can be bought with 18 pounds of sugar worth 8 cents a pound ? 12. I exchanged 24 yards of ribbon worth 12 cents a yard for apples worth 72 cents a bushel : how many bushels were obtained ? 13. A farmer gave 16 bushels of potatoes worth 60 cents a bushel for rice at 24 cents à pound : how many pounds did he obtain ? 14. A man paid for 12 tons of coal worth $5 a ton, with wood at $4 a cord : how many cords were required ? 15. How many chickens at $.60 each, must be given for 4 bushels of apples worth $.90 a bushel ? 16. A farmer exchanged 20 cows worth $35 each for 5 horses : what was the value of each horse ? 17. A farmer bought three pieces of calico each containing 50 yards at 8 cents i yard and paid for it with apples at $4 a barrel : how many barrels were required ? 18. A farmer's daughters put up seven tubs of butter, each containing 32 pounds at 28 cents a pound, and exchanged it for 4 patterns of silk of 16 yards each : how much was the silk a yard ? 19. How many tons of hay worth $16 a ton will pay for 30 cords of wood worth $4 a cord ? 20. From 5 pieces of cloth, each containing 25 yards, worth $4 a yard, how many suits can be made worth $20 a suit ? 21. How many bushels of corn, worth 40 cents a bushel, must be exchanged for 80 pounds of butter worth 32 cents a pound ? 22. A grocer sold 18 boxes of soap, each containing 64 pounds at 10 cents a pound, and received in payment 80 barrels of potatoes, each containing 3 bushels : what were the potatoes worth a bushel ? 23. How many boxes of tea, each containing 28 pounds at 75 cents à pound, must be given for 25 firkins of butter of 56 pounds each, at 30 cents a pound? 24. How many bales of cloth, cach containing 45 pieces, and each piece containing 32 yards, worth $2 a yard, must be given for 120 horses, worth $120 each? 25. How many days' work at $1.50 a day will pay for 26 visits of a physician, at $.75 a visit ? 26. A grocer sold 6 boxes of soap, each containing 45 pounds at 8 cents a pound, and received in payment 15 barrels of apples, each containing 3 bushels : what were the apples worth a bushel ? 27. How many bushels of ats at $.48 a bushel must be given in exchange for 336 yards of cotton cloth at 11 cents a yard ? 28. A miller bought 8 loads of wheat, each containing 30 bags of 3 bushels each, worth $1.25 a bushel, and paid for it with flour worth $9 a barrel : how much flour was required ? 29. A grocer exchanged sugar worth 6 cents a pound for 12 bushels of potatoes, at $.60 a bushel : how many pounds were required ? 30. A bought 5 pieces of muslin, each containing 40 yards at 12 cents a yard, and paid for it with wheat at $2 per bushel : how many bushels were required ? 31. A farmer sold 20 barrels of apples, each containing 3 bushels, at $.80 a bushel : how many pieces of calico, each containing 40 yards, at 10 cents a yard, were required to pay for the apples ? 32. A grocer gave 10 boxes of sugar, each containing 120 pounds, for 6 loads of apples, each load containing 15 bags, and each bag 2 bushels, at 40 cents a bushel : what was the sugar worth a pound ? 33. Illustrate cancellation by an original problem. Common Fractions. In any ART. 108.—James being presented with a large apple, wishes to share it equally with a playmate. If he cuts the apple into two equal parts, what is each part called ? Now, suppose each boy divides his portion into two equal parts, how many are there of such parts ? What is the name of each part? Of two parts? Of three ? Of four ? If a pear is sliced into three equal pieces, what is each one called ? Suppose that each piece is cut into two equal parts, how many are there? What is each called ? How many halves are there in a melon ? In a banana ? In a dollar ? In a foot ? In a mile ? thing? How many thirds ? How many fifths ? How many fourths ? How many sevenths ? tenths ? fifteenths ? How many twentieths ? You will observe that in writing the fraction one half, I place the figure 1 above a short, horizontal line, and the figure 2 below it. The 2 below the line shows into how many parts the apple, banana, dollar, or whatever we are talking about, is divided. It therefore gives the name or denomination to the fraction, and is called the denominator; as 1. The figure 1 above the line shows the number of the parts that we have taken, and is therefore called the numerator; as How many How many ART. 109.—In the following fractions, name the numerator and the denominator, and tell into how many parts the unit, or object we are considering, is divided, and how many of those parts are meant: Write the following fractions : 1. Nine seventeenths; eleven nineteenths; three thirtyfirsts. 2. Thirty eighty-fourths; forty-three seventy-ninths. 3. Seventy-two one-hundredths; eighty-eight one hundred forty-fifths. 4. One hundred thirty-three two hundred twenty-fifths. 5. One hundred ninety-five two hundred thirty-thirds. 6. Two hundred eighteen three hundred twenty-fourths. 7. Three hundred forty-eight four hundredths. 8. Five hundred twenty-two five hundred eightysixths. 9. Six hundred eleven seven hundred twenty-ninths. 10. Eight hundred thirteen nine hundred twentieths. 3 ART. 110.—Mrs. Jones has 3 boys, to each of whom she gives of a melon. How many sixths does she give to all ? How many times t of a melon in i of a melon ? If I write ; on the board, and wish to multiply it by 3, how can it be done ? Then, if I multiply the numerator of a fraction, what effect has it on the value of the fraction ? Thus : 1 * 3 - 8 = 3; 7 = How many sixths are there in anything? Three sixths is what part of anything ? We have seen that if the numerator of is multiplied by 3, the result is & or . Now, instead of multiplying the numerator of the fraction ở by 3, let us divide the denominator by 3. The result is , which you perceive is the same we obtained by multiplying the numerator by 3. We conclude, therefore, that if the nu as merator of a fraction is multiplied by a number, the fraction itself is multiplied by that number; and also, if the denominator of a fraction is divided by a number, the fraction itself is multiplied by that number. How many sixths of a melon did Mrs. Jones give to her 3 boys ? Since she divided the å of a melon among the 3 boys, how much did each boy receive ? You observe, therefore, that when the numerator of a fraction is divided by a number, the fraction itself is divided by that number. Thus: 3+3 = ; * x3 = i. Suppose that instead of dividing the numerator, as we have just done, we multiply the denominator of the fraction & by 3. What is the result? How many eighteenths are in anything? How many eighteenths in $ of anything? You perceive that and have the same value, and we conclude, therefore, that if the denominator of a fraction be multiplied by any number, the fraction itself is divided by that number. If I multiply the numerator and denominator of t by 3, what is the result ? But we have just learned that 4 and is have the same value. We conclude, therefore, that it does not change the value of a fraction to multiply the numerator and denominator by the same number. Thus : * = * ; * * j = d. We will now divide the numerator and denominator of is by 3. What is the result? You will notice that the fractions is and d have the same value. We conclude, therefore, that it does not change the value of a fraction to divide its numerator and denominator by the same number. ANALYSIS OF FRACTIONS. ART. 111.-All fractions, as well as all 1. mbers, are derived from the unit 1: hence, all numbers, whether integral or fractional, are referred in analysis to the num |