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Fractions.

205. DENOMINATE FRACTIONS. 1. One half inch is what part of a foot ? 2. Two and inches are what part of a foot ? 3. Five and 1 inches are what part of a foot ? 4. One half foot is what part of a rod ? 5. Three and one half feet are what part of a rod ? 6. Ten and one half feet are what part of a rod ? 7. Sixty-four rods are what part of a mile? 8. Ninety-six rods are what part of a mile? 9. One hundred eighty rods are what part of a mile? 10. One and one half quarts are what part of a peck ? 11. Two and one half quarts are what part of a gallon? 12. Twenty-four quarts are what part of a bushel ? 13. Fourteen ounces are what part of a pound? 14. Seven and one half ounces are what part of a pound? 15. One and one fourth ounces are what part of a pound? 16. Six hundred pounds are what part of a ton ? 17. Four hundred fifty pounds are what part of a ton ? 18. Six hundred twenty-five pounds are what part of a ton? 19. Seventy-five square rods are what part of an acre ? 20. Forty-five square rods are what part of an acre ? 21. One hundred square rods are what part of an acre? 22. Thirty-two cubic feet are what part of a cord ? 23. Fifty-six cubic feet are what part of a cord ? 24. One hundred cubic feet are what part of a cord ? 25. Seven and one half minutes are what part of an hour? 26. Forty minutes are what part of an 8-hour day? 27. Ninety minutes are what part of an 8-hour day?

Algebraic Fractions.

(Review pages 77 and 78).

206. FRACTIONS IN EQUATIONS.

EXAMPLE I.

+ 2 x = 30 2

Multiplying both members of the equation (See page 77, Art. 170, Statement 5) by 2, the denominator of the fraction in the equation, we have

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* Observe that the equation might have been cleared of fractions by multipli. ing both its members by 6, the 1. c. m. of 2 and 3.

Algebra.

207. PROBLEMS LEADING TO EQUATIONS CONTAINING

ONE UNKNOWN QUANTITY WITHOUT FRACTIONS.

3 x =

EXAMPLE John and Henry together have 60 oranges, and Henry has three times as many as John. How many has each?

Let x = the number John has,
then

the number Henry has,
and x + 3 x = the number they together have.

But they together have 60.
Therefore x + 3 x = 60.
Uniting

60.
Dividing

15. Multiplying 3 x 45. Therefore John has 15 oranges and Henry has 45 oranges.

4 x =

x =

PROBLEMS. 1. The sum of two numbers is 275, and the greater is four times the less. What are the numbers ?

2, Robert has a certain sum of money and Harry has five times as much; together they have $216. How many dollars has each?

3. One number is four times another, and their difference is 270. What are the numbers?

4. Peter has a certain number of marbles and William has 8 more than Peter; together they have 96 marbles. How many has each ?

Sarah has a certain number of pennies and her sister has nine more than twice as many ; together they have 93. How many has each ?

6. Two times Reuben's money plus three times his money equals 175 dollars. How many dollars has he?

Geometry.

208. CONSTRUCTION PROBLEMS-TRIANGLES.

a

b

1. Draw a triangle whose base, ab, is 3 inches long. Make the angle a, 55° and the angle 6, 359. The angle c should be

degrees. Measure the three sides.

Observe that the longest side is opposite the greatest angle and the shortest side opposite the smallest angle.

Measure

2. Draw a triangle two of whose sides are equal. and compare the angles opposite the equal sides.

Observe that a triangle, two of whose sides are equal, has two angles equal; and conversely if two angles of a triangle are equal, two of the sides are equal.

3. If two triangles have the three sides of one equal to the three sides of the other, each to each, do you think the two triangles are alike in every respect ?

4. If two triangles have the three angles of one equal to the three angles of the other, each to each, do you think the two triangles are necessarily alike in every respect?

5. Draw two triangles, the angles of one being equal to the angles of the other, and the sides of one not being equal to the sides of the other.

6. Is it possible to draw a triangle whose sides are equal, but whose angles are unequal?

3

1

5

5

9

8

209. MISCELLANEOUS REVIEW. 1. Without a pencil, change each of the following frac

7 3 7

7 tions to hundredths:

5

8 10' 25' 50' 12' 33; 163 60 30 45 60 60 60 60

..t 500 200' 900 150 250

125 66 2. Butter that cost 25¢ a pound was sold for 29¢ a pound. The gain was equal to what part of the cost ? The gain was equal to how many hundredths of the cost ?

3. The taxes on an acre of land which was valued at $600, were $12. The taxes were equal to what part of the valuation ? The taxes were equal to how many hundredths of the valuation ?

4. Mr. Jones purchased 500 barrels of apples. He lost by decay a quantity equal to 75 barrels. What part of his apples did he lose? How many hundredths of his apples did he lose ?

5. Regarding a month as 30 days and a year as 360 days, what part of a year is 7 months and 10 days ? How many hundredths of a year in 7 months and 10 days? How many thousandths of a year ? How many ten-thousandths of a year?

'6. One cord 48 cubic feet is what part of 4 cords 16 cubic feet? Change the fraction to hundredths; to thousandths ; to ten-thousandths.

7. One mile 240 rods is what part of 3 miles 160 rods? Change the fraction to hundredths; to thousandths; to tenthousandths.

8. From a bill of $175 there was a discount of $14. The discount is equal to how many hundredths of the amount of the bill ?

* Since 100 is 3 of 150, take of 60.

+ 100 is 14 times 663.

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