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3. What is the interest of 24 £. 7 s. 6 d. for 2 yrs. 4 mo.?

4. Oct. 12, 1861, gave my note on demand, with interest, for $480. Feb. 6, 1862, paid $120. What remained due Aug. 24, 1862 ?

5. I held a note for $500, which bore interest from May 10, 1859. Sept. 16, 1860, received $140; July 28, 1862, received $50. What remained due Sept. 4, 1862?

6. If I pay $45 interest for the use of $500 for 3 years, what is the rate per cent?

7. How long must $204 be on interest at 7 per cent. to amount to $217.09 ?

8. What principal will gain $9.20 in 4 mo. 18 ds., at 4 cent.?

per

9. What sum, at 7 per cent., will amount to $221.075 in 3 yrs. 4 m.?

10. What is the compound interest of $600 for 1 yr. 4 mo., interest payable semi-annually?

11. What is the present worth of a note for $488.50, due in 2 yrs. 5 mo. 15 ds., at 9 per cent.?

12. What is the discount of $105.71, due 4 yrs. hence?

13. What commission must be paid for collecting $17380, at 3 per cent.?

14. What amount of stock can be bought for $9682, and allow per cent. brokerage?

15. What is the value of 20 shares bank stock, at 8 per cent. discount, the par value of each share being $150 ?

16. What sum will be received from a bank for a note of $260, payable in 4 months?

17. What is the bank discount on $320 for 90 days?

18. What is the face of a note which yields $112.803, when discounted at a bank for 60 days?

19. A house, valued at $4750, is insured at 2 of 1 per cent.; what is the premium?

20. What is the duty, at 15 per cent. ad valorem, on 20 bags of coffee, each containing 115 lbs., valued at 42 cts. per lb.?

RATIO.

357. Ratio is the relation which one number bears to a other number of the same kind.

Ratios are of two kinds, Arithmetical and Geometrical. 358. Arithmetical Ratio is ratio of numbers with respect to their difference; as 6 4 = 2.

GEOMETRICAL RATIO.

359. Geometrical Ratio is ratio of numbers with respect to their quotient; as 2:4: = , read 2 is to 4, or the ratio of 2 to 4 = = 1; 6: 3 = 2, read 6 is to 3, or the ratio of 6 to 3 = 2.

360. The first term of a ratio is called the Antecedent, the second, the Consequent; both together are called a Coup let.

What is the antecedent in the first illustration in Article 359? the consequent in the second? the ratio in the first? the consequent in the first? the ratio in the second ?

361. When the terms of a ratio are equal, the ratio is one of equality; when the antecedent is greater than the consequent, it is a ratio of greater inequality; when the antecedent is less than the consequent, it is a ratio of less inequality.

362. It will be readily seen that ratios, being expressions for division, are similar to fractions. They can at any time be written in a fractional form, the antecedent taking the place of the numerator, and the consequent that of the denominator. The principles applicable to fractions apply also to ratio. Hence, Multiplying the antecedent,

or dividing the consequent, S multiplies the ratio.

Dividing the antecedent,

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or multiplying the consequent, Multiplying or dividing both terms of a ratio by the same number,

divides the ratio.

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363. Ratios, like fractions, may be simple, complex, or compound. A ratio is simple when each term is a simple number; it is complex when either term contains a fraction; it is compound when it is the indicated product of two or more ratios.

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1. Write the ratio of 2 to 3; of 7 to 10; of to §; of 2 × 7

to 5 X 4.

2. Multiply the ratio 3: 4 by 2.

3. Divide the same by 2.

4. Reduce the ratio 6: 8 to lower terms.

5. Write any ratio of equality; of greater inequality; of less inequality.

365. ILL. Ex. Reduce :

5

to a simple ratio.

21

OPERATION.

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5. Multiplying each term of the ratio: 15 by 3 X 7, we

2 × 3 × 7 15 X 3 X 7

3

%

=14:45, Ans. Hence,

To reduce a complex ratio to a simple one: Reduce each term to its simplest form, then multiply each by the least common multiple of the denominators, and cancel.

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PROPORTION.

366, Proportion is an expression of equality between two ratios; thus, 2:34:6, read 2 is to three as 4 is to 6; that is, 2 is the same part of 3 that 4 is of 6. 2 is of 3, and 4 is 3

of 6.

367. The first and fourth terms of a proportion are called the extremes, and the second and third are called the means. The first ratio is called the first couplet, and the second ratio the second couplet. Read the following proportions:

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Name the extremes of the first proportion; the means of the second; the antecedents of the third; the consequents of the fourth; the sec ond couplet of the first proportion.

368. Inverse Proportion. Four terms are directly proportional when the first is to the second as the third is to the fourth. They are inversely proportional when the first is to the second as the fourth is to the third, or when one ratio is direct and the other inverse. Thus, the amount of work done in any given time is directly proportional to the men employed; i. e., the more men, the more work; but the time occupied in doing a certain work is inversely proportional to the men employed; i. e, the more men, the less time.

369. A compound proportion is an equality between a compound ratio and a simple ratio, or between two compound ratios.

370. Three terms are in proportion when the first is to the second as the second is to the third. The second term is called a mean proportional between the other two; thus, in the proportion, 3:6 6:12, 6 is a mean proportional between 3 and 12.

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371. The performance of arithmetical examples by pro portion depends upon the following important principle:~

In every proportion the product of the means equals the product of the extremes.

ILLUSTRATION.

2:34:6

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2X3X6

3

2X6

4X3X6

6

4X3

Writing the given proportion in a fractional form, we have. Multiplying each fraction by the product of the denominators, and cancelling, we have 2 X 6=4×3. But 2 and 6 are the extremes, and 4 and 3 the means; hence the product of the extremes equals that of the means.

372. From the above, it follows that whenever an extreme in a proportion is wanting, it can be found by dividing the product of the means by the given extreme; and whenever a mean is wanting, it may be found by dividing the product of the extremes by the given mean.

Supply the terms wanting in the following proportions:

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373. In the proportion 2: 44: 8, 42 = 2 × 8,.. 4= √2X8 (Arts. 91, 92); hence a mean proportional between twe numbers equals the square root of their product.

Supply the mean proportionals between the following numbers and write the proportions:

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374. ILL. Ex., I. If 15 boxes of oranges cost $60, what will 17 boxes cost?

OPERATION BY ANALYSIS.

4

$60 X 17

15

If 15 boxes cost $60, 1 box will cost of $60, and 17 boxes will cost 17 × 1 of

= $68, Ans. $60. Cancelling and multiplying, the re

OPERATION BY PROPORTION.

15:17

4

17 × $60

15

$60: Ans.

$68, Ans.

sult is $68.

$60, the price of 15 boxes, must bear the same relation to the price of 17 boxes that 15 bears to 17. We have then three terms of a proportion (15:17= $60), and can find the fourth by multiplying the second

and third together, and dividing the product by the first. Hence we derive the following

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