3. How many pints in gills? In 2 gills? In gills? In 13o gills? 4 4. How much is of a dollar? A. $1. Is ? A. 1 and f 14. Is 10? Is 16? Is ? Is 24? Is 25? 8 17 Q. What is the finding how many whole things are contained in an improper fraction called? A. Reducing an improper fraction to a whole or mixed number. 1. James, by saving of a dollar a day, would save in 33 days; how many dollars would that be? OPERATION. 16)33 Ans. 2 dollars. In this example, as † make 1 dollar, it is plain, that as many times as 16 is contained in 33, so many dollars it is, 16 is contained 2 times and 1 over; that is, 216 dollars. RULE I. What, then, is the rule for reducing an improper fraction to a whole or mixed number? A. Divide the numerator by the denominator. More Exercises for the Slate. 2. A regiment of soldiers, consuming of a barrel of pork a day, would consume in 23 days 28 of a barrel; how many barrels would that be? A. 5 barrels. 1 3. A man, saving of a dollar a day, would save in 365 days 5 365; how many dollars would that be? A. $73. 4. Reduce 101 to a mixed number. A. 20. 5. Reduce to a mixed number. A. 721. 87 6. Reduce to a mixed number. A. 4. 8. Reduce 167 to a mixed number. A. 131. 12 9. Reduce to a mixed number. A. 231. 6384 10. Reduce 1728 to a whole number. A. 144. ¶ XXXVI. To reduce a Whole or Mixed Number to an Improper Fraction. 1. How many halves will 2 whole apples make? Will 3? Will 4? Will 6? Will 20? Will 100? 2. How many thirds in 2 whole oranges? In 2? In 2? In 3? In 3? In 8? In 12? 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? 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. 1781. A. 874. A. 39. A. 38. 2. What improper fraction is equal to 20? 3. What improper fraction is equal to 72? 4. What improper fraction is equal to 4? 5. What improper fraction is equal to 123? 6. What improper fraction is equal to 16? 7. What improper fraction is equal to 17? 8. What improper fraction is equal to 144? A. 1729. 9. Reduce 30 pounds to 20ths. As zo of a pound = 18., σ A. 197. A. 189. 28., the question is the same as if it had been stated thus: n 30£ 5 s. how many shillings? A. 305605 shillings. 10. In 144 weeks, how many 7ths? A. 191101 days. 11. In 26 pecks, how many 8ths? A. 2211 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 2, and multiply both the 2 and 4 by 2, we obtain ; what, then, is equal to ? 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 2, and, dividing the 2 and 4 in 2 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? A. Reducing the fraction to its lowest terms. 6. Reduce 7. Reduce 50 to its lowest terms. 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? 8 ΤΣ How do you get the in this example ? A. By dividing 15 and 60, each, by 5. How do you get the 1o 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 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 that will divide the terms without a remainder. More Exercises for the Slate. 2. Reduce of a barrel to its lowest terms. 3. Reduce of a hogshead to its lowest terms. 6 90 4. Reduce of a tun to its lowest terms. 5. Reduce of a foot to its lowest terms. 30 144 1728 324 48 9 6. Reduce of a gallon to its lowest terms. 7. Reduce 142 of an inch to its lowest terms. ↑ XXXVIII. 1. If 1 apple cost much is 2 times? 2. If a horse eat of a cent, what will 2 apples cost? How 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 3 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 = 2: 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 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 3 4. At of a dollar a yard, what will 4 yards of cloth cost? 4 times are 12 of a dollar, Ans. But, by dividing the denominator of by 4, as above shown, we immediately have † 3 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. TS 1. If a horse consume of a bushel of oats in one day, how many bushels will he consume in 30 days? A. 1-6 bushels. 10. How much is 530 times? A. 11130—48331. 23 Divide the denominator in the following. 11. How much is 42 times ? A. 11. |