8. Invoice of 75 Drums Caustic Soda, shipped per S. S. Germania for New York, by the undersigned on acc't and risk of Messrs. Smith & Bro.

What was the cost of this invoice, including duty, when delivered to Messrs. Smith & Bro.? Ans. $1994.25 gold.

9. Invoice of Mdse., purchased, paid for in London, and shipped by David Taylor & Sons, per steamer Ohio to Philadelphia, for % and risk of Messrs. Snow, Gilbert & Co.

NOTE.-The 8th and 9th problems are taken from actual custom-house transactions, and indicate the exact forms used. In problem 8, the letters at the left are the trade mark on the drums: indicates numbers of the drums from 1 up to 75; gross weight is 421 cwt. 3 qr. 3 lb., tare, 14 cwt. 3 qr. 23 lb.; net weight is 406 cwt. 3 qr. 8 lb. or 45564 lb., at 13s. 3d. per cwt.; from which a discount of 2 per cent. must be subtracted, giving the amount paid, and adding a commission of 21⁄2 per cent, we have the cost up to shipment, which must be reduced to United States currency, reckoning $4.8665 £1. The duty being specific is, however, reckoned only on the weight. In the 9th example, having deducted discount and added price of case and the commission, and reduced the amount of the invoice to United States rioney, 45 per cent. of the nearest exact number of dollars will be the duty.

3. What is the relation of 12 to 3? Of 16 to 4?

to 5? Of 24 to 6? Of 30 to 5?

Of 18 to 6? Of 20

4. What is the relation of 3 to 6? Of 4 to 12? Of 6 to 24? Of i to 35? Of 8 to 57? Of 9 to 62?

5. The measure of the relation of two numbers is called their ratio. 6. What is the ratio of 12 to 4?

Ans. The ratio of 12 to 4 is three. 7. What is the ratio of 18 to 9? Of 25 to 5? Of 48 to 8? Of 63 to 7? Of 64 to 4? Of 70 to 10? Of 80 to 8? 8. What is the ratio of 3 to 6?

9. What is the ratio of 4 to 12? Of 3 to 18? Of 5 to 3 Of 9 to 108? Of 11 to 132? Of 12 to 144?

10. What is the ratio of to? Of} to}? Of to ? Of.5 to .25? Of.2 to .04? Of .03 to .12?

11. The ratio of two numbers may be expressed by writing the colon between them; thus 8: 4 denotes the ratio of 8 to 4.

12. Required the value of 12:6; of 28:7; of 42:6; of 24: 12; of 12:24.

13. How does the ratio of 8 to 4 compare with the ratio of 12 to 6? Ans. They are equal.

14. What number has the same ratio to 12 that 18 has to 6? 15. What number has the same ratio to 20 that 40 has to 10? 16. The ratio of 9 to 36 is the same as the ratio of 15 to what number?

17. 25 is to 5 as 40 to what number? 24 is to 12 as 15 is to what number?

18. When we express the ratio of two numbers equal to the ratio of two other numbers, as, 24 is to 4 as 36 is to 6, we have a proportion. 19. What proportion can we derive from the two ratios 40 to 8 and 60 to 12?

20. How many numbers do we have in a proportion? How many ratios? Are the ratios equal or unequal?

21. The equality of two ratios may be expressed by writing the symbol between them; thus 8:4 12: 6.

=

22. Write the proportion 16 is to 8 as 24 is to 12; also 15 is to 45 as 18 is to 54.

SECTION IX.

RATIO AND PROPORTION.

RATIO.

547. Ratio is the measure of the relation of two similar quantities; thus, the ratio of 8 to 4 is 2.

548. The Symbol of ratio is the colon (:); thus, 8 : 4 signifies the ratio of 8 to 4. Ratio is also expressed by writing the numbers in the form of a fraction; thus, &.

549. The Terms of a ratio are the two numbers compared, called respectively the antecedent and the consequent. 550. The Antecedent is the number compared with the consequent; thus, in the ratio 8:4, 8 is the antecedent.

551. The Consequent is the number with which the antecedent is compared; thus, in 8: 4, 4 is the consequent. 552. A Ratio is found by dividing the antecedent by the consequent; thus, in 8: 4, the ratio is §, or 2.

553. A Simple Ratio is the ratio of two numbers, as 6:3. A Compound Ratio is the product of two or more simple ratios; as (3:4)× (5:6), or x.

554. A Compound Ratio is usually expressed by writ

ing the simple ratios one under another; thus, 5:6
(3:4)

555. Ratio exists only between similar quantities, and is always an abstract number.

NOTES.-1. The symbol of ratio (:) is supposed to be a modification of the symbol of division.

2. Ratio is usually defined as the relation of two numbers. This is indefinite, for the ratio is the measure of the relation.

3. A few authors divide the second term by the first, calling it the French Method. The method and name are both founded in error; nearly all the French mathematicians, like the German, English, etc., divide the first term by the second.

PRINCIPLES.

1. The ratio equals the quotient of the antecedent divided by the consequent.

Thus, if the antecedent is represented by a, and the consequent by c, and the ratio by r, we have a÷c=r, or=r.

a

с

2. The antecedent is equal to the product of the consequent and ratio.

a

For, since =r, multiplying by we have a cxr.

с

3. The consequent is equal to the quotient of the antecedent divided by the ratio.

3. 90 to 16?

Ans. 5. 7.

10

:?

Ans. .

4. 488 to 61?

:? Ans. 8. 8. 1? 23:21? Ans. 11. 25 :

9. What is the value of the compound ratio

(2:4) ? 23:95

SOLUTION. This compound ratio equals (2: 4) X (3:9), which equals

12. The antecedent is 24, the consequent 8; what is the ratio?

Ans. 3.

13. The consequent is 8 and ratio 9; what is the antecelent?

Ans. 72. 14. The antecedent is 36 and ratio 4; what is the consequent? Ans. 9.

15. The consequent is 15 and ratio ; what is the antecedent? Ans. . 16. The antecedent is 15 and ratio ; what is the consequent? Ans. 1.

17. Can you express the ratio between $24 and 6 lb.? Why not?

18. The antecedents of a ratio are 5 and 6, and the consequents 10 and 14; what is the ratio? Ans. 4.

556. A Proportion is the expression of equality between equal ratios, the terms of the ratios being indicated.

557. The Symbol for proportion is the double colon, (), which expresses an equality of ratios; thus, 8:46:3, means the same as 8:4 = 6:3.

558. A Proportion is read in two ways; thus, 8:4:: 6:3 is read "the ratio of 8 to 4 equals the ratio of 6 to 3;" or "8 is to 4 as 6 is to 3."

559. The Terms of a proportion are the four numbers used in the comparison. The first and fourth terms are

the Extremes; the second and third are the Means.

560. The Couplets are the two ratios compared. The first couplet consists of the first and second terms. The second couplet consists of the third and fourth terms.

561. Proportion may be Simple or Compound. In Simple Proportion both the ratios compared are simple; in Compound Proportion one or both of the ratios are compound.

562. A Simple Proportion is the expression of the equality of two simple ratios.

563. The Principles of proportion are the truths relating to a proportion. They enable us to find any one term when the other three are given.

PRINCIPLES.

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

= {, and multiplying

In any proportion, as 6:3::8:4, we have these equals by 4 and 3 we have 6 × 4 =8 × 3; that is, the product of the two means 8 and 3, equals the product of the two extremes 6 and 4. 2. Either extreme equals the product of the means divided by the other extreme.

For, from the proportion 6:3:: 8:4, we have 6 x 4 = 3 × 8; hence, 63 x 84, or 4 = 3 × 86. Therefore, etc.

3. Either mean equals the product of the extremes divided

by the other mean.

For, from the proportion 6:3::8:4, we have 6 × 43 × 8; hence, 36 × 48, or 86× 43. Therefore, etc.