28. Three hundred seventy-five thousandths of a lot of dry goods, valued at $1000, was destroyed by fire; how much would a firm lose who owned .12 of the entire lot!

Ans. $180.

30. If 7.5 tons of hay are worth 375 bushels of potatoes, and 1 bushel of potatoes is worth $.331, how much is 1 ton of hay worth? Ans. $16.663.

31. A person invested a certain sum of money in trade; at the end of 5 years he had gained a sum equal to 84 hundredths of it, and in 5 years more he had doubled this entire amount. How many times the sum first invested had he at the end of the 10 years? Ans. 3.68 times.

10

32. A miller paid $54 for grain, 3% of it being barley at $.624 per bushel, and of it wheat at $1.87 per bushel; the balance of the money, he expended for oats at $.37 per bushel. many bushels of grain did he purchase?

How

Ans. 40.

33. A merchant tailor bought 27 pieces of broadcloth, each piece containing 19 yards, at $4.31 a yard; and sold it so as to gain $381.87, after deducting $9.62 for freight. How much was the cloth sold for per yard? Ans. $5.06.

34. Bought 1356 bushels of wheat @ $1.18%, and 736 bushels of oats @ $.41; I had 870 bushels of the wheat floured, and disposed of it at a profit of $235.871, and I sold 528 bushels of the oats at a loss of $13.621. I afterward sold the remainder of the wheat at $1.12 per bushel, and the remainder of the oats at $.31 per bushel; did I gain or lose, and how much?

Ans. I gained $171.07).

35. The sum of two fractions is 125, and their difference is 116; what are the fractions?

704

36. A manufacturer carried on business for 3 years. The first year he gained a sum equal to of his original capital; the second year he lost of what he had at the end of the first year; the third year he gained of what he had at the end of the second How much had he gained in Ans. $10594.70

year, and he then had $28585.70. the 3 years?

CONTINUED FRACTIONS.

268. If we take any fraction in its lowest terms, as 12, and divide both terms by the numerator, we shall obtain a complex fraction, thus:

Reducing, the fractional part of the denominator, in the same manner, we have,

13 1

Expressions in this form are called continued fractions. Hence, 269. A Continued Fraction is a fraction whose numerator is 1, and whose denominator is a whole number plus a fraction whose numerator is also 1, and whose denominator is a similar fraction, and so on.

270. The Terms of a continued fraction are the several simple fractions which form the parts of the continued fraction. Thus, the terms of the continued fraction given above are,,, and.

CASE 1.

271. To reduce any fraction to a continued fraction. 1. Reduce to a continued fraction.

OPERATION.

109 1

=

339

3+1

9 +1

ANALYSIS. We divide the denominator, 339, by the numerator, 109, and obtain 3 for the denominator of the first term of the continued fraction. Then in the same manner we divide the last divisor, 109, by the remainder, 12, and obtain 9 for the denominator of the second term of the continued fraction. In like manner we obtain 12 for the denominator of the final term. Hence the following

12

RULE. I. Divide the greater term by the less, and the last divisor by the last remainder, and so on, till there is no remainder

II. Write 1 for the numerator of each term of the continued fraction, and the quotients in their order for the respective denom

inators.

EXAMPLES FOR PRACTICE.

1. Reduce 288, to a continued fraction. 1793

1

Ans. 6+1

272. To find the several approximate values of a continued fraction.

An Approximate Value of a continued fraction is the simple fraction obtained by reducing one, two, three, or more terms of the continued fraction.

38

273. 1. Reduce 163 to a continued fraction, and find its approximate values.

ANALYSIS. We take 4, the first term of the continued fraction, for the first approximate value. Reducing the complex fraction formed by the first two terms of the continued fraction, we have for the second approximate value. In like manner, reducing the first three terms, we have for the third approximate value. By exam ining this last process, we perceive that the third approximate value, 7 , is obtained by multiplying the terms of the preceding approximation,, by the denominator of the third term of the continued frac tion, 2, and adding the corresponding terms of the first approximate value. Taking advantage of this principle, we multiply the terms of

by the 4th denominator, 5, in the continued fraction, and adding the corresponding terms of, obtain, the 4th approximate value, which is the same as the original fraction. Hence the following

RULE. I. For the first approximate value, take the first term of the continued fraction.

II. For the second approximate value, reduce the complex fraction formed by the first two terms of the continued fraction.

III. For each succeeding approximate value, multiply both numerator and denominator of the last preceding approximation by the next denominator in the continued fraction, and add to the corresponding products respectively the numerator and denominator of the preceding approximation.

NOTES.1. When the given fraction is improper, invert it, and reduce this result to a continued fraction; then invert the approximate values obtained therefrom.

2. In a series of approximate values, the 1st, 3d, 5th, etc., are greater than the given fraction; and the 2d, 4th, 6th, etc., are less than the given fraction.

EXAMPLES FOR PRACTICE.

1. Find the approximate values of 67.

155*

Ans. 1, 17, 164' 349

3. What are the first three approximate values of 2831?

20357

4. What are the first five approximate values of 23?

5. Reduce to the form of a continued fraction, and find the

value of each approximating fraction.

COMPOUND NUMBERS.

274. A Compound Number is a concrete number expressed in two or more denominations, (10).

275. A Denominate Fraction is a concrete fraction whose integral unit is one of a denomination of some compound number Thus, of a day is a denominate fraction, the integral unit being one day; so are of a bushel, of a mile, etc., denominate iractions.

276. In simple numbers and decimals the scale is uniform, and the law of increase and decrease is by 10. But in compound numbers the scale of increase and decrease from one denomination to another is varying, as will be seen in the Tables.

MEASURES.

277. Measure is that by which extent, dimension, capacity or amount is ascertained, determined according to some fixed standard.

NOTE. The process by which the extent, dimension, capacity, or amount is ascertained, is called Measuring; and consists in comparing the thing to be measured with some conventional standard.

Measures are of seven kinds:

1. Length.

2. Surface or Area.

3. Solidity or Capacity.

4. Weight, or Force of Gravity.

5. Time.

6. Angles.

7. Money or Value.

The first three kinds may be properly divided into two classesMeasures of Extension, and Measures of Capacity.

MEASURES OF EXTENSION.

278. Extension has three dimensions-length, breadth, and thickness.

A Line has only one dimension — length.

-

A Surface or Area has two dimensions-length and breadth.