1 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 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 †, and, di viding the 2 and 4 in 2 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. Q. What is the process of changing into its equal call 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 15. 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 num ber 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 lowesi terms? A. Divide both the terms of the fraction by any num per that will divide them without a remainder, and the quo tients 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. 2. Reduce 4 of a barrel to its lowest terms. 3. Reduce 4. Reduce of a tun to its lowest terms. A. 4. 90 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 ? 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 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. 1. How can you multiply a fraction by a whole number? A. 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. dollars. 6 of a dollar a yard; wnat 2. If 1 pound of butter cost of a dollar, what will 205 pounds cost? A. 15—3015=30 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 11⁄2? 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. 19=174. A. 942-2354-2354. ? A. 351–326. 10. How much is 530 times ? A. 11130—48331. Divide the denominator in the following. 11. How much is 42 times ? A. 11. 12. How much is 13 times? A. 259–1248. 13. How much is 60 times ? A. §=21. 14. At 2 dollars a yard, what will 9 yards of cloth cost 9 times 2 are 18, and 9 times are 8=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 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 reDeated 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, à 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 20? of 20? of 8 of 8? of 40? of 40? of 40? Q. We found in Multiplication, T 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 operation 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.) of 12? 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. ↑ XL. To divide a Fraction by a Whole Number 1. If 3 apples cost of a cent, what will 1 apple cost? How much is ÷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 of a dollar, how much is it a yard? The foregoing examples have been performed by simply di viding 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, x2-f. 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 be fore; 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 |