ROLE. Q. What, then, is the rule for reducing a mixed or whola umber to an improper fraction? A. Multiply the whole number by the denomi nator of the fraction. Q. What do you add to the product? A. The numerator. Q. What is to be written under this result? A. The denominator. More Exercises for the Slate. 2. What improper fraction is equal to 20? A. 1281. 4. What improper fraction is equal to 4? A. 7. A. 38. A. A. 19. 7. What improper fraction is equal to 17? A. 189. 8. What improper fraction is equal to 144? A. 1739. 9. Reduce 30 pounds to 20ths. As of a pound =1 s., 2s., the question is the same as if it had been stated thus In 30£ 5 s. how many shillings? A. 605605 shillings. 10. In 14 weeks, how many 7ths? A. 121-101 days. 11. In 26 pecks, how many 8ths? A. 211211 quarts. 1XXXVII. To reduce a FRACTION TO ITS LOWEST TERMS. Q. When an apple is divided into 4 parts, 2 parts, or 2, are evidently of the apple: now, if we take ), and multiply the 1 and 2 both by 2, we shall have again; why does not this multiplying alter the value? A. Because, when the apple is divided into 4 parts, or quarters, it takes 2 times as many parts, or quarters, to make one whole apple, as it will take parts, when the apple is divided into only 2 parts, or halves: hence, multiplying only increases the number of parts of a whole, without altering the value of the fraction. Q. Now, if we take †, and multiply both the 2 and 4 by 2, we ob equal to ? saint; what, then, is A., or Q. Now it is plain that the reverse of this must be true; for, if we vide both the 4 and 8 in by 2. we obtain †, and, dividing the 2 and 4 in by 2, we have; what, then, may be inferred from these remarks respecting multiplying or dividing both the numerator and denominator of the same fracuon? A. That they may both be multiplied, or divided, by the same number, without altering the value of the fraction. Q. What are the numerator and denominator of the same fraction called? A. The terms of the fraction. Q. What is the process of changing into its equal called? A. Reducing the fraction to its lowest terms 5)15= 12 =1 of an hour, and 15 minutes are ; make, reduced to its lowest terms? Q. How do you get the example? in this A. By dividing 15 and 60 each by 5. A. By dividing 3 and 12, each, by 3. Q. How do you know that is reduced to its lowest terms? A. Because there is no number greater than 1 that will divide both the terms of without a remainder From these illustrations we derive the following RULE. Q. How do you proceed to reduce a fraction to its lowest terms? A. Divide both the terms of the fraction by any number that will divide them without a remainder, and the quotients again in the same manner. Q. When is the fraction said to be reduced to its lowest terms? A. When there is no number greater than 1 that will divide the terms without a remainder. ¶ XXXVIII. TO MULTIPLY A FRACTION BY A WHOLE NUMBER. 1. If 1 apple cost of a cent, what will 2 apples cost? How much is 2 times? 2. If a horse eat of a bushel of oats in one day, how many bushels will he eat in 2 days? In 3 days? How much is two times ? 3 times ? 3. William has of a melon, and Thomas 2 times as much; what is Thomas's part? How much is 2 times †? 2 times ? 2 times? 3 times ? 6 times? Q. From these examples, what effect does multiplying the numero or by any number appear to have on the value of the fraction, if the denominator remain the same? A. It multiplies the value by that number. Q. 2 times is; but, if we divide the denominator 4 (in è̟) By 2, we obtain ; what effect, then, does dividing the denominator by Buy number have on the value of a fraction, if the numerator remain the same ? 4. It multiplies the value by that number. Q. What is the reason of this? A. Dividing the denominator makes the parts of a whole so many times larger; and, if as many are taken, as before, (which will be the case if the numerator remain the same,) the value of the fraction is evidently increased so many times. Again, as the numerator shows how many parts of a whole are taken, multiplying the numerator by any number, if the denominator remain the same, increases the number of parts taken; consequently, it increases the value of the fraction. are 4. At of a dollar a yard, what will 4 yards of cloth Cost? 4 times of a dollar, Ans. But, by dividing the denominator of by 4, as above shown, we immediately have in its lowest terms. From these illustrations we derive the following RULE. Q. How can you multiply a fraction by a whole number ? A. Multiply the numerator by it without changing its denominator. Q. How can you shorten this process? A. Divide the denominator by the whole number, when it can be done without a remainder Exercises for the Slate. 1. If a horse consume of a bushel of oats in one day, how many bushels will he consume in 30 days? A. = 6 bushela 2. If 1 pound of butter cost of a dollar, what will 205 pounds cost? A. 12=3018=304 dollars. 3. Bought 400 yards of calico, at did it come to? A. 1200 = $150. 4. How much is 6 times? 5. How much is 8 times §? of a dollar a yard; what Divide the denominator in the following. 11. How much is 42 times ? A. 11. 12. How much is 13 times 38? 13. How much is 60 times To? A. 259 = 1248 A. 14. At 2 dollars a yard, what will 9 yards of cloth cost? times 2 are 18, and 9 times are 1, which, added to 18, makes 19 dollars. A. This process is substantially the same as T XXVII., by which the remaining examples in this rule. may be performed. 15. Multiply 3 by 367. 16. Multiply 6 by 211. 17. Multiply 3 ¶ XXXIX. A. 1192. A. 1450. by 42. A. 12988=1294. TO MULTIPLY A WHOLE NUMBER BY Q. When a number is added to itself several times, this repeated addition has been called multiplication; but the term has a more extensive application. It often happens that not a whole number only, but a eertain portion of it, is to be repeated several times; as, for instance, If you pay 12 cents for a melon, what will of one cost? of 12 cents in 3 cents, and to get, it is plain that we must repeat the 3, 3 times, making 9 cents, the answer; when, then, a certain portion of the multiplicand is repeated several times, or as many times as the numerator shows, what is it called? A. Multiplying by a fraction. Q. How much is of 12 of 81 of 401 of 121 of 20 of 201 of 8 of 401 of 40? of 40? Q. We found in Multiplication, ¶ X., that when two numbers are to be multiplied together, either may be the multiplier; hence, to multiply a whole number by a fraction, is the same as a fraction by a whole number; consequently, the operations of both are the same as that de scribed in T XXVII.; what, then, is the rule for muitiplying a whole number by a fraction? (For answer, see ¶ XXVII.) Exercises for the Slate. 1. What will 600 bushels of oats cost, at of a dollar a bushel? A. $112. 2. What will 2700 yards of tape cost, at † of a dollar a yard ? 4. $337. |