remainder. They sell the same for 1425 dollars; what must A receive; also, what is B's gain? Ans. A receives $78638. $123. 23. 7 men and a boy own a house in company. Each man's share is worth $480, and the boy's share is equal to of one man's. What is the value of the house? Ans. $3520. Illustration.-When two fractions of the same unit or quantity, or of two equal units or quantities, have equal denominators, a fractional unit, (that is, one of the parts contained in the numerator,) in one numerator, is equal to a fractional unit in the other, and the difference between the numerators will be the difference between the parts of an unit or a quantity expressed by one fraction, and the parts expressed by the other. If we have and of a dollar, the 5 twelfths express more than the 2 twelfths. The same may be expressed as follows: let a dollar be divided into 12 equal parts, and let A have 5 of these parts and B 2. A must have 3 parts more than B. Fractions having unequal denominators cannot be subtracted until they are reduced to a common denominator. The difference between and, is neither nor. equal, and equal, and the difference is; but equals, and equals RULE. Prepare the fractions as in addition. Subtract the less numerator from the greater, and write the difference over one of the denominators. 1. What is the difference between 3 and ?? -the answer. 2. What is the difference between 2 and 3? 3. Subtract from . Ans. 1. Ans. ; 4. A and B travel together; A pays of the expense, and B ; what part of the expense does B pay more than A? 26 5. Subtract from of 1. Ans. 13 Rem. . 23 Explanation.- of, which must be used instead of the compound fraction. 6. What is the difference between and ? Ans. 23 7. If one man own of of a ship, and another, of, what is the difference of their shares? 8. Subtract from 3. Ans. Rem. 9. If a man pay $29 towards a debt of $465, what will remain unpaid? Ans. $175. 10. How much more brandy in that cask which contains 67, gal. than in that which contains 513 gal.? Ans. 16. 11. What number is equal to the difference between 58,3 and 60? Explanation. The easiest method of working questions like this, is to reduce the fractions to a common denominator by themselves, without bringing the mixed numbers to improper fractions. Care should be taken that the same fraction which is taken from one whole number after being reduced, should be placed to the right hand of the same whole number. The numbers after the fractions are reduced to a common denominator, will be 58 and 6017. Ans. 21 12. If 45 gal. of gin be drawn from a cask containing 677 gal., how many gallons will remain in the cask? Ans. 63 Explanation. In operation 1, we suppose 4 the denominator added to 1 the numerator of the upper fraction, and subtract the numerator of the under fraction from 5 their sum. By adding 4 to the upper numerator, we increase the upper number by 1, for the denominator equals 1. To balance this 1 we add 1 to 7, the whole number in the subtrahend: In operation 2, 1 is taken from 9 and supposed divided into 4 equal parts, and these 4 parts added to the upper numerator, which makes the upper fraction the larg est. The subtracting can now be performed as in the preceding examples, and the remainder is found to be the same as in the first operation. The reason for not adding 1 to the 7 in operation 2, is because we have not increased the upper number. 14. A man owing 100 dollars, paid all but 15 dollars, how much did he pay? • Ans. 8433. 15. What is the difference between 1 and ? Ans.. 16. A farmer sells 113 bushels of grain from a chest containing 17 bushels; how much remains? 1 7 1 123 Explanation.-There being no fraction in the larger number, we subtract the numerator from the denominator, which is the same as if we had set 5 fifths in the upper number and taken 3 fifths from them: but adding 5 fifths to 17 would make 18 in the minuend, therefore 1 must be added to 11 the whole number in the subtrahend. 17. A man sells 19 acres from a farm containing Ans. 80 100; how many acres remain ? 18. What number must be added to 88, so that the sum shall be 96? Ans. 7. 19. What fraction of a pound is equal to the difference between of a pound and of a shilling? Explanation:-First reduce the fraction of a shilling to the fraction of a pound, (See Explanation page 152,) then proceed as with other fractions? Ans. 1. 20. What part of an hour is the difference between of a day and of an hour? Ans. 3. DIVISION OF VULGAR FRACTIONS. RULE. Multiply the whole number by the denominator and divide the product by the numerator. 1. How many times can be subtracted from 3? 3 4 1) 1 2 Ans. 1 2 Illustration.1 fourth can be taken from 1 fourth once, and from 4 fourths 4 times, but 4 quarters equal an unit, or 1. If can be taken 4 times from 1 unit, it can be taken 3 times 4, or 12 times, from 3 units. Multiplying 3 by 4 reduces the whole number to fourths, for we place 4 in the product once for each unit in the whole number, as was explained in multiplication of simple numbers. The question is really no other than this, -How many times can 1 fourth be taken from 12 fourths. Had the divisor been instead of, the divisor would have been 3 times as large and the quotient could have been only of what it was before, viz. 4 instead of 12. Multiplying the whole number by the denominator of any simple fraction reduces the whole number into the same parts of an unit as those expressed by the fraction, and the numerator being composed of parts of an unit equally small, the quotient must express the number of times the parts contained in the numerator can be taken from the number of parts contained in the whole number. If the denominator be 8, multiplying the whole number by 8 the product will express each unit in the whole number divided into 8 equal parts. Should the divisor be an improper fraction, and the numerator contain more units than the dividend, the quotient must be less than the dividend; but if the divisor be less than 1, which is the case when it is a proper fraction, the quotient will be greater than the dividend. It may at first appear singular that the quotient should ever be greater than the dividend. This apparent mystery vanishes when the student reflects, that a part of one can be taken from 1 more than once. 2. How many times can be taken from 6? 6 x 32 9 the answer. 3. How many times can be subtracted from 12? Ans. 15. 4. What is the quotient of 20, divided by ? 13 Ans. 23 I 5. If a merchant sell cloth to the amount of 66 dollars at of a dollar per yard, how many yards does he sell? Ans. 110. 6. A flock of sheep was sold for 100£. the price of each sheep being is of a pound, how many sheep in the flock? Ans. 190. 7. If a yoke of oxen eat of a bushel of corn in one day, how long will 21 bushels last them? Ans. 73 days. Note 6.-If the fraction be a compound one, reduce it to a simple one. 8. At of of a dollar per bushel, how many bushels of corn can be bought for 125 dollars? Ans. 200. |