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3. The time at St. Louis is 1 hr. 4 min. 564. sec. earlier than that of New York, whose longitude is 74° 1' 6". What is the longitude of St. Louis ?

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4. The time at Louisville is 5 hr. 16 min. 38 sec. earlier than that of Dublin, whose longitude is 6° 20' 30" west. What is the longitude of Louisville ?

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85°

30'

00", long.of Louisville.

REM.— When it is 9 A.M. at New York it is 10 A.M. 15° east of New York and 15° west of New York the time is 8 A.M.

5. The difference in time of Washington and St. Petersburg is 7 hr. 9 min. 191 sec. What is the difference in longitude?

Ans. 107° 19' 483" = difference in longitude.

6. The longitude of Cincinnati is 84° 24' west. When it is 10 o'clock A.M. at Cincinnati, it is 22 min. 8 sec. past 10 at Buffalo. What is the longitude of Buffalo?

Ans. Longitude of Buffalo 78° 52'.

RATIO.

Two fractions can be formed of any two integral num. bers, the one a proper fraction, the other an improper fraction. These fractions are also called Ratios.

As the product of any number multiplied by the proper fraction is less than the number itself, the fraction is called a Diminishing Ratio; and as the product of any number multiplied by the improper fraction is greater than the number itself, it is called an Increasing Ratio.

PROBLEMS.

1. If 5 lbs. sugar cost 50 cents, what will 9 lbs. cost ?

It is evident that 9 lbs. will cost more than 5, and just as much more as is indicated by the increasing ratio formed by the two like terms, 5 lbs. and 9 lbs.

10

If 5 lbs. cost 50 cts., 9 lbs. will cost $0 cts. xf = 90 cts.; this may be further demonstrated thus,

5 lbs. = 50 cts.
1 lb. = 10 cts.
9 lbs. = 90 cts.

In a problem of ratios, the one ratio is given, and one of the terms of the other ratio, to get the second term; thus, in the above:

Given, 5 lbs. sugar and 50 cents.
Required, 9 lbs.

sugar

and ? The ratio of the money will be the same as of the sugar. As the required sugar is more than the given, the ratio must be increasing; that is, . .. 50 xf = 90, the ratio of the required money to the gain, 88 = , the same as

of the sugar.

EXAMPLES.

er! oo

1. If 5 bushels of wheat cost $6.25, what will 8 bushels cost?

Given 5 bu. and $6.25.
Required, 8 bu. and

$6.2$ x 8 = $10.00. .
8 25 ) 1000 5) 40

8
5
625

25 The ratios of the wheat and of the money is the same. REM.--Ratios can only be formed by two like terms.

2. If 5 bushels of oats cost $1.50, what will 21 bushels cost ? 2 왕

t; the ratio is the same.
Given 5 bu. and $1.50.
Required, 21 bu. and
$8.$0 x 41

= $6.30. REM.-Write the given terms in a line and the like term of the required immediately under that of the given. One term of the required is wanting, and the given like term may be called the term of demand, and should be placed first and multiplied by the ratio, having for its numerator the required term of the ratio, and for its denominator the given term.

As a general thing, an increase in the required term of the ratio will take more of the unknown to accomplish it; an increased amount of goods will cost a greater sum of money; an enlarged piece of work, an additional sum of money; and the greater the work, the longer time to perform it, etc. In examples of this kind, the ratios are direct, and the required term of the ratio holds the place of the numerator and the given term that of the denominator, and the product of the ratio and the odd given term is the term required.

3. If a man travel 40 miles in 8 hours, how many

miles will he travel at that rate in 18 hours ?

Given 40 miles and 8 hours.
Required, ? miles and 18 hours.

5

40 miles x 148

= 90 miles.

4. If 15 bushels of wheat yield 3 barrels of flour, how many bushels will yield 10 barrels of flour?

Given 15 bu. and 3 barrels of flour.
Required, ? and 10 barrels of flour.

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5. If a man travel 30 miles in 2 days, how long will it take him to travel 240 miles ?

Given 30 miles and 2 days.
Required, 240 miles and ?

6. If a staff 4 feet long cast a shadow 3 feet, what is the height of a steeple which casts a shadow 90 feet?

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%. If the interest of $100 for one year is $5, what would be the interest of $500 for the same time?

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Given

$100 and Required, $500 and ? 8. If of a barrel of flour cost $4, what will 4 barrels cost ?

Given barrel and $4.
Required 44 barrel and ?

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REM. 4 is a multiplier, and f is a divisor; the f must be inverted.

9. If 44 bushels of wheat cost $5.40, what will 8j bu. cost ? 91 bu.? 2312 313 ? 471? 391 ? 581? 974 ? 1064?

10. If 8 bushels of wheat cost $10, what will be the cost of 9j bu. ? 104 bu.? 15% bu. ? 371 bu.? 95% bu.? 125 bu. ? 1504 bu. ? 2794 bu.?

11. If 5 acres of land cost $230, what is the cost of 64 acres ? 7 acres ? 124 acres ? 133 acres ? 173? 183 ? 194 ? 205 ? 378? 493?

12. If 2} acres of land cost $110, what will of an acre cost? 1 acre? 4 acre? } acre ? acre? acre?

acre? 11 acres ? 13 acres?

13. If ia of a yard of cloth cost io of a dollar, what will # of a yard cost ? Given

yd. and $ %.
Required,
yd. and

?
1o XX

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