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Geometry.

208. CONSTRUCTION PROBLEMS TRIANGLES.

C

a

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

be

The angle c should

degrees. Measure the three sides.

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?

7. Is it possible to draw a quadrilateral whose sides are equal but whose angles are unequal?

209. Miscellaneous Review.

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

tions to hundredths:

3 1 5 7 3 7 5 7 9 5 8 8' 10' 25' 50 12 33 16

60 30 45 60 60 60 60 500 200 900 150 250 125 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?

How many

5. Regarding a month as 30 days and a year as 360 days, what part of a year is 7 months and 10 days? 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?

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 =

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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.

Percentage.

213. CASE I.

Find 17 per cent (.17) of 8460.

NOTE. We may find § of a number by finding 3 times 1 fourth of it; that is, by multiplying it by . So we may find .17 of a number by finding 17 times 1 hundredth of the number; that is, by multiplying by .17.

Operation. 84'60

.17

592.20

846.0

1438.20

Explanation.

One per cent (1 hundredth) of 8460 is 84.60; 17 per cent (hundredths) of 8460 is 17 times 84.60, or 1438.20.

PROBLEMS.

1. Find 17% of 6420; of 5252; of 31.40. 2. Find 35% of 6420; of 5252; of 31.40. 3. Find 43% of 6420; of 5252; of 31.40. 4. Find 25% of 6420; of 5252; of 31.40. 5. Find 50% of 6420; of 5252; of 31.40. 6. Find 30% of 6420; of 5252; of 31.40. 7. Find 35% of 6420; of 5252; of 31.40. 8. Find 65% of 6420; of 5252; of 31.40. (a) Find the sum of the

twenty-four results.

9. A sold goods for B. As remuneration for his services. he received a sum equal to 12% of the sales. He sold $2146 worth of goods. How much did he receive?

10. C is a collector of money. For this service he charges a commission of 6%; that is, his pay is 6% of the amount collected. He collected for D $375. How much should he pay over to D, and how much should he retain as pay for collecting?

Percentage.

214. CASE II.

673.20 is 17 per cent (.17) of what number?

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Observe that dividing by .17 is finding 100 of the dividend, just as dividing by & is finding of the dividend, and dividing by is finding of the dividend.

NOTE. Sometimes the process may be shortened by writing the per cent as a common fraction and reducing it to its lowest terms; then using the reduced fraction instead of the one whose denominator is 100.

PROBLEMS.

1. 360 is 15% of what number?
2. 360 is 25% of what number?

3. 360 is 50% of what number?
4. 360 is 75% of what number?
5. 360 is 40% of what number?
(a) Find the sum of the five results.

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