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Ex. 154.

Ex. Find the cost of a draft on New York for $1000, at

of 1% premium.

1% of $1000-$2.50 (premium).

$1000 + $2.50 = $1002.50 (cost).

$1002.50. Ans.

1. Find the cost of a draft on New York for $1200, at 1 of 1% discount.

2. Find the cost of a draft on St. Louis for $2000, at 4 of 1% premium.

3. Find the cost of a draft on New Orleans for $2400, at 1% premium.

4. Find the cost of a draft on Chicago for $3200, at %

discount.

Ex. Find the cost of a draft on Cincinnati for $1000, payable in 30 dys. after sight, exchange being 1% premium, and interest 6%.

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5. Find the cost of a draft for $800, payable 30 dys. after sight, when exchange is 1% premium, and interest 6%.

6. Find the cost of a draft for $1900, payable in 30 dys., when exchange is at par, and interest 44%.

7. Find the cost of a draft for $1450, payable in 60 dys., when exchange is 1% discount, and interest 5%.

8. Find the cost of a draft for $1000, payable 60 dys. after sight, when exchange is 1% discount, and interest 7%.

CHAPTER XII.

PROPORTION.

245. The relative magnitude of two numbers is called their ratio, and is expressed by the fraction which the first. number is of the second.

Thus the ratio of 2 to 3, commonly written 2:3, is expressed by the fraction.

246. The first term of a ratio is called the antecedent, and the second term the consequent.

247. If both terms of a ratio be multiplied or divided by the same number, the ratio is not altered.

Thus, if both terms of the ratio 21:31 be multiplied by 6, the

resulting ratio is 15: 20, and the two ratios are equal, for Since 18, the simplest expression for 21: 33 is 3: 4.

=

21 15 31

20

248. If the numerator and denominator are interchanged, the fraction is said to be inverted; likewise, if the antecedent and consequent of a ratio are interchanged, the resulting ratio is called the inverse of the given ratio.

Thus, if the fraction is inverted, the resulting fraction is §, and the inverse of the ratio 4:5 is 5: 4.

249. If two quantities are expressed in the same unit, their ratio will be the same as the ratio of the two numbers by which they are expressed.

Thus the quantity $7 is the same fraction of $9 as 7 is of 9.

250. Since ratio is simply relative magnitude, two quantities different in kind cannot form the terms of a ratio; and two quantities of the same kind must be expressed in a common unit before they can form the terms of a ratio.

Thus no ratio exists between $5 and 20 dys.; and the ratio of 3 t. to 5000 lbs. can be expressed only when both quantities are written as tons or pounds.

251. When two ratios are equal, the four terms are said to be in proportion, and are called proportionals.

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Thus 6, 3, 18, 9 are proportionals; for § 18.

252. A proportion is written by putting the sign double colon between the ratios.

or a

Thus 6:3 = 18:9, or 6 : 3 :: 18 : 9, means, and is read, the ratio of 6 to 3 is equal to the ratio of 18 to 9.

253. The first and last terms of a proportion are called the extremes, and the two middle terms are called the means.

254. Test of a proportion. When four numbers are proportionals, the product of the extremes is equal to the product of the means.

This is seen to be true by expressing the ratios in the form of fractions, and multiplying both by the product of the denominators.

Thus the proportion 5: 3 :: 15: 9 may be written; and, if both be multiplied by 3 × 9, the result will be 5 × 9 = 3 × 15.

255. Either extreme, therefore, will be equal to the product of the means divided by the other extreme; and either mean will be equal to the product of the extremes divided by the other mean. Hence, if three terms of a proportion be given, the fourth may be found.

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(2) 20 is to 24 as what number is to 30?

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(3) 18 is to 32 as 45 is to what number?

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As these fractions are equal, their reciprocals are equal;

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256. When three terms of a proportion are given, the method of finding the fourth term is called the Rule of Three.

It is usual to arrange the quantities (that is, to state the question) so that the quantity required for the answer may be the fourth term. Hence the quantity which corresponds to that of the required answer must be the third term. (1) If 5 t. of hay cost $87.50, what will 21 t. cost?

2

Since the cost of 21 t. is required, $87.50 is the third term. Since 21 t. will cost more than 5 t., 21 t. is the second term and 5 t. the first term.

That is, 5 t. 21 t.:: $87.50: What quantity?

A difficulty presents itself here, inasmuch as no meaning can be given to the product of the means ($87.50 multiplied by 21 t.). Since, however, the ratio of 5 t.: 21 t. the ratio of 5: 21, the ratio 5: 21 may be substituted for 5 t.: 21 t.

Then 5:21 $87.50: What quantity?

::

=

That is, What quantity

21 x $87.50,

5

$367.50. Ans.

(2) When a post 11.5 ft. high casts a shadow on level ground 17.4 ft. long, a neighboring steeple casts a shadow 63.7 yds. long. How high is the steeple? Height is required; the height 11.5 ft. is therefore the third

term.

Since the shadow of the steeple is the longer, the height of the steeple must be the greater; therefore the second term must be the greater of the two remaining quantities expressed in the same unit. 63.7 yds. 191.1 ft.

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257. In solving problems by the Rule of Three,

Make that quantity which is of the same kind as the required answer the third term.

The numbers by which the other two quantities are expressed, when expressed in a common unit, will be the first and second terms.

If, from the nature of the question, the answer will be greater than the third term, make the greater of these two numbers the second term; if less, make the smaller of these numbers the second term, and the other the first term.

Divide the product of the second and third terms by the first term, and the quotient will be the answer required.

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