222. 1. What is the difference between 4 and 4 ? 2. What is the difference between and ? How is one fraction subtracted from another, each having the same denominator? 3. A gentleman who owned a sail-boat sold of it. What part did he still own? 4. A boy having $1, gave $1 for a neck-tie. What had he left? ANALYSIS. He had left the difference between $ and $. is equal to, and equals; less are. 5. A man owning of an acre of ground, sold of an What part remained? acre. How is one fraction subtracted from another having a different denominator? 6. Subtract from; from; from 7. Find the difference between and ; and 3. 8. From a piece of cloth containing 12 yards, 5 yards were cut. How many yards remained? ANALYSIS.-The difference between 12 yards and 5 yards. from leaves, and 5 from 12 leaves 7. Hence 73 yards remained. 9. If a ton of coal costs $74, and a cord of wood $41, what is the difference in their cost? How is one mixed number subtracted from another? 10. What is the value of 31-21? 81-21? 611-j? 223. PRINCIPLE.-Fractions can be subtracted only when they have a common denominator, and when they express parts of like units. Hence ANALYSIS.-Reduce the given fractions to equivalent fractions having the -=H. ANALYSIS.-Reduce and to equivalent fractions having the least common denominator. As cannot be taken from, take 1 or from 134, leaving 133, and add it to, making. Then from leaves, and 76 from 133, leaves 57. Hence 5717 is the result. 6. From 36 take 101. take 1. take 7. From 112111⁄2 take 56. 8. From 204 take 39. RULE I. Reduce the given fractions to equivalent fractions having the least common denominator, and write the difference of the numerators over the common denominator. II. When there are mixed numbers, subtract the fractional and integral parts separately, and add the results. If the mixed numbers are small, they may be reduced to improper fractions and subtracted according to the usual method. 19 A farmer having 208 acres of land, sold 92 acres. How many acres had he left? 225. 1. How much less than 2, is 1+†? 2. How much greater than 2, is f+1+1†? to 3. What is the difference between 4 and 24? 54 and 741 4. Mr. Smith sold of his farm to one man, another, and to a third. What part had he left? 5. Paid $6 for a ton of coal, and $34 for a load of wood. What change must be returned for a ten-dollar bill? 6. What is the difference between 14+ and 5%? 7. What is the difference between + and }+{ ? 12. 12—(8—2§)+13=?\| 17. 125+9—1—21=? 226. 1. The sum of two numbers is 1241, and the less is 36. What is the greater? 2. What number added to 1474 will make 2168? 3. What number added to 3074+210 will make 700g? 4. What number must be added to the difference of 186 and 214 to make 104214? 5. What fraction added to the sum of, fa, and fs, will make ? 6. What must be added to 4, that the sum may be ? 7. Bought a quantity of barrel staves for $160, and of lumber for $1136. Sold the staves for $2054 and the lumber for $1240%. What was the whole gain? 8. A man bought a ton of hay for $15, a barrel of flour for $9, and a barrel of apples for $3. What change should be given to him for 3 ten-dollar bills? Complete the following equations: 9. +4-4+4= ? 10. 88+2-5=? 12. 414+56—247-4|1=? 13. 120-51+90jf—z=? 11. 48-(164-31)= ? 14. 342-(21+‡—9)=? 15. 176+1327-267-174 =? 16. $1000-$500 + $10755% + $91% = ? MULTIPLICATION. 227. When one factor is a fractional number. 1. What part of a mile is 3 times 2. What part of a dollar is 4 times 3. How many times 1 is 3 times ? of a mile? of a dollar? 4 times? 4. At $ a pound, what will 4 pounds of tea cost? ANALYSIS.-Four pounds will cost 4 times $5, or $30, equal to $21. 5. At $ a bushel, what is the cost of 6 bushels of oats? Of 7 bushels? Of 8 bushels? Of 9 bushels ? 6. If a horse eat of a bushel of grain in a day, how much will 4 horses eat? 6 horses? 8 horses? 10 horses? 7. What cost 12 baskets of pears, at $4 a basket? With each class of oral questions in Art. 227, the pupil may solve the corresponding written examples on pages 118 and 119. BRAR UNIVERSITY 8. What cost 9 pounds of butter at $ a pound? 9. Show that multiplying the numerator of by 4 multiplies the fraction by 4. 10. Show that dividing the denominator of by 4 multiplies the fraction by 4. How many ways to multiply a fraction by an integer? 11. Multiply by 5; † by 6; by 5; by 8. 12. At $4 a box, what will 5 boxes of raisins cost? ANALYSIS. They will cost 5 times $43. 5 times $ are $39, and 5 times $4 are $20. $20 + $3 - $234. Hence, etc. = 13. At 7 cents a pound, what will 9 pounds of rice cost? 14. At $9 a barrel, what is the cost of 6 barrels of flour? Of 8 barrels ? Of 9 barrels ? Of 10 barrels ? 15. What will 8 yards of cloth cost, at $5 16. Multiply 7 by 9; 94 by 6; 10g by 7; 17. What is of 12 yards? of 24 men? 18. Multiplying by 1, 1, 1, dividing by what integers? a yard ? 12 by 8. of $30? †, etc., is the same as When a fractional part of an integer, or of a fraction, is to be taken, the word of, and not times, should be used. 19. At $7 a ton, what will of a ton of coal cost? ANALYSIS.-It will cost of $7, or 3 times of $7. of $7 is $14, and 3 times $1 are $51. Hence, etc. 20. At $12 a gross, what will of a gross of butts cost? 21. What will of a ton of hay cost, at $15 a ton? 22. What is of 6 ? of 2? of 8? of 9? 2 23. At $5 a yard, what will of a yard of cloth cost? 24. If a man can build a wall in 28 days, in what time can he build of it? of it? § of it? 25. If an acre of land produce 45 bushels of corn, how much will of an acre produce? ? ? ? of? |