1. Write the numbers in a horizontal line, omitting such of the smaller numbers as are factors of the larger, and divide by any prime factor that will exactly divide one or more of the given numbers, and write the quotients and undivided numbers, if any, in a line beneath. Find the L.C.M. 8. Of 144 and 180. 10. Of 225 and 375. 2. In like manner, divide the quotients and undivided numbers, until the results are prime to each other. The product of the divisors and numbers in the last line will be the L. C. M. 11. Of 16, 40, and 96. 14. Find the L. C. M. of 25, 60, 100, and 125. 15. Find the L.C. M. of 7, 15, 21, 25, and 35. 17. What is the smallest sum of money which can be exactly expended for books at $5, or $3, or $4, or $6 each? 18. What is the shortest piece of rope that can be cut exactly into pieces either 15, 18, or 20 feet long? 19. What is the product of the L. C. M. of 12, 16, 24, and 32, multiplied by their G.C.D.? 20. Divide the L. C.M. of 7, 42, 6, 9, 10, and 630, by the G. C.D. of 110, 140, and 680. 21. What is the smallest sum of money which can be exactly expended for sheep at $8, or cows at $28, or oxen at $54, or horses at $162 each? 22. What is the smallest quantity of grain that will fill an exact number of bins, whether they hold 36, 48, 80, or 144 bushels? 23. What is the least number of acres in a farm that can be exactly divided into lots of 10 acres, 14 acres, 16 acres, and 20 acres each ? FRACTIONS. INDUCTIVE EXERCISES. 97. 1. If any unit, as an apple or a yard, is divided into 2 equal parts, what is each part named? (64.) 2. If into 3 equal parts? If into 5 equal parts? 3. What are the largest equal parts that can be made of a whole? The next largest? The next? 4. What is 1 of 4 equal parts of a unit called? 2 of 3 equal parts? 3 of 5 equal parts? 5 of 7 equal parts? 5. What is meant by 1 fourth of anything? 3 fourths? 2 fifths? 3 eighths? 5 ninths? 6. What are 3 of the 7 equal parts of a week called ? 7. What are 5 of the 8 equal parts of a yard called? DEFINITIONS. 98. A fraction is one or more of the equal parts of a unit, or of anything regarded as a whole.* Thus, 1 half, 2 thirds, and 4 fifths are fractions. 1. A fractional unit is one of the equal parts into which the integer or thing is divided. Thus, in the expression three fourths of a mile, the fractional unit is one fourth of a mile. Fractions are classified into common and decimal. * Properly speaking, the value of a fraction is less than 1. Hence, when the value is greater than 1, it is called an improper fraction (100, 2). One third is written "6 Five sixths Seven eighths 99. A common fraction is expressed by two numbers, called the numerator and the denominator, the former written over the latter, with a line between them. Thus, ་་ Nine tenths Twelve thirty-fifths 6 5 is written "6 7 9 8 7 1. The denominator of a fraction, written below the line, shows the number of equal parts into which the unit is divided, and also names the parts. Thus, in the fraction, 8 is the denominator, and shows that the unit is divided into eight equal parts, named eighths. 11 9 2. The numerator of a fraction, written above the line, shows the number of equal parts taken to forn fraction. 7 10 Thus, in 7,7 is the numerator, and shows that 7 of the eight equal parts are taken, or expressed by the fraction. 3. The terms of a fraction are its numerator and denominator. Thus, 6 and 7 are the terms of the fraction §. 4. The reciprocal of a fraction is 1 divided by that fraction, or it is the fraction inverted. Thus, the reciprocal of & is 1÷3 = }; of, it is 12. 5. The value of a fraction is the quotient of its numerator divided by its denominator. Thus, 12 = 4. Name the fractional unit, the terms, the reciprocal, and the value of each of the following fractions: 18 24 220 12 100. Common fractions are classified as proper or improper, according to their value. 1. A proper fraction is a fraction whose numerator is less than its denominator. Hence, its value is less than 1. Thus, t, §, and 11 are proper fractions. 2. An improper fraction is a fraction whose numerator equals or exceeds its denominator. Hence, its value is equal to or greater than 1. Thus, 8, 10, and 21 are improper fractions. 101. A mixed number is a number composed of an integer and a fraction united. Thus, 125 is a mixed number, equivalent to 12+5. 102. Since fractions indicate division, all changes in the terms of a fraction will affect the value of the fraction according to the "General Principles of Division” (72), as shown in the following illustrations I. II. III. 416 4 many. 2. The value of each fractional unit is twice as large, but the number of un its remains the same. Hence, In both cases, the value of the fraction is multiplied (72, I). 1. The value of each fractional unit remains the same, but the number of units is one half as many. 160 ૭ 2 4 6 × 2 = [4 × 2 6 × 2 4÷2 = = = 2. The value of each fractional unit is one half as large, but the number of units remains the same. Hence In both cases, the value of the fraction is divided (72, II). = 6 = 4130 216 4 12 8 12 1. The value of each fractional unit remains the same, but the number of nits is twice as 213 1. The value of each fractional unit is only one half as large, but the number of units is twice as many. 2. The value of each fractional unit is twice as large, but the number of units is one half as many. Hence, In either case, the value of the fraction is not changed (72, III). The above may be reduced to one general principle, viz.: A change in the NUMERATOR, by multiplication or division, produces a LIKE change in the value of the fraction; but such a change in the DENOMINATOR produces an OPPOSITE change in the value of a fraction. REDUCTION. 104. Reduction of fractions is the process of changing their form, without altering their value. 1. A fraction is changed to higher terms, when its numerator and denominator are expressed in larger numbers; to lower terms, when expressed in smaller numbers; and to its lowest terms, when the numerator and denominator are prime to each other. 2. All higher terms of a fraction are multiples of its lowest terms. 105. To change fractions to higher or lower terms. EXERCISES. INDUCTIVE 1. One half is equal to how many fourths? ANALYSIS.-Since 1 is equal to 4 fourths, is equal to 1 half of 4 fourths, or 2 fourths. |