity. Ans. 2grs. Sina. 10. How much is of a hhd. of wine? Ans. 54gal. 11. What is the value of of a day? Ans. 16h. 56min. 55sec. 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. $8. 2. What part of an hundred weight is Sqrs. 14lb. Sqrs. 14lb. 9816. Ans. 98 =1 Ans. 3. What part of a yard is 3qrs. Sna. ? 25 Ans. 53 6. What part of a mile is 6fur. 26po. Syds. 2ft. ? fur. po. yd. ft. feet. 6 26 3 2=4400 Num. a mile #5280 Denom. 7. Reduce 7oz. 4pwt. to the fraction of a pound troy. Ans. 2. What part of an acre is 2 roods, 20 poles ? Ans. 9. Reduce 54 gallons to the fraction of a hogshead of wine. Ans. & 10. What part of a hogshead is 9 gallons? Ans. 11. What part of a pound troy is 10oz. 10pwt. 10grs.? DECIMAL FRACTIONS. Ans. 505 378 A Decimal Fraction is that whose denominator is an 5 unit, with a cypher, or cyphers annexed to it, Thus, 1, 1802 165, &c. &c. 5 45 0210009 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; 5,45; 75%,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 will make up the defect; so write 1 thus, ,05; and thus, ,006, &c. 6 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 Cyphers 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 5 or 3. 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. ADDITION OF DECIMALS. RULE. 1. Place the numbers, wnether 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 Thus, 41,653 24,009 1,6 Sum, 103,312 which is 103 integers, and 3 parts of an unit. Or, it is 103 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. For since dollar is the money unt; 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 doliars, 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. 5. Add the following sums of Dollars together, viz. $12,34565+7,891+2,34+14,+,0011 Ans. $36,57775, or $36, 5di. 7cts. 775 mills. 6. Add the following parts of an acre together, viz. ,7569+,25+,654+,199 Ans. 1,8599 acres, 7. Add 72,5+32,071+2,1574+871,4+2,75 S. Add 30,07+200,71+59,4+3207,1 Ans. 480,8784 Ans. $497,28 9. Add 71,467+27,94+16,084+98,009+86,5 10. Add ,7509+,0074+,69+,8408+,6109 11. Add ,6+,099+,37+,905+,026 Ans. 300 Ans. 2,9 Ans. 2 12. To 9,999999 add one millionth part of an unit. and the sum will be 10. 13. Find the sum of Twenty-five hundredths, Three hundred and sixty-five thousandths, Six tenths, and nine millionths, Answer, 1,215009 SUBTRACTION OF DECIMALS.. RULE. Place the numbers according to their value; then subtract as in whole numbers, and point off the decimals as in Addition. tself. Ans. ,04816 Ans. 55,3 Ans. 194,7925 13. From an unit, or 1, subtract the millionth part of Ans. 999999 Ans. 106,9993 MULTIPLICATION OF DECIMALS. RULE. 1. Whether they be mixed numbers, or pure decimals, place the factors and multiply them as in whole numbers. 2. Point off so many figures from the product as there are decimal places in both the factors; and if there be not so many places in the product, supply the defect by prefixing cyphers to the left hand. |