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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 multiplying 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.

x =

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

the number John has,
then 3 x = 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

х

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 ?

5. 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.

с

a

b

1. Draw a triangle whose base, ab, is 3 inches long. Make the angle a, 55° and the angle 6, 35o. 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.

2. Draw a triangle two of whose sides are equal. Measure 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

5

3

7

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8

8

150

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

1
7

5 7 9 tions to hundredths:

5

10' 25° 50' 12' 33' 163 60 30 45 60 60 60 60

.+ 500 200 900 250' 125' 663

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 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 of 150, take of 60.

+ 100 is 14 times 66].

PERCENTAGE.

210. Per cent means hundredth or hundredths. Per cent may be expressed as a common fraction whose denominator is 100, or it may be expressed decimally ; thus, 6 per cent =

} tro or .06; 284 per cent or .284 ; } per cent

100 or .001.

28}

100

Note.-Instead of the words per cent sometimes the sign (%) is used; thus 6 per cent may be written 6 %.

211. The base in percentage is the number of which hundredths are taken ; thus, in the problem, Find 11% of 600, the base is 600 ; in the problem, 16 is what per cent of 800 ? the base is 800; in the problem, 18 is 3% of what? the base is not given, but is to be found by the student.

Observe that whenever the base is given in problems like the above, it follows the word of.

212. There are three cases in percentage and only three.

Case I. To find some per cent (hundredths) of a number, as: find 15 % of 600.

Case II. To find a number when some per cent of it is given, as : 24 is 8 % of what number?

Case III. To find what per cent one number is of another, as : 12 is what % of 400 ?

Observe that a thorough knowledge of fractions is the necessary preparation for percentage. The work in percentage is work in fractions, the denominator employed being 100.

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