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2. What is the cost of a draft on Chicago for $4200 at 3% discount?

Process.
$4200 x .003 $31.50, discount.
$4200 – $31.50 = $4168.50, cost.

, 3. Find the cost of a draft for $1000, payable in 60 days after sight, when exchange is 3% premium, and interest 6%.

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Process.

Draft $1000.00
Discount for 60 da., at 6% = 10.00

$990.00
Premium at 1 %

2.50 Cost of draft $992.50 4. The draft was for $580; the time 30 da. after sight; the exchange is at a premium of 3%. Find the cost.

5. What will be the cost in Philadelphia of a draft on Boston for $1800, payable 60 days after sight, exchange being at a premium of 2% ?

6. What must be paid in Detroit for a draft of $3000 on Boston at 30 days, exchange being 1% premium ?

7. If exchange on Chicago is 11% premium, what will be the cost in Savannah, Ga., of a sight draft for $3000?

8. What will be the cost of a sight draft on New York for $6400, at 14% premium ?

9. Find the cost of a draft on Omaha for $1400, payable in 60 days, when exchange is 1% premium, and interest 5%?

10. Find the cost of a draft on Baltimore for $1237.50, payable in 30 da. after sight, exchange being $% discount, and interest 5%?

11. A San Francisco merchant bought goods in New York valued at $5284. What will be the cost of a 3 mo. draft for the amount on New York at % premium ?

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12. What must be paid for a draft of $900 on New Orleans at 90 da., exchange at % discount, and interest 5%?

13. If a Boston firm owes a bill in Chicago of $8750, what must they pay for a draft on Chicago, exchange at 3% premium?

14. What must be the face of a draft to pay $500, exchange being at 11% premium ?

Process. $1.00 + .015 = 1.015, cost of a draft for $1.00.

Hence $500 = $1.015 = $492.61 = the draft that $500 will buy.

15. Find the face of a draft, drawn at 30 da., that will pay $369.72, exchange being at 37% discount.

Analysis. Discount of $1.00 for 30 da. at 6%=.005. $1.00 —.005 =.995. .995 — .0325 = .9625, face that will pay $1.00. $369.72 = .9625 = $384.125, face that will pay $369.72.

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16. How large a sight draft can be purchased on Chicago for $6836 when the rate of exchange is 1 % premium ?

17. What is the face of a 90-day draft on Philadelphia bought for $4600 at 6%, exchange 11% premium ?

18. What is the face of a draft on St. Paul for 60 days which may be bought for $2000, exchange being 1% discount and interest 7%?

19. How large a draft on sight on San Francisco can be purchased for $3500 if exchange is at 3% premium?

20. If I pay $325.05 for a draft payable 60 days after sight, what is the face of the draft if exchange is 1% discount and interest 6% ?

21. A draft payable 90 days after sight was bought for $2756 when exchange was 1% discount and interest 6%. What was its face?

22. If exchange is 3% premium, how large a draft will $1201.50 buy?

23. If exchange is at 14% premium, what bill of exchange can be bought for $762, in current funds, supposing a discount of 1% is charged on the funds ?

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RATIO AND PROPORTION.

1. Ratio is the relation which one quantity has to another of the same kind, and is expressed by a common fraction, as and 14. These fractions express the ratio of 2 to 3 and 12 to 7. The same ratios may be expressed thus : 2:3 and 12:7.

2 and 12 are called antecedents. 3 and 7 are called consequents.

2. Since ratios have a fractional form, they are governed by the principles that govern fractions. f = f; therefore 2:3 and 6 : 9 express the same ratio.

3. Ratio cannot exist between two quantities of different kinds. If it be required to find the ratio between 5 pounds and 25 ounces, the pounds must first be reduced to ounces, or the ounces to pounds. 5 lbs.

The ratio of 80 oz. to 25 oz. = = 16

. 4. A proportion is an equation composed of two ratios. Since g = 18, the expression is called a proportion, and the terms 8, 4, 18, and 9 are called proportionals. The proportion may also be written thus: 8:4 = 18:9; or, 8:4:: 18 : 9, and is thus read : “The ratio of 8 to 4 equals the ratio of 18 to 9”; or, “8 is to 4 as 18 is to 9."

5. The first and last terms of a proportion are called extremes ; the second and third terms are called means.

80 oz.

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4 X 9

X X

9

6. The proportion 8: 4 = 18:9 is also =1 Reduc

,.. ing these fractions to a common denominator, we have 8 *: 184. Since the fractions are equal and the denominators are equal, the numerators are equal; that is, 8 X 9= 18 X 4. But 8 and 9 are the extremes, and 18 and 4 are the

Hence we have the following fundamental principle : The product of the extremes equals the product of the

means.

means.

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7. No four terms are proportional unless, when arranged as means and extremes, they conform to the foregoing principle. 2, 5, 6, and 12 are not proportionals, since they cannot be arranged to make the product of the extremes equal the product of the means.

8. If any one of the terms of a proportion is wanting, it may readily be found. For convenience we will call the unknown term x, and find what number x equals.

(a.). If 6, 8, and 9 are the first, second, and third terms of a proportion, what is the fourth term ? By using a we have 6:8= 9:x. The product of the extremes being equal to the product of the means, we have the equation, 6 times x= - 8 times 9; or, 6:2

72. Since 6 times x = 72, once x = 72 = 12. Hence 12 is the fourth term, and the completed proportion is 6:8 = 9:12.

. (6.) Let the second term be wanting, as in 6 : x = 3: 7.

. Applying the principle we have 3 times x = 6 times 7; or, 3x = 42. 42 = 14, the second term. The proportion completed is 6 : 14 = 3:7. Hence the formulæ :

Product of Means (A.) Required Extreme

Given Extreme

Product of Extremes (B.) Required Mean

Given Mean To find any required term the other three terms must be given ; hence arose the old name, Rule of Three.

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EXERCISES. Find the value of x in : 1. 8:12 = 16 : x. 9. 20 : 6

12 : x.
2. 18:10 =x: 30. 10. 280 : 16

140 : X.
3. 16 : x = 12: 24. 11. 1:5 = x:6.
4. x : 5 8: 20. 12. j :x=1:10.
5. x: 16 20: 80. 13. 6:4 = 10 : x.
6. 18: x =
14: 42. 14. .20 :.05

11 : x.
7. 12:10 = X: 22. 15. 1.7 :1.9
8. 16 : 8 = 24 : x. 16. 2.8 : 3.9

= .07 : X.

1.5 : x.

The Principles of Proportion Applied to Practical

Problems.

1. Most practical problems under this rule involve the use of concrete (denominate) numbers, and in each example there are only two different kinds of quantities.

2. Every problem furnishes two like quantities and a third quantity of like denomination with the required answer.

ILLUSTRATIONS.

1. If 8 lb. of sugar cost 40 cents, what will 20 lb. cost?

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Process. (a.) The ratio of the pounds equals the ratio of the costs.

(6.) The ratio of the pounds is 8 : 20; hence the ratio of costs is 40 : x, since 20 lb. cost more than 8 lb. Therefore

: we have: 8 : 20 Or, 40 : x = 8: 20.

Analysis.

If 8 lb. 8x 800.

1 lb. 100, the cost of 20.

20 lb. 100 cts. NOTE.—The pupil will observe that because x is to be greater than its antecedent, the second term must be greater than its antecedent. It is only in this way that the equality of ratios can be preserved.

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40 : x.

40 cts., 5 cts. ;

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