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254. Examples for the Slate. (176.) 16} is of what number? (177.) 251 is of what number? (178.) of zi is of what number? (179.) g of 63 is of what number? (180.) of f is of what number?

(181.) 24 x 7} is 3} times what number (or } of what number)?

(182.) 182 = (12 x 2}) is 37 times what number?

183. An author's copyright on a book was $ 54.57. If this was zo of the whole profit, what was the whole profit ?

184. Mr. Smith owns îi of an acre of land; his neighbor Mr. French owns s as much, which is ☆ of what Mr. Brown owns. What does Mr. Brown own ?

185. If of my property is in real estate, 4 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 2} 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 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 ß 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. f is what part of Z? is what part of 33 ? g. } is what part of t. Note. Change 1 and } to sixths. h. What part of 10 is 3}? is 21 ?

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 $300 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 27 ? (198.) Of 267 is 89 ? (203.) 1 is what part of 175 ? (199.) Of 8 is ? (204.) 1} is what part of i? (200.) Of 11 is 33 ? (205.) 24 is what part of 33?

206. What part of 100 is 33} ? 663 ? 87} ? 37}? 12}? 62} ? 61 ? 561 ?

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, 31 is an aliquot part of 10.

259. Oral Exercises.

Find such aliquot parts of the following numbers as are indicated below:

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a. Of the number 30 find }; }; £; ; di Yo ; *; . b. Of the number 60 find į; }; $; }; b; lo; Ib; 15; od c. Of the number 100 find }; }; $; }; }; }; lo; is. d. Of the number 100 find 3; 4; 3; 4; &; ; ; ; 1's. e. Of the number 144 find }; }; ; &; }; }; 12; 3; 4. f. Of the number 200 find }; }; 4 ; }; &; }; lo; Iz; }. g. Of the number 1000 find }; }; ; };d; ; to; 3; .

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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 per foot ?

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.

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. ?

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 $6.663 per year? f. Of 750 feet of boards at $12 per thousand ? g. Of 80 pounds of butter at 37} per pound ?

h. How many pounds of cheese at 163¢ 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 163¢ ? at 25¢ ? at 50¢ ? at 37}\ ?

261. Examples for the Slate. (207.) 987 - 163 = ? (210.) 496 = 12!= ? (208.) 864 : 33}= ? (211.) 684 x 663= ? (209.) 572 x 62}= ? (212.) 487 x 37}= ?

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 inethod ? 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 7, te, and 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

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