12. Paris is 79° 23′ 15′′ E. from Washington. How much is Paris time ahead of Washington time?

Ans. 5 hr. 17 min. 33 sec.

13. If Albany is 3° 18′ 15′′ E. from Washington, how much is Albany time ahead of Washington time?

Ans. 13 min. 13 sec.

14. If Boston is 5° 59′ 36′′ E. from Washington, what time is it at Washington, when it is noon at Boston?

Ans. 11 o'c. 36 min. 1.6 sec. A. M.

15. If Philadelphia is 1° 53′ 24.6" E. from Washington, what time is it at Washington, when it is noon at Philadelphia ? Ans. 11 o'c. 52 min. 26.36 sec. A. M. 16. If Santiago, Chili, is 6° 22' 30" E. from Washington, what time is it at Santiago, when it is noon at Washington? Ans. 25 min. 30 sec. P. M. 17. If New Orleans is 90° 2' 30" W. from Greenwich, what time is it at Greenwich, when it is noon at New Orleans? Ans. 6 hr. 10 sec. P. M. 18. If Copenhagen is 12° 34' 40" E. from Greenwich, what time is it at Copenhagen, when it is noon at Greenwich? Ans. 50 min. 18.67 sec. P. M.

19. If Rome is 89° 32′ 12′′ E. from Washington, what time is it at Washington, when it is 3 o'c. 58 min. 8.8 sec. P. M. at Rome ? Ans. 10 o'c. A. M.

NOTE.-In subtracting the difference of time from the time at Rome, 3 o'c. may be called the 15th (12 o'c. + 3 hr.) hour of the day.

20. An observer in longitude 5° 44′ 45′′ E. of Washington, noted the beginning of an eclipse of the moon at 36 min. 5 sec. after 12 A. M. on the 10th of March, 1876. What time did the same event occur in Washington?

Ans. At 13 min. 6 sec. A. M., Mar. 10, 1876.

CHAPTER V.

RATIO AND PROPORTION

259. Ratio, in Arithmetic, is the relation which one number bears to another of the same kind.

This relation is found by dividing one of the numbers by the other. The quotient is also the value of the ratio.

The divisor may be regarded as the measure of the number divided. EXAMPLE 1.-What is the relation of 5 to 25?

Ans. 5 is of 25.

EXAMPLE 2.-What is the relation of 18 to 6?

Ans. 18 is 3 times 6.

1

In Ex. 1, we notice that 25 is the measure of 5, andof the ratio.

In Ex. 2, 6 is the measure of 18, and 3 the value of the ratio.

260. Any number and its measure, or any two numbers that are compared, are the Terms of a ratio. They are also a couplet.

The Terms are named from the order of their position, Antecedent (going before), and Consequent (following).

261. The Antecedent is the first term, or the number to be divided, or measured.

262. The Consequent is the second term, or the divisor, or measure.

Ratio is indicated in two ways, viz.:

263. FIRST.-By a fraction, whose numerator is the antecedent, and whose denominator is the consequent.

264. SECOND.-By writing the consequent after the antecedent, with a colon between them.

Thus, the ratio of 4 to 5 is written either 4, or 4: 5, and is read, "four is to five." We may change the former expression into the latter, or the reverse.

The terms of a ratio must not only express the same kind of quantity, but also the same denomination. Thus, the ratio of 3 quarts to 2 pecks is not; but if we reduce the 2 pk. to qt., the ration is.

Oral Exercises

EXAMPLE. What is the relation of 4 to 5 ?

SOLUTION.-Since we find the relation of one number to another by dividing the former by the latter (259), the relation of 4 to 5 is §; or, 45; or, 4: 5.

PROBLEMS.

1. What is the relation of 3 to 6? 5 to 8?

7 to 9 ?

Since the ratio of two numbers is expressed by a fraction

whose numerator is the antecedent, and whose denominator is the consequent, it follows that

265. FIRST.-Dividing the antecedent, or multiplying the consequent, divides the value of the ratio (162).

266. SECOND.—Multiplying the antecedent or dividing the consequent, multiplies the ratio (161).

267. THIRD.—Multiplying or dividing both terms of the ratio by the same number does not affect the value of the ratio (163).

In reference to the number of its terms, a ratio is either Simple or Compound.

268. A Simple Ratio is one that has only two terms; as 45; 6: 8; ; ; &c.

269. A Compound Ratio is one which has two or more pairs of terms. Thus x; ××4; (2:4)× (78); (35) × (7: 10) × (5: 13) are compound ratios.

Written Exercises

EXAMPLE.-Reduce 3: 5 and 10: 13 to a simple ratio.

3: 5 =

10: 13 =

SOLUTION.
2

+ } # × 1 = 15, or 6 : 13.

EXPLANATION.-For convenience, we express the ratios in their fractional form

and proceed as in multiplication of fractions, obtaining the simple ratio, or 6 13. Hence the

RULE. Write the given ratios in their fractional form, and multiply them together as in multiplication of fractions; the product will be the simple ratio required.