locomotive. How many hours after starting will they meet? And at what distance will they meet from the starting point? They will meet in 5 hours, Ans. at a distance of 75 miles. 14. One hundred miles of railroad track are to be laid with heavy rail, requiring 116 tons to the mile. After receiving iron at 52 dollars per ton to lay 58 miles, the price per ton was increased so as to make the whole cost of the entire road 612944 dollars. What was the latter price per ton of the iron? Ans. 54 dollars. FRACTIONS. 32. A fraction is a part of a unit. Several methods are used to express fractions or parts of units, which give rise to several distinct kinds of fractions. Those usually employed in arithmetic are VULGAR or COMMON FRACTIONS, and DECIMAL FRACTIONS. What is a fraction? What two methods are usually employed to express fractions? VULGAR FRACTIONS. 33. Vulgar fractions consist of two distinct parts or terms, the one written above the other, with a straight horizontal line between them, as in division, (ART. 23.) The number above the line is called the numerator. The number below the line is called the denominator. The denominator shows how many parts the unit is divided into; and the numerator shows how many parts are used. Thus is a vulgar fraction, whose numerator is 5 and denominator 8: it is read five eighths. A vulgar fraction may be considered a concise method of expressing division, (ART. 23,) where the numerator corresponds to the dividend, and the denominator to the divisor. Thus is the same as 5 divided by 8, and it may therefore be read one eighth of fire, or, as above, five eighths of one. In the same way indicates that 1 is divided into 9 equal parts: it is read one ninth of one. After the same manner, is read one seventh of three, or three sevenths of one. is read one eleventh of six, or six elevenths of one. &c. &c. &c. The fraction denotes that 5 is to be divided by 7. When the numerator is equal to the denominator, the value of the fraction is a unit. When the numerator is less than the denominator, the value is less than a unit, and the expression is called a proper fraction. When the numerator is greater than the denominator, the value is greater than a unit, and the expression is called an improper fraction. Thus, each of the expressions, 3, §, 18, 13, &c., is equal to a unit. Each of the expressions, 2, 4, 4, 5, 4, to, &c., is a proper fraction. Each of the expressions, †,,, H, £, 17, &c., improper fraction. is an When a whole number and fraction are connected, the expression is called a mixed number. Thus, 4, 37, 57, 29, &c., are mixed numbers. The whole number is called the integral part of the expression, and the fraction is called the fractional part. When several fractions are connected by the word of, the expression is called a compound fraction. The expressions, of 3 of 5, of of of 1, of of of, &c., are compound fractions. Any number may be made to assume the form of an improper fraction, by writing under it a unit for the denominator. Thus, 2, 3, 4, 5, 7, &c., are the same as f, t, t, t, 7, &c. †, †, Fractions sometimes occur, in which the numerator, or denominator, or both, are themselves fractional; such expressions are called complex fractions. A fraction is said to be inverted when the numerator and denominator exchange places. Thus the fractions, ,,,, 4, 7, when inverted, become,,,, I, J. What is a vulgur fraction? Which is the numerator of a vulgar fraction? Which the denominator? What does the denominator show? What does the numerator show? In the vulgar fraction five eighths, which is the numerator, and which the denominator? How is it read? What may a vulgar fraction be considered a concise way of expressing? In a vulgar fraction, which part corresponds to the dividend, and which to the divisor? What is the value of the fraction, when the numerator is equal to the denominator? When is the value less than a unit? What is the fraction then called? When is the value greater than a unit? What is the fraction then called? Give examples of proper fractions. Give examples of improper fractions. When a whole nutnber and fraction are connected, what is the expression called? Give ex amples. When several fractions are connected by the word of, what kind of a fraction is it then called? Give examples. When the numerator, or denominator, or both, are already fractional, what are they called? Give examples. When is a fraction said to be inverted? Give examples. REDUCTION OF FRACTIONS. 34. In division, the divisor, dividend and quotient are so related, that the product of the divisor and quotient is always equal to the dividend. Hence, the divisor and quotient may be interchanged; that is, if the dividend be divided by the quotient, the result will be the divisor. It is also obvious, that, with the same divisor, twice as great a dividend will give twice as great a quotient; thrice as great a dividend will give thrice as great a quotient; and in general, the effect of multiplying the dividend by any number is to multiply the quotient by the same number. On the o ́her hand, if the dividend remain the same, multiplying the divisor by any number produces the same effect as dividing the quotient by the same number. Consequently, if we multiply both dividend and divisor by the same number, it will produce no change in the quotient. Again, it is obvious, that with the same divisor, half as great a dividend will give but half as great a quotient; one-third as great a dividend will give one-third as great a quotient; and in general, the effect of dividing the dividend by any number, is to divide the quotient by the same number. On the other hand, if the dividend remain the same, dividing the divisor by any number produces the same effect as multiplying the quotient by the same number. Consequently, if we divide both dividend and divisor by the same number, it will produce no change in the quotient. If, now, we call to mind that the value of a fraction is the quotient arising from dividing the numerator by the denominator, we readily infer the following PROPOSITIONS. I. That, multiplying the numerator by any number is the same as multiplying the value of the fraction by the same number. II. That, multiplying the denominator by any number is the same as dividing the value of the fraction by the same number. III. That, multiplying both numerator and denominator by any number does not alter the value of the fraction. IV. That, dividing the numerator by any number is the same as dividing the value of the fraction by the same number. V. That, dividing the denominator by any number is the same as multiplying the value of the fraction by the same number. VI. That, dividing both numerator and denominator by the same number does not alter the value of the fraction. GREATEST COMMON DIVISOR. 35. The greatest common divisor of two or more numbers, is the greatest number which will divide them without any remainder. |