а 9. Reduce 6 furlongs, 26 poles, 11 feet, te the fraction of à mile. Ans. 4. - 10. Reduce 2qrs., 23 na. to the fraction of a yard. Ans. 1:11. What part of a hogshead is 54 gallons Ans. QUESTIONS ON VULGAR FRACTIONS. What is a vulgar fraction ? A. A broken pumber, or ur finished division, of which the numerator is the remainder, and the denominator, the divisor. How is a yulgar fraction set down ? A. By two numbers, one directly above the other, with a line between. What is the number called, above the liņeļ A. Numerator, because it numbers or declares the number of parts contained in the fraction. What is the number called, below the line A. Denominator, because it denotes or declares into how many parts a unit is divided. How is a • fraction reduced to its lowest terms or expressed in the least number of figures ? A. By dividing the numerator and denominator by any number that will divide them without a remainder, and, then the quotients in the same manner, till no number will divide the terms of the fraction without a remainder; the last result will be the fraction in its lowest terms. How do you find the value of a fraction in known parts of the integer ? A. By multiplying the numerator of the fraction by the parts or inferiour denominations of an integer, and dividing the product after each multiplication by the denominator, and the quotient will express the value of the fraction in the inferiour' denominations of the integer. How do you reduce inferiour denominations to the fraction of some superiour denomination retaining the same value? A. By reducing the given sum to the lowest denomination mentioned for a pumerator, and reducing an integer to the same dénomination for a denominator, then write the fraction down with the numerator above the denominator, and then reduce the fraction to its lowest terms for the answer required. In reducing a fraction to its lowest terms, do your quotients, after the division, express the same value of your divi: dends? A. They do, because the numerator, in the new fraction, bears the same proportion to the denominator of the fraction as the numerator of the dividend bears to the denominator of the dividend, consequently the value of the fraction is not altered. DECIMAL FRACTIONS, A decimal fraction is a fraction whose denominator is à unit with as many ciphers annexed as the numerator has decimal places or figures. A decimal fraction is usually expressed by writing the numerator only, with a comma or point prefixed at the left hand of the fraction; thus ,5 tenths is the same as tv; and ,75 hundredths is the same as 77&c.In reading a decimal, we always express the number of parts in the fraction, also the number required to make a unit; thus when we read 5 tenths, the five shows the number of parts 100 a 60 in the fraction, and the tenths in the expression, show that it requires ten of those parts to make a unit. Whole numbers and decimals may be written in the same line, with a point between them, called the separatřix; thus, 357 is written 35,3; and 3117 is written 3,47. The denominator, the student will recollect, is repeated in the expression when it is not written, for we say, 3 tenths, and, 47 hundredths, and so on, without having a denominator expressed in the fraction. Decimals decrease in a tenfold proportion, counting . from the left to the right; thus ,5 is only one tenth the value it would express in the place of units, by taking away the de cimal point; and ,05 is only one tenth as much as ,5; so it will be perceived that they diminish in a tenfold proportion as they recede from the place of units, NOTE.--Ciphers prefixed to decimals at the right, neither increase nor diminish their value, thus ,5 ,50, ,500 being two to, 100% are all equal in value, because whenever we annex a cipher to the decimal, the denominator which is understood, assumes one, so that it is multiplying the numerator and denominator by the same number, consequently the proportion between them must remain the same: Btit čiphers prefixed at the left diminish the value of the decimal in a tenfold proportion, thus ,5 ,05, ,005 are the same as the Tous Tivo in value, for in the first example ,5 shows that a unit is divided into 10 parts, and that the fraction contains 5 of those parts; and the second example ,05 shows that one is divided into 100 parts, and the fraction contains only 5 of those parts, &c. This will appear very plain to the student from what has been said of Federal Money, which is purely decimal money, of which the dollar may be considered as the unit, and the inferiour denominations the decimal parts; thus, 6 dollars and 4 dimes are expressed $6,4 or 56:40 The learner must know that it takes 10 tenths or 10 dimes to make a unit in dollars, and 8 dollars and 45 cents are ex. pressed $8,45, or $84; in qut first example $6,4, it will be discovered that our decimal ,4 tenths, shows that a unit is divided into 10 parts, and the fraction contains 4 of those parts; and when a dollar is divided into ten parts, the parts must be dimes, and in the last example, 88,45, the decimal (,45) shows that it takes 100 parts, to make a unit, and when a dollar is divided into 100 parts, the parts are 5 a gents, consequently the ,45 hundredths are cents, of which it takes 100 to make a unit, and so of mills it takes 1000 to make a unit. Such being the nature of Federal Money, it is plain that tenths represent dimes; hundredths, cents; and thousandths, mills; but we commonly express the decimals where the unit is a dollar, in cents and mills; or taken together they represent thousandths of a dollar. Considering decimals in this light, the student must understand them, because he can see and handle the inferiour denominations of a dollar, that is, the decimal. parts of a dollar. TABLE. It !! || | || ll Hundredş, Tens. 36 TOUT = 36 ,00 5 6,0 0,00 3 705zuotovo 70 5,0 0:0.0.0.1... 705, and 1 Millionth. 075 36, and 5 Thousandths. 6, and 3 Hundred Thousandths. 7 Tenths. Hundredths. Thirty-six, and eight tenths=36-4-36,8 the decimal ex. pression. In like manner write the following sums": ADDITION OF DECIMALS: RULE.- Place the given 'numbers according to the value of their places, whether mixed or pure decimals; so that tenths may stand under tenths, and hundredths under hundredths, &c. Add the same as in whole numbers, and point off as many places for decimals, at the right hand, as shall equal the greatest number of decimal places in any of the given numbers. Or set the decimal point in the amount, exactly under those in the given numbers. EXAMPLES, 1. Add , 3 dimes and ,9 dimes together, or 3.tenths and 9 tenths. Ans. $1,2di. or 1,2. dà. Dem. It is plain, that 3 tenths and 9 ,3 ,3 tenths make 12 tenths, or one unit and ,9 two tenths, because ten tenths are equal to a unit or one, and the reason of $1,2 Ans. 1,2 Ans. our pointing off directly below the given points, is manifest, because whenever our tenths exceed 9 tenths, they must equal something in units; and the number of figures must increáse, and that increase must be units; for when we suppose these decimals to be dimes, (which correspond. with tenths,) ten of which make a dollar or unit, we then have $1 or 1 unit, and 2 dimes, or two tenths. 2. Add 1,7, 3,45, 6,75, 1,705, ,50, ,05 together. Ans. '14,155. 1,7 DEM.—The learner will perceive, that we add 3,45 the same as in whole numbers, and the reason 6,75 is plain, "for as the parts diminish in a tenfold proportion from the left to the right,, so they must 1,705 increase in a tenfold proportion from the right ,50 to the left, which the student may perceive from ,05 Federal Money, in which it takes ten mills, which occupy the place of thousandths, to make a cènt; Ans. 14,155 ten cents, which occupy the place of hundredths, to make a dime; and ten dimes, which occupy the place of tenths, to make a dollar. or ** 1 | 1 3. 6. 67 ,156 7,46 4,0005 3,05 1,01 2,009 7. Add three hundredths, five tenths, forty-five hundredths, eleven thousandths, three ten-thousandths, and four millionths together. Ans. ,991304. 8. Add together the following sums, viz.: forty-five thousandths, four tenths, four hundredths, and five thousandths. Ans. ,49. 9. Add six tenths, six hundredths, six thousandths, six ten thousandths, five hundredths, and eleven thousandths. Ans. ,7276. 10. Add 105,7, 19,4, 1119,05, 648,006, and 19,041, together, Ans. 1911,197. 11. Add one thousand and one thousandth, three hundred and eleven thousandths. Ans. 1300,012. SUBTRACTION OF DECIMALS. RULE.Place the given numbers the same as in addition of decimals, with the 'less under the greater, and subtract the same as in whole numbers. Set the decimal point in the remainder directly under those in the given numbers. EXAMPLES. 1. From ,65 take ,32, or from 6 dimes and 5 cents, take 3 dimes and 2 cents. Ans. ,33, or 3 dimes and 3 cents. Dem. It is plain, if we take 2 hundredths 5 hundredths, 3 hundredths will remain; anu if we take 3 tenths from 6 tenths, 3 tenths must remain, ,65 consequently the difference between the or ,3 2. tenths, and 3 hun33 Ans. ,3 3 Ans. dredths. When we consider the decis mal to be Federal Money, with which it agrees exactly in principle, it will appear still more plain to the young and if we take 3 dimes from 6 dimes; will remain t'emain; difference then between the given numbers, is 3 dimes and 3 cents, or as we commonly express the decimal of a dollar, it stan 33 cents. ercents. ,32 given numbers is e espress it, 33 hun |