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 number 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. 90 630 4. Reduce of a tun to its lowest terms. A. 4. ¶ XXXVIII. To multiply a Fraction by a Whole 1. If 1 apple cost much is 2 times? Number. of a cent, what will 2 apples cost? How 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 tor 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 18—2 of a dollar, Ans. But, by dividing the denominator off by 4, as above shown, we immediately have in its lowest terms. From these illustrations we derive the following RULE. I. 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. 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. bushels. 6 of a dollar a yard; what 2. If 1 pound of butter cost 2% of a dollar, what will 205 pounds cost? A. 15-3018-302 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 ? 6. How much is 12 times? 7. How much is 13 times 8 How much is 314 times § ? ? 9. How much is 513 times? A. 14=177. A. 120—236=24. A. 19—665–63. 850 A. 942-2354-2351. 10. How much is 530 times ? A. 11130—4833}. Divide the denominator in the following. 11. How much is 42 times? A. 11. 12. How much is 13 times? A. 252–1218. 13. How much is 60 times To? A. §=24. 14. At 2 dollars a yard, what will 9 yards of cloth cost? 9 times 2 are 18, and 9 times are 8-13, which, added to 18, makes 19 dollars. A. This process is substantially the same as ¶ XXVII., by which the remaining examples in this ruli may be performed. 15. Multiply 3 by 367. 16. Multiply 6 by 211. 17. Multiply 3 by 42. A. 11922. A. 1450§. ¶ 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. 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, 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.) 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. 4. Multiply 272 by 15. 5. Multiply 999 by 21. 6. Multiply 20 by 54% 1 XL. A. 4284. To divide a Fraction by a Whole Number. 1. If 3 apples cost of a cent, what will 1 apple cost? How much is 2÷3? 2. If a horse eat or of a bushel of meal in 2 days, how much will he eat in one day? How much is ÷2? 3. A rich man divided § of a barrel of flour among 6 poor men; how much did each receive? How much is §-6? 4. If 3 yards of calico cost of a dollar, how much is it a yard? How much is ÷÷3? 5. If 3 yards of cloth cost yard? of a dollar, how much is it a The foregoing examples have been performed by simply dividing their numerators, and retaining the same denominator, for the following reason, that the numerator tells how many. parts any thing is divided into ; as, are 4 parts, and, to divide 4 parts by 2, we have only to say, 2 in 4, 2 times, as in whole numbers. But it will often happen, that the numerator cannot be exactly divided by the whole number, as in the following examples. 6. William divided of an orange among his 2 little brothers; what was each brother's part? We have seen, XXX".II., that the value of the fraction is not altered by multiplying both of its terms by the same number; hence, ×2—§. Now, & are 6 parts, and William can give 3 parts to each of his two brothers; for 2 in 6, 3 times. A. & of an orange apiece. Q. In this last example, if (in ) we multiply the denominator 4 by 2, (the whole number,) we have, the same result as before; why is this? A. Multiplying the denominator makes the parts so many times smaller; and, if the numerator remain the same, no more are taken than before; consequently, the value is lessened so many times. From these illustrations we derive the following RULE. 1. When the numerator can be divided by the whole number without a remainder, how do you proceed? A. Divide the nu nerator by the whole number, writing the denominator under the quotient. II. When the numerator cannot be thus divided, how do you proceed? A. Multiply the denominator by the whole number, writing the result under the numerator. Exercises for the Slate. 1. If 8 yards of tape cost of a dollar, how much is it a yard? How much is÷8? Note. When a mixed number occurs, reduce it to an improper fraction, then divide as before. among five men. A. 64—27÷5—27=15% 7. Divide $6 11. Divide 8 by 6. 12. Divide 114 by 280. ↑ XLI. To multiply one Fraction by another. 1. A man, owning of a packet, sells of his part; what part of the whole packet did he sell? How much is of §? 3 x 5 = 15 Ans. The reason of this operation will appear from the following illustration. 4x8=32 Once is, and of is evidently divided by 4, which is done, T XL., by multiplying the denominator 8 by the 4, making 32; that is, of 1=32. of Again, if much, that is, 32. Again, if of Ins., as before. be, then of § will be 5 times as be, then I will be 3 times =}}. |