144. Fractions that have the same denominators are called Similar Fractions. 145. Fractions that have not the same denominators are called Dissimilar Fractions. 146. The denominator of similar fractions is called a Common Denominator. 147. When similar fractions are expressed in their lowest terms, they have their Least Common Denominator. WRITTEN EXERCISES. 148. 1. Reduce 2, 3, and 11, to similar fractions. 8 11 12 = = 8 x 3 = = 18 EXPLANATION. Since the fractions are to be 24 changed to other fractions having a common denominator, the terms of each fraction must be multiplied by some number which will cause them to have the same denominator. (Prin., Art. 133.) 15 24 11 x 2 22 12 × 2 = 24 By examining the denominators 4, 8, and 12, it is evident that the denominators of all the fractions can be made 24, and the fractions will then be similar. To make the denominators 24, the terms of the first fraction must be multiplied by 6; the terms of the second, by 3; the terms of the third, by 2. And thus, the fractions are changed to the similar fractions 1, 14, 27. 15 26. Reduce,,, and to similar fractions having their least common denominator. 33 4 12 16 21 4 4 16 21 2 2 8 1 1 1 4 EXPLANATION. -The least common denominator cannot always be easily found by inspection. It may then be found as in the margin. Since the least common denominator must be the smallest number that will contain each of the denominators, it must contain each of the 3×2×2×4=48 prime factors of the denominators and no other factors. The prime factors are found as in the margin. 3 is a prime factor of 3 and 12, and consequently a factor of the least common denominator. Dividing by 3, and writing below the quotients and numbers of which 3 is not a factor, we have, 1, 4, 4, 16. Dividing by 2, and again by 2, the factors of the denominator are found to be the divisors 3, 2, 2, and the factor 4 in the last row. Their product is 48, the least common denominator. The fractions thus 32 36 28 become 33, 18, 18, 13. NOTE. In finding the factors of the least common denominator a number that is a factor of another number may be disregarded. Thus, since 3 and 4 are factors of 12, they might have been disregarded, and the factors of 12 and 16 only found. Change to similar fractions having their least common denominator: ADDITION. 149. 1. James spent of a dollar for an arithmetic, for a slate, and for a geography. How much did he spend for all? 2. I bought of a yard of silk, but afterward I was compelled to buy of a yard more. How much did I buy? 3. A boy caught a fish that weighed of a pound, and his sister caught one weighing of a pound. How much did both weigh? 4. What is the sum of $3, $ 1, and $1 ? 5. A boy worked of a day for A, and of a day for B. How long did he work for both? 6. A man planted of an acre with potatoes, and § of an acre with corn. How much land was planted with both? 7. A grocer sold 11⁄2 dozen eggs to one person, to another, and 13 dozen to another. How many sell to all? 8. A man sold three lots, the first containing acre, the second much land did he sell? dozen did he of an How 9. A girl paid $13 for a sled, $1⁄2 for a book, and $11 for a doll. How much did her purchases cost? 10. James had $31, Henry had $43, and Samuel had $33. How much had they all? 11. A dressmaker deposited in a savings-bank $33 at one time, $43 at another, and $3 at another. What was the sum of the deposits? 12. A fruiterer sold Mr. A 33 dozen bananas, Mr. B 25 dozen, Mr. C 45 dozen. How many dozen did he sell them all? 13. A carpenter worked 45 days one week, 3 days the next, and 5 days the next. How many days did he work in the three weeks? 14. The expenses of a party were $34 for railroad tickets, $2 for carriages, $576 for provisions. How much were the expenses? 15. What must be done to dissimilar fractions before they can be added? 150. PRINCIPLE. Only similar fractions can be added. WRITTEN EXERCISES. 151. 1. What is the sum of ğ, 3, and 7? The least common denominator of the given fractions is 40, and } = 18; } = }}; and 12. Hence the sum is 77, or 127. 2. What is the sum of 31, 51, and 41. 31=3 6 EXPLANATION.-Since the numbers are composed of 51-521 both integers and fractions, they may be added separately and their sums united. Thus, the sum of the = 4=424 fractions is 3, or 14; the sum of the integers 12; and the sum of both 1374. 13 24 RULE. Reduce the given fractions to similar fractions, add their numerators, and write the sum over the common denominator. When there are mixed numbers, or integers, add the fractions and integers separately, and then add the results. If the sum is an improper fraction, reduce it to an integral or mixed number. |