Again, divide { of a pear into 6. equal parts. If a whole pear were divided into 6 equal parts, each part would be expressed by But since the half of the pear was divided, each part will be expressed by į of , or is: In the division of fractions we should note the follow. ing principles : 1st. When the dividend is just equal to the divisor, the quotient will be 1. 2d. When the dividend is greater than the divisor, the quotient will be greater than 1. 3d. When the dividend is less than the divisor, the quotient will be less than 1. 4th. The quotient will be as many times greater than 1, as the dividend is greater than the divisor. 5th. The quotient will be as many times less than 1, as the dividend is less than the divisor. ಅ ಅ CASE I. 159. To divide a fraction by a whole number. Divide the numerator or multiply the denominator by the whole number. EXAMPLES. 1. Divide by 2. In the first operation we OPERATION. divide the fraction by mul. 4 4 4 2 tiplying the denominator 2 3X2 6 (Art. 120): in the second 4 2)4 2 we divide the numerator :: 3 3 3 (Art. 119), giving the same result in both cases. 158. What does division of whole numbers explain? In division of fractions, may the divisor exceed the dividend? How will the quotient then compare with 1? If an apple be divided into 2 equal parts, what will express each part ? If half an apple be divided into 4 equal parts, what will express one of the parts? What is one-half of one-half? What is one-sixth of one-half? What principles do you note in the division of fractions ? When will the quotient be 1 ? When greater than 1? When will the quotient be less than 1 ? When greater than 1, how many times greater? When less than 1, how many times less ? or 10 10 Let it be required to divide by The true quotient will be expressed by the complex fraction , Let the terms of this fraction be now multiplied by the denominator with its terms inverted : this will not alter the value of the fraction (Art. 122), and we shall then have, H_**_ =*=*=quotient. 1 It will be seen that the quotient is obtained by simply. multiplying the numerator by the denominator with its terms inverted. This quotient may be further simplified by cancelling the common factors 5 and 8, giving for the true quotient. SECOND METHOD OF PROOF. Let us first divide the dividend by OPERATION. 5. This is done by multiplying the 10 -5=12 denominator (Art. 120), which gives 120x8=120 10. But the divisor being but of 159. In how many ways may a fraction be divided by a whole namber? 160. How do you divide one fraction by another? How may the quotient of one fraction divided by another be expressed? If any fraction be multiplied by the fraction which arises from inverting its terms, to what will the product be equal ? In the second method of proof, after dividing by 5, is the quotient too small or too large, and how much? How then do you find the true quotient? 80 126 5, this quotient is 8 times too small, since the eighth of a number will be contained in the dividend 8 times more than the number itself. Therefore, by multiplying 1 by 8, we obtain for the true quotient. Hence, to divide one fraction by another, Reduce compound and complex fractions to simple ones, also whole and mixed numbers to improper fractions : then multiply the dividend by the divisor with its terms inverted, and the product reduced to its simplest terms will be the quotient sought. EXAMPLES. 64 23 1. Divide by 2. Divide 37 by 3. Divide 16 of } by 41. 4. Divide 44,33 by 33. 5. Divide 371} by a 6. Divide by is. 7. Divide of į by of 8. Divide 5 by to 9. Divide 5205} by of 91. 10. Divide 100 by 4. 11. Divide of by k. 12. Divide of 50 by 45. 13. Divide 14 of } by 3} of 6. 545 14. Divide 344 by 93 € Ans. 64 63 APPLICATIONS. a per pound? 523 1. If 77b. of sugar cost of a dollar, what is the price 47= 4 of $1 ; or of 11° cents=29=1 Ans. Bå cents. 2. If of a dollar will pay for 1021b. of nails, how much is the price per pound? Ans. $ 3. If 4 of a yard of cloth cost $3, what is the price per yard? Ans. $54. Ans. $2125 94 4. If $21} will buy 71barrels of apples, how much are they per barrel ? 5. If 47 gallons of molasses cost $25, how much is it per quart ? Ans. 6. If ifhhd. of wine cost $2503, how much is the wine per quart ? Ans. *150*=88170cts. 7. If eight pounds of tea cost 7 of a dollar, how much is it per pound? Ans. 95 cts. 8. In 8 weeks a family consumes 165% pounds of butter: how much do they consume a week? Ans. 1963 lb. 9. If a piece of cloth containing 176 yards costs $3759, what does it cost per yard ? Ans. 10. If I pay s dollar a pound for tea, how many pounds can I have for 4284 dollars ? Ans. 48961). 11. Bought flour at 70 dollars a barrel, and laid out 129 dollars for the article: how many barrels did I buy ? Ans. 1633 12. Paid 666} cents for marbles at 6 cents apiece: how many did I buy ? Ans. 1113 13. If raisins are 2816 cents a pound, how much can I have for 17 11 cents ? Ans. 14. How many barrels of four can I buy for 16111 dollars if I pay 14 dollars a barrel ? Ans. 11124 Ibar. 15. Divide 5205} dollars among of 90 persons: what will each have ? Ans. 72126 16. At 27 dollars an acre, how much land can I buy for 1 of a dollar ? Ans. ažo acre. 17. How many apples can I buy for 21 of 1 of 2 cents, if I pay of 2 of i cents apiece ? 18. Bought of a lot of land for 5040 dollars, and having sold á of what was bought, I gave of the remainder to a charitable society, and divided the residue among 9 poor persons: what was the share of each ? Ans. 37. 19. Of an estate valued 15000 dollars, the widow has į, the oldest son of remainder, and the residue was divided among 9 children: what was the share of each of the 9 children ? Ans. $37011 53 a Ans. 26. DECIMAL FRACTIONS. 161. If the unit 1 be divided into 10 equal parts, the parts are called tenths, because each part is one-tenth of unity. If the unit 1 be divided into one hundred equal parts, the parts are called hundredths, because each part is one hundredth of unity. If the unit 1 be divided into one thousand equal parts, the parts are called thousandths, because each part is one thousandth of unity: and we have similar expressions for the parts, when the unit is divided into ten thousand, one hundred thousand, &c., equal parts. The division of the unit into tenths, hundredths, thou. sandths, &c., forms a system of numbers called Decimal Fractions. They may be written thus : Four-tenths, 18 Six-tenths, 16 Forty-five hundredths, 100 125 thousandths, 1047 ten thousandths, From which we see, that in each case the denominator gives denomination or name to the fraction; that is, determines whether the parts are tenths, hundredths, thousandths, &c. 162. The denominators of decimal fractions are sel. dom set down. The fractions are usually expressed by means of a comma, or period, placed at the left of the numerator. 6 45 125 1000 1047 10000 161. When the unit 1 is divided into 10 equal parts, what is each part called? What is each part called when it is divided into 100 equal parts? When into 1000? Into 10,000, &c.? How are decimal fractions formed? What gives denomination to the fraction ? 162. Are the denominators of decimal fractions generally sot down? How are the fractions expressed? Is the denominator understood ? What is it? What is the place next the decimal point called? The next? The third, &c. ? Which way are decimals numerated ? |