16. Find the least common multiple of 255 and 257. 26. Find the contents of the smallest vessel that may be filled by using a 3-quart, a 4-quart, a 5-quart, or a 6-quart measure. 27. What is the shortest length that can be measured by either of four measures which are respectively 10 in., 15 in., 27 in., and 30 in. long? 28. A can walk round a race-course in 12 min., B in 15 min., and C in 18 min. If they start together and keep walking each at his own rate, how many minutes will elapse before they are all three together at the startingpoint, and how many times will each have made the circuit? 29. A lady desires to purchase a quantity of cloth that can be cut without waste into parts 4, 5, or 6 yards long. What is the least number of yards that she can buy for that purpose? GREATEST COMMON DIVISOR OF FRACTIONS. 519. 1. Give several fractions which are contained in 3 an integral number of times. 2. What relation do the numerators of these divisors bear to the numerator of ? 3. What relation do the denominators of these divisors bear to the denominator of ? 4. Find several fractions which are common divisors of and. Of and 3. Of and 3. Of and 3. 3 5. What relation do the numerators of these common divisors bear to the numerators of the fractions which they divide? 6. What relation do the denominators of these common divisors bear to the denominators of the fractions which they divide? 7. Since the fraction which will exactly divide the given fractions is greatest when its numerator is as large as possible, and its denominator as small as possible, how are the terms of the greatest common divisor of fractions obtained from the terms of the fractions? 520. PRINCIPLE. The greatest common divisor of two or more fractions is the greatest common divisor of their numerators divided by the least common multiple of their denominators. 7. The sides of a triangular lot are 115 feet, 128 feet, and 1343 feet long. How many rails of the greatest length possible will be needed to fence it, the rails lapping 6 inches at each end, and the fence to be 7 rails high? LEAST COMMON MULTIPLE OF FRACTIONS. 522. 1. Give several fractions or integers which will contain an integral number of times, 1, 3, 4, 16. 3 2. What relation do the numerators of these multiples bear to the numerators of the given fractions? 3. What relation do the denominators of these multiples bear to the denominators of the given fractions? 4. Find several common multiples of and 4. Of and 3. Of 1 and 3. 5. What relation do the numerators of these common multiples bear to the numerators of the fractions which they contain? 6. What relation do the denominators of these common multiples bear to the denominators of the fractions which they contain? 7. Since the number that will exactly contain the given fractions is least when its numerator is as small as possible and its denominator as large as possible, how are the terms of the least common multiple of fractions obtained from the terms of the fractions? 523. PRINCIPLE. The least common multiple of two or more fractions is the least common multiple of their numerators divided by the greatest common divisor of their denomi 7. The pendulum of one clock makes 25 beats in 28 seconds, and that of another clock 30 beats in 34 seconds. If the clocks are started at the same moment, when first after starting will the clocks beat together again? CIRCULATING DECIMALS. 525. 1. When a cipher is annexed to a number, by what is the number multiplied? After the cipher has been annexed, by what numbers can the number be divided by which it could not be divided before? 2. Since the only new factors by which a number multiplied by 10 can be divided are 2 and 5, when a common fraction in its lowest terms is being reduced to a decimal, if the denominator contains only the factors 2 or 5, will the division be exact or not? 3. If the denominator contains other factors besides 2 or 5, what can be said of the division? 4. If any fraction, as, is reduced to a decimal by annexing ciphers to the numerator and dividing by the denominator, how many possible remainders can there be? 5. Since in each instance the remainder with a cipher annexed forms the new dividend, and since there can be but 6 different dividends, what may be concluded regarding the repetition of the decimal figures? 6. What, then, may be inferred regarding the repetition of the decimal figures in any infinite decimal? 526. A decimal that contains a definite number of decimal places is called a Finite Decimal. 527. A decimal that never terminates is called an Infinite Decimal. All infinite decimals have a figure or set of figures which are repeated indefinitely. Such decimals are called Circulating Decimals. 528. The figures or set of figures repeated in an infinite or circulating decimal are called the Repetend. Thus, the common fraction tion by .142857142857 + etc. in the second 142857. is expressed by .3333 + etc.; the fracIn the first fraction the repetend is 3; A repetend is indicated by placing a dot over the repeated figure; or over the first and last figures of the set that is repeated. Thus, .333 + is written .3; .142857142857 + is written .142857; .1666+ is written .16. 529. A decimal expressed wholly by a repetend is called a Pure Circulating Decimal. Thus, .3333+ and .142857142857 + are pure circulating decimals. 530. A decimal expressed only in part by a repetend is called a Mixed Circulating Decimal. Thus, .1666 + etc., .4535353 + are mixed circulating decimals. 531. PRINCIPLES. -1. Any fraction in its lowest terms whose denominator contains no other prime factors besides 2 or 5 can be reduced to a finite decimal. 2. Any fraction in its lowest terms whose denominator contains other prime factors besides 2 or 5 will produce a circulating decimal. EXERCISES. 532. Tell by inspection which fractions will produce finite decimals and which circulating decimals. 18. Reduce to a decimal. 999 999. 333. 73 757 |