RULE. Q. 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. 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. A. 1201. A. 874. A. 38. A. 38. 2. What improper fraction is equal to 20? A. 197. 9. Reduce 30 pounds to 20ths. As of a pound 1 s., 22s., 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. 101-101 days. 11. In 26 pecks, how many 8ths? A. 211211 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 =2; what, then, is equal to ? A. 2, 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, 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 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. Q. What is the process of changing into its equal called? Q. How do you get the in this example? 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 th 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. More Exercises for the Slate. 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 numera tor 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) oy 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 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. 3 4. At of a dollar a yard, what will 4 yards of cloth cost? 4 times are 1 of a dollar, Ans. But, by by 4, as above shown, we dividing the denominator of 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 bushels. 2. If 1 pound of butter cost of a dollar, what will 205 pounds cost? A. 15—3015—30 dollars. = Divide the denominator in the following. 11. How much is 42 times ? Ꭿ. 11. 115 14. At 2 dollars a yard, what will 9 yards of cloth cost? g times 2 are 18, and 9 times are =1, which, added to 18, makes 19 dollars. A. This process is substantially the same as ¶ XXVII., by which the remaining examples in this rule may be performed. A. 1192}. 15. Multiply 34 by 367. 17. Multiply 3 by 42. A. 12988-129. ¶ 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 melon, 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. Q. How much is of 12? of 12? of 20? of 20? of 8 ? of 8?of 40? of 40? 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 described in T XXVII.; what, then, is the rule for multiplying 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? A. $337. 3. Multiply 425 by 5. A. 2210. A. 4284. |