254. Examples for the Slate. Owns. (176.) 161 is of what number? (181.) 21 ~ 7% is 3} times what number (or 1 of what num. ber)? (182.) 182 = (12 x 2y) is 31 times what number? 183. An author's copyright on a book was $ 54.57. If this was go of the whole profit, what was the whole profit? 184. Mr. Smith owns jf of an acre of land; his neighbor Mr. French owns as much, which is of what Mr. Brown What does Mr. Brown own ? 185. If } of my property is in real estate, & in trade, and the balance, which is $ 33000, is in stocks: what is the value of my property ? 186. A man sold a lot of land for $ 1440, which was 21 times what it cost him. What did it cost him ? 187. Having lost I of my money in trade, I now have $2476.50. What had I at first? 188. A person against whom I had an account has failed, and I have lost i of what he owed me. If I receive $ 1584.72, how much did he owe me? 189. A body of 4800 troops had as many cavalry as infantry. What was the number of each ? 190. A lot of land yielded 4140 bushels of grain in two years, yielding $ as much the second year as the first. What was the yield each year? 191. What number is that to which if of itself be added the sum will equal 275 ? 192. In counting his fowls, a farmer finds that he has 396 in all, which is } more than he had the previous year. How many had he then ? TO FIND WHAT FRACTION ONE NUMBER IS OF ANOTHER. Oral Exercises. 255. ILLUSTRATIVE EXAMPLE I. 1 is what part of 5? Answer. 1 is } of 5 because it is one of the five equal parts into which 5 may be divided. a. 1 is what part of 7 ? of 9? of 10? Why? In comparing 1 with any number to see what fraction it is of that number, what do you take as the numerator? as the denominator ? b. 1 is what part of 7 ? 2 is what part of 7 ? Why? c. What part of 9 is 2? What part of 10 is 7 ? d. What part of 200 is 20 ? 50 ? 25? 40 ? e. 1 peach is what part of 7 peaches ? 3 pears of 13 pears? f. } is what part of 3? is what part of 38 ? g. } is what part of }. NOTE. Change · and to sixths. h. What part of 10 is 3} ? is 24 ? 256. From the foregoing illustrations we may derive the following Rule. To find what fraction one number is of another, Make the number which is the part the numerator of a fraction, and the number with which it is compared the denominator. i. If a piece of work can be performed in 9 days, what part of the work can be performed in 7 days? j. If Mr. Chase has $54 and spends $18 for a coat, what part of his money does he spend ? ? k. Stock originally worth $50 a share now sells for $40. What part of the original value does it bring ? 1. When goods which cost 75 cents sell for $1, what is the gain? What part of the cost is the gain? m. A and B hired a pasture together; A pastured 12 cows in it and B 13 cows. What part of the price should each pay ? 257. Examples for the Slate. 193. A man owing $316, paid $84 of the debt. What part of the debt did he pay ? 194. What part of 2724 square feet is 9 square feet? 195. I bought a house for $3000 and sold it for $4500. What part of the original cost was the gain ? 196. Four men were hired to work on a farm. A worked 7 days, B worked 5 days, C 8 days, and D 4 days. They received $72. What was each man's share ? (201.) 124 is what part of 19 ? (197.) Of 75 is 30 ? (202.) 1 is what part of 2% ? (198.) Of 267 is 89 ? (203.) 1 is what part of 135 ? (199.) Of 8 is ? (204.) 1} is what part of it? (200.) Of 11 is 33 ? (205.) 27 is what part of 33 ? 206. What part of 100 is 33}? 663 ? 874? 371? 124? 621? 64? 564 ? What part To solve Examples by using Aliquot Parts of Numbers. 258. What is one of the three equal parts of 9 ? of 10 ? One of the equal parts of a number is an aliquot part of the number. Thus, 3} is an aliquot part of 10. 259. Oral Exercises. Find such aliquot parts of the following numbers as are indicated below: a. Of the number 30 find }; }; }; }; di to; ; b. Of the number 60 find !; ; ; ; di Hoi bi si to c. Of the number 100 find }; }; 4; }; ; }; ibi Ibo d. Of the number 100 find } ; ; $; $; 6 ; $; $; $; 1%. e. Of the number 144 find }; }; }; }; }; $; Ibi ; . f. Of the number 200 find ); }; $; $; d; }; lo; Ibi . g. Of the number 1000 find ); }; }; }; &; bi to; $; . 260. By using the aliquot parts of numbers, the work of multiplying and dividing may often be shortened, thus : ILLUSTRATIVE EXAMPLE. What is the cost of 25350 ft. of gas at 3} mills Operation. — 3}m.=} of 10 mills, or of a cent. 25350 ft. at 1 cent a foot costs $ 253.50, and at } of a cent a foot it must cost } of $253.50, or $84.50. Ans. $84.50. per foot ? a. Oral Exercises. Find the cost Of 1872 lbs. of butter at $ 0.33} per lb. ? b. Of 64 bu. potatoes at $ 0.87} per bu.? c. Of 44 yds. of silk at $ 1.12} per yd.? d. Of fencing 50 rods of road, both sides, at $ 3.75 per rod ? e. Of insuring a house 5 years at $ 6.66$ per year ? f. Of 750 feet of boards at $ 12 per thousand ? Of 80 pounds of butter at 37{per pound ? h. How many pounds of cheese at 1634 a pound can be bought for $10 ? i. For $20 how many yards of cloth can be bought at $1 a yard? at 12}¢ ? at 16$¢ ? at 25% ? at 50% ? at 37}$ ? 261. Examples for the Slate. (207.) 987 - 163= ? (208.) 864 : 33}= ? (209.) 572 x 62}= ? (210.) 496 - 12}= ? 262. Questions for Review. What is a FACTOR of a number ? What is a composite number ? a prime number ? a prime factor ? What is an even number ? an odd number? What numbers are divisible by 2? 3? 4 ? 5? 6 ? 8? 9 ? 11 ? How can you find the PRIME FACTORS of a number? A composite number equals what product ? Find the prime factors of 180 and explain the process. How can you make sure that a number is prime ? What is CANCELLATION ? Why should arithmetical processes first be indicated by signs? Explain the use of the parenthesis. When are numbers prime to each other? What is a common factor of two or more numbers ? the GREATEST COMMON FACTOR? Find the g. c. f. of three numbers by the first method given ; explain and give the rule. Find the g. c. f. of two numbers by the second method given; give the rule. In what cases would you find the g. c. f. by the second method ? When do we make use of the g. c. f. of numbers ? What is a multiple ? a common multiple of two or more numbers ? the LEAST COMMON MULTIPLE? When do we make use of the l. c. m. ? Explain the first method of finding it; the second. What does the 1. c. m. of numbers prime to each other equal ? What is a FRACTIONAL UNIT ? a fractional number? What name is applied to both ? Name and define the terms of a fraction. Explain the expression How do you change fractions to smaller terms ? to larger terms ? When is a fraction expressed in its smallest terms? How do you change improper fractions to integers or mixed numbers? How do you change integers or mixed numbers to fractions ? When are fractions said to have a coMMON DENOMINATOR ? For what operations upon fractions do we first change them to others having a common denominator? Change 24 to, and I to fractions having a common denominator, and explain. How do you ADD FRACTIONS ? Take three fractions of different denominators, add and explain. How do you add mixed numbers ? How do you SUBTRACT FRACTIONS ? Give a general rule for the addition of fractions. Give a general rule for the subtraction of fractions. Let 44 be the minuend and 11%the subtrahend; subtract and explain. How do you MULTIPLY A FRACTION by an integer ? a mixed num. ber by an integer ? Explain, by an example, the method of multiplying an integer by a fraction. Multiply a fraction by a fraction; explain and give the rule. How do you multiply a mixed number by a mixed number or a fraction? How can you simplify the expressions called compound fractions ? How do you DIVIDE A FRACTION by an integer ? a mixed number by an integer ? an integer by a fraction ? Explain, by an example, the method of dividing a fraction by a fraction, and give the rule. How can you simplify the expressions called complex fractions ? How do you find what fraction one number is of another? What is an aliquot part of a number? What effect does multiplying both terms of a fraction by the same number have upon it? Why? What effect does dividing both terms |