2. If § 567 gain § 81, what will be the gain on $ 1701? 3. If the interest on $ 450 for one year, is $ 27, what will be the interest on $1150 for the same time. Ans. $ 69? 4. When a pipe of wine costs $ 252, what is the value of 171⁄2 gallons? Ans. $ 35? 5. Bought 721 yards broad cloth, at the rate of $ 65 for every 13 yards, what did the whole come to ? Ans. 3605 ? VULGAR FRACTIONS. A vulgar fraction is any assignable part of a unit or integer, expressed by two numbers, placed one above the other with a line drawn between them, as one fourth, two thirds, &c. The number above the line is called the numerator, and that below the line, the denominator. A fraction is said to be in its lowest terms, when it is expressed by the least numbers possible, as 4, when reduced to its lowest terms will be; and is equal to &c. CASE I. To reduce fractions to their lowest terms. RULE. Divide both the numerator and denominator, by any number which will divide them without a remainder, and the quotients again in the same manner, till it appears that there is no number greater than 1, which will divide them again. To reduce the value of a fraction to the known parts of an integer. RULE. Multiply the numerator by the common parts of the integer and divide by the denominator. EXAMPLES. 1. What is the value of of a pound sterling? Ans. 0 15s. O 3. Reduce of an hundred weight to its proper quantity. Ans. 3qrs. 3lb. loz. 124dr. 4. Reduce of lb. troy to its proper quantity. Ans. Toz. 4pwt. 5. Reduce of a mile to its proper quantity. AB Ans. 6 fur. 16 po. 6. Reduce of a month to its proper quantity. Ans. 2 wks. 2 d. 19 h. 12 m. CASE III. To reduce a fraction of one denomination to that of another, but greater, retaining the same value. 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 form the fraction required. EXAMPLES. 1. Reduce 15s. 8d. 2qrs. to the fraction of a pound. 754 Num. 754 Denominator. 3116. Ans. 2. What part of a pound sterling is 13s. 4d. ? 6. What part of a mile is 6 fur. 26 po. 3 yds. 2 ft. ? feet. 6 26 3 2= 4400 Num. a mile = 5280 Denom. 7. What part of a hhd. of wine is 54 gallons? Ans. Ans. 8. What part of a day is 16 h. 36 min. 55 9. What part of a shilling is 44d. ? sec. 10. What part of an acre is 3 roods, and 20 rods? Ans. 12. What part of a cord is 116,4 feet? Ans. 127 144 DECIMAL FRACTIONS. A DECIMAL FRACTION is that whose denominator is an unit, with a cyper, or cyphers annexed to it, thus 1 &c. &c. As the integer is always divided either into 10, 100, 1000 &c. equal parts; consequently the denominator will always be either 10, 100, 1000, 10,000, &c. which being implied, need not be expressed; for the true value of a Decimal Fraction is properly expressed by writing the numerator, only with a point before it on the left. But if the denominator has not so many places as the denominator has cyphers, prefix so many cyphers on the left as will make up the deficiency. { } write { .05 .006 .0007 &c. Decimals are reckoned from the left hand towards the right, and the value of each figure is determined by its distance from the place of units, if it be in the first place after unit,s (or seperating point) it signifies tenth's; if in the second, hundreth's &c.decreasing towards the right in a tend fold 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, 500, 5000, are all of the same value, and each equal to or 1. But cyphers placed at the left hand of decimals, decrease their value in a ten fold proportion, by removing them farther from the decimal point. Thus, .5 .05 .005 .0005 &c. are 5 tenth, 5 hundredth, 5 thousandth paats, &c. respectively ADDITION OF DECIMALS. RULE. 1. Place the numbers, whether mixed, or pure decimals, under each other, according to their local value. 2. Find their sum as in whole numbers, and point off so many places for decimals, as are equal to the greatest number of decimal parts in any of the given numbers. Note. The point prefixed is called the SEPERATRIX, from its-separating the Integral from the Decimal part. EXAMPLES. 1. What is the sum of 276, 59.213, 72014.9, 417, 5032 and 2214.298? 276. 72014.9 417. 5032. 2214.298 Ans. 79993.411 Hence we may observe that the Denominations of FEDERAL MONEY, as determined by an Act of Congress Aug. 8, 1786, are in a decimal ratio, and subject to one, and the same law of notation, and consequently of operation. For since a dollar is the integer or unit, and a dime being the tenth, and a cent the hundredth, and a mill the thousandth part of a dollar, it is evident that any number of dollars, dimes, cents, and mills, is simply the expression of dollars, and deoimal parts of a dollar: Thus, 15 dolls. 8 dimes, 3 cents, 5 mills is expressed in decimals. 15,835, or 15.835 2. What is the sum of the following expressions of money, viz. §7530, § 16.201, § 3.0142, $ 957.13, § 6.72819, § .03014 ? Ans. 8513.10353 3. Required the sum of .014, .9816, .32, .15914, .72913, and .0047, Ans. 2.20857 4. What is the sum total of 27.148, 918.73, 14016, 294304, 7138, and 221.7? Ans. 309488.2918 5. Required the sum of $12.984, 21.3918, 2700.42, 3.153, 27.2, and 581.06, Ans. 3646.2088 SUBTRACTION OF DECIMALS, RULE. 1. Set the less number under the greater in the same manner as in addition. 2. Then subtract as in whole numbers, and place the decimal point in the remainder directly under the other points. |