3. A father, dividing one whole apple among his children, gave them of an apple apiece; how many children were there? 4. James, by saving of a dollar a day, found, after several days, that he had saved 13 of a dollar; how many 8ths did he save? and how many days was he in saving them? 5. How many 7ths in 2 whole oranges? In 24? In 24? In 34? This rule, it will be perceived, is exactly the reverse of the last, and proves the operations of it. RULE I. What, then, is the rule for reducing a mixed or whole number to an improper fraction? A. Multiply the whole number by the denominator of the fraction. II. What do you add to the product? A. The numerator. III. What is to be written under this result? A The denominator. More Exercises for the Slate. A. 1281 A. 874. 12 A. 32. A. 38. A. 197. A. 189. 2. What improper fraction is equal to 20? 3. What improper fraction is equal to 7219? 4. What improper fraction is equal to 4? 5. What improper fraction is equal to 12? 6. What improper fraction is equal to 1611⁄2? 7. What improper fraction is equal to 17: 8. What improper fraction is equal to 1447? A. 179. 9. Reduce 30 pounds to 20ths. As of a pound =1s., 228., the question is the same as if it had been stated thus in 30£ 5 s. how many shillings? A. 905-605 shillings. 10. In 14 weeks, how many 7ths? A. 11-101 days. 11. In 26 pecks, how many 8ths? A. 241=211 quarts. ↑ XXXVII. To reduce a Fraction to its lowest Terms Q. When an apple is divided into 4 parts, 2 parts, or, 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 4, and multiply both the 2 and 4 by 2, we obtain; what, then, is equal to? A., or . Q. Now it is plain that the reverse of this must be true; for, if we divide both the 4 and 8 in by 2, we obtain 4, and, dividing the 2 and 4 in 4 by 2, we have ; what, then, may be inferred from these remarks respecting multiplying or divid ing both the numerator and denominator of the same fraction? 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. into its equal call Q. What is the process of changing ed? A. Reducing the fraction to its lowest terms. 7. Reduce to its lowest terms. Operation by Slate illustrated. 1. One minute is of an hour, and 15 minutes are 18, what part of an hour will make, reduced to its lowest terms? How do you get the in this example? A. By dividing 15 and 60, each, by 5. How do you get the? A. By dividing 3 and 12, each, by 3. 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. I. How do you proceed to reduce a fraction to its lowest terms? A. Divide both the terms of the fraction by any nuniber that will divide them without a remainder, and the quotients again in the same manner. II. When is the fraction said to be reduced to its lowest terms? A. When there is no number greater than 1 tnat will divide the terms without a remainder. More Exercises for the Slate. ↑ XXXVIII. of a hogshead to its lowest terms of a foot to its lowest terms. 1. If 1 apple cost of an inch to its lowest terms. A. §. A. Z. A. 4. A. 12. A. . A. To multiply a Fraction by, a Whole 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 2 times? 3 times ? 3. William has of a melon, and Thomas 2 times as much; what is Thomas' part? How much is 2 times ? 2 times ? 2 times? 3 times? 6 times? Q. From these examples, what effect does multiplying the numerator 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) by 2, we obtain ; what effect, then, does dividing the denominator by any number have on the value of a fraction, if the numerator remain the same? A. It multiplies the value by that number. Q. What is the reason of this? A. Dividing the denomina lor 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 nu merator 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. 4. At of a dollar a yard, what will 4 yards of cloth cost? 4 times are of a dollar, Ans. But, by dividing the denominátor off by 4, as above shown, we immediately have in its lowest terms. From these illustrations we derive the following RULE. 1. How can you multiply a fraction by a whole number? 4. Multiply the numerator by it, without changing its denominator. II. How can you shorten this process? A. Divide the denominator by the whole number, when it can be done without a remainder. 1. If a horse consume of a bushel of oats in one day, how many bushels will he consume in 30 days? A. bushels. =6 2. If 1 pound of butter cost of a dollar, what will 205 pounds cost? A. 15-305-30 dollars. 3. Bought 400 yards of calico, at § of a dollar a yard; what did it come to? A. 1200-$150. 4. How much is 6 times? 5. How much is 8 times 11⁄2? 6. How much is 12 times? 7. How much is 13 times ? A. 19=174. A. 2742—3183. 850 8 How much is 314 times? A. 942-2354-2351. 9. How much is 513 times? A. 3501–326TT. 10. How much is 530 times? A. 11130—48331. 23 Divide the denominator in the following. 11. How much is 42 times ? A. 11. 12. How much is 13 times 8? A. 259–1248. 13. How much is 60 times To? A. §=2). 14. At 2 dollars a yard, what will 9 yards of cloth cost? 9 times 2 are 18, and 9 times are f=1}, which, added to 18, makes 19 dollars. A. This process is substantially the same as ¶ XXVII., by which the remaining examples in this ruls may be performed. 15. Multiply 3 by 367. 16. Multiply 63 by 211. 17. Multiply 36 by 42. A. 11922. A. 1450§. A. 12988-1291. ¶ XXXIX. To multiply a Whole Number by a Fraction. 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 certain portion of it, is to be repeated several times, as, for instance, If you pay 12 cents for a me.on, what will of one cost? of 12 cents is 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. How much is of 12 of 12?of 20?of 20?" of of 40? of 40? of 40? Q. We found in Multiplication, TX., 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 described in ¶ XXVII.; what, then, is the rule for multiplying a whole number by a fraction? (For answer, see ¶ XXVII.) 8 of 8?of 40? 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? A. $337. 3. Multiply 425 by 5. A. 2210. |