13. A gentleman wishes to set 20 trees at equal distances on a line 18 r. 7 ft. in length; how far apart must the trees be? Ans. 16. 14. In 14 E. E. how many yards ? Ans, 17 and 2 qr., 15. In 14 E. Fr. how many yards ? Ans. 21. 16. In 14 E. FI. how many yards ? Ans. 10 and 2 qr. . 17. Divide 735 dollars between A and B, giving to A 2 shares and to B 3. 2 number of A's shares, 3 number of B's shares, 5) 7 3:52 5 sum of the shares. $ 1.4 7 the value of 1 share. 1 4 7 3 1 4 7 2 $2 9 4 A's part. $4 4 1 B's part. Proof. A's part 2 94 Explanation.--Had A and B's part 4.4 1 B owned the money equally between them, we should have 7 3 5 divided by 2, to have found the part of each, but as one owns a greater part than the other, we must add the shares of the whole together, and divide by the sum, the quotient will be the value of 1 share; and multiplying the value of one share by the number of shares which one person owns must give his part. 18. Three men, A, B and C, buy a ticket in company, for which they pay $10. A pays 1 dol., B 3 dol: and the remainder. If the ticket draw $12000, what will be the share of each? Ans. A, $1200, B, $3600 and C, $7200. 19. A retailer draws off a pipe of brandy into casks containing 10 gal. 1 pt.; how many casks does it fill? Ans. 12, and 4 gal. 2 qt. over. 20. If a tun of wine be drawn into quart and pint bottles, of each kind an equal number, how many of each kind will be required ? Ans. 672. Explanation.-Were we to draw the wine in the last question, into a measure containing 3 pints, each measure full would fill a bottle of each kind, consequently, as many times as we can take 3 from the number of pints contained in a tun, there will be so many bottles of each kind." 21. Three men A, B, and C perform a journey in company, the expense of which was 91 dols. B is to pay 3 times as much as A, and C three times as much as B; what does each man pay? Ans. A pays $7, B $21, and C $63. SECTION VII. FRACTIONS. DEFINITIONS. A Fraction is a part of an unit, or a part of one single thing of any kind, expressed by two numbers, one written above the other, with a line between them; thus, is a fraction expressed by the numbers 6 and 11. A Denominator is the under number of a fraction; 11 is the denominator in the fraction of A Numerator is the upper number of a fraction ; 6 is the numerator in the fraction of A Simple Fraction is a fraction having but one numerator and one denominator; as, ž. A Compound Fraction has more than one numerator and denominator, with the word of written between its parts; as, į of į. The first part of a compound fraction is a fraction of that part which follows it; thus, in the fraction of 1, the is a fraetion of the , and the is a fraction of an unit, A Whole Number is the name given to a simple number to distinguish it from a fraction, when they are written in connection. In the expression 154, 15 is the whole number and the fraction. A Mixed Number is a whole number and a fraction; as, 3. An Improper Fraction has its numerator equal to, or greater, than its denominator; as, j, or. A Proper Fraction has its numerator less than its denominator; as, , or. The Terms of a Fraction is a word denoting both the numerator and denominator of the same fraction, when both are to be expressed by one word, EXPLANATION OF FRACTIONS. It is frequently necessary to express parts of a quantity less than that part denoted by an unit. If we wish to represent by figures one barrel of flour and a part of another barrel, the figure 1 will represent the whole barrel, and the part of the other must be represented in a different manner. Suppose we have one barrel and three-quarters of another ;-to show into how many parts that barrel is divided, of which we have only a part, we must make use of one figure, expressing the number of parts into which it is divided : and this number will be the denominator of a fraction. We must also make use of a figure denoting the number we have of those parts into which the barrel is divided ; and this figure will be the numerator of the fraction. The one barrel and three-quarters of flour would be expressed thus, 1 ; the figure 1 at the left hand represents the whole arrel, and the figure 4 denotes how many parts another barrel is divided into, and is the denominator; the figure 3 shows how many of those parts we have, and is the numerator. The denominator is so called because it denotes how many parts one, or an unit, is divided into. The numerator is so called because it expresses, or numbers the parts we possess. Again, let it be required to express one yard of cloth, and two-thirds of another yard. To exhibit the fraction in a clearer light, we shall make use of the following Plate : 1 PLATÉ 1:. 1 and yards of cloth. In plate 1, let that A part of the plate una der A represent one yard of cloth, the figure 1, standing above it, signifies the same thing. Let c the part of the plate C represent another yard of cloth, equal to that represented by A; but let us suppose it divided into three equal parts, that is, into thirds. The parts above c, d, and e, will each represent one-third of a yard, and the three pieces a whole yard. We do not possess the whole yard denoted by C, but suppose a whole yard divided into the three equal parts, c, d, e, and this supposed yard is used as a denominator to show what each part in the numerator is equal to. The figure 3, the denominator of the fraction f signifies the same three parts as the letters c, d, and e, that is, the supposed yard. Let the part of the plate B denote two pieces of cloth, each piece just as large as the piece c, or d, in the părt C. Then the part a will represent one-third part of a yard, and the parts a and b will represent and these two parts are denoted by 2, the numerator of the fraction. We supposed in the first place, that we had one yard, and two-thirds of another. The part of the plate A denotes the whole yard, and the part C represents a supposed yard, of which we have only a part. As we have two-thirds of another yard, we may suppose the third c and the third d, renioved, and placed above the horizontal line, in the places a, b ; then e would show what part is remaining to make up another full yard. But as a and b are just equal to c and d, we may as well let c and d remain, and a and b will show what part of another yard we have. In plate 2, let a and b denote two single things of any kind, or two units ; and let d represent a suppos. ed thing or unit, just as large as a. If d be divided into 5 equal parts, one of the parts will be a fifth part of the whole unit denoted by d, and the 5 parts will be equal to a or b, that is, equal to an unit. The partn is divided into three equal divisions, each equal to one of the divisions in d, therefore one of the divisions in n is one fifth part of an unit, and the three divisions in nare three-fifths of an unit. d and n rep resent a fraction, d the denominator, and n the numerator. The whole would be thus expressed by figures, 23. The figure 2 represents a and b, 5 the same as d, and 3 the same as no If any particular quantity were denoted by the several parts of the plate, we should possess the parts represented by a, b, and n, all the parts above the line h I. |