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below par? 15. The rate per cent. is a decimal carried to how many places? 16. What are decimal expressions lower than hundredths? 17. How is interest, (when the time is 1 year,) commission, insurance, or any thing else rated at so much per cent. without respect to time, found? 18. When the rate is 1 per cent., or less, how may the operation be contracted? 19. How is the interest on $1, at 6 per cent. for any given time, found by inspection? 20. How is interest cast, at 6 per cent., when there are months and days in the given time? 21. When the given time is less than 6 days, how is the interest most readily found? 22. If the sum given be in pounds, shillings, &c., how is interest cast? 23. When the rate is any other than 6 per cent, if there be months and days in the given time, how is the interest found? 24. What is the rule for casting interest on notes, &c. when partial payments have been made, and what is the principle on which the rule is founded? 25. How may the principal be found, the time, rate per cent., and amount being given ? 26. What is understood by dis27. by present worth? 28. How is the principal found, the time, rate per cent., and interest being given ? 29. How is the rate per cent. of gain or loss found, the prices at which goods are bought and sold being given? 30. How is the rate per cent. found, the principal, interest, and time being given? 31. How is the time found, the principal, rate per cent., and interest being given? 32. What is simple interest? 33. compound interest? 34. How

count?

is compound interest computed ?

7

EXERCISES.

Ans. 19'677.

1. What is the interest of $273'51 for 1 year 10 days, at per cent.? 2. What is the interest of $486 for 1 year, 3 months, 19 days, at 8 per cent. ?

3. D's note of $203'17 was given Oct. est after three months; Jan. 5, 1809, he was there due May 2, 1811?

Ans. $50'652. 5, 1808, on interpaid $50; what Ans. $174'53. 17, 1800, on in

4. E's note of $870'05 was given Nov. terest after 90 days; Feb. 11, 1805, he paid $186'06; what was there due Dec. 23, 1807?ut

Ans. $1041'58.

5. What will be the annual insurance, at § per cent., on a house valued at $1600 ?

Ans. $10.

6. What will be the insurance of a ship and cargo, valued

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8. What is the value of $800 United States Bank stock,

at 112 per cent. ? Ans. $900. 9. What is the value of $560'75 of stock, at 93 per cent.? Ans. $521'497 10. What principal at 7 per cent. will, in 9 months 18 days, amount to $422'40?

Ans. $400. 11. What is the present worth of $426, payable in 4 years and 12 days, discounting at the rate of 5 per cent. ? In large sums, to bring out the cents correctly, it will sometimes be necessary to extend the decimal in the divisor to five places. Ans. $354'506.

12. A merchant purchased goods for $250 ready money, and sold them again for $300, payable in 9 months; what did he gain, discounting at 6 per cent. ? Ans. $37'081.

13. Sold goods for $3120, to be paid, one half in 3 months, and the other half in 6 months; what must be discounted for present payment? Ans. 68'492. 14. The interest on a certain note, for 1 year 9 months, was $49'875; what was the principal ?

Ans. $475.

15. What principal, at 5 per cent., in 16 months 24 days, will gain $35 ? Ans. $500. 16. If I pay $15'50 interest for the use of $500, 9 months and 9 days, what is the rate per cent.?

17. If I buy candles at $167 per lb., and sell them at 20 cents, what shall I gain in laying out $100 ?

Ans. 1976. 18. Bought hats at 4 s. apiece, and sold them again at 4 s. 9 d.; what is the profit in laying out 100 £.?

Ans. 18 £. 15 s.

19. Bought 37 gallons of brandy, at $1'10 per gallon, and sold it for $40; what was gained or lost per cent.? 20. At 4 s. 6 d. profit on 1 £., how much is gained in laying out 100 £., that is, how much per cent. ? Ans. 22 £. 10 s. 21. Bought cloth at $4'48 per yard; how must I sell it to gain 12 per cent. ? Ans. $5'04.

22. Bought a barrel of powder for 4 £.; for how much must it be sold to lose 10 per cent.? Ans. 3 £. 12 s.

23. Bought cloth at 15 s. per yard, which not proving so good as I expected, I am content to lose '17 must I sell it per yard?

per cent.; how Ans. 12 s. 4 d. 24. Bought 50 gallons of brandy, at 92 cents per gallon, but by accident 10 gallons leaked out; at what rate must I sell the remainder per gallon to gain upon the whole cost at the rate of 10 per cent.? Ans. $16265 per gallon.

25. A merchant bought 10 tons of iron for $950; the freight and duties came to $145, and his own charges to $25; how must he sell it per lb. to gain 20 per cent. by it? Ans. 6 cents per lb.

EQUATION OF PAYMENTS.

¶ 92. Equation of payments is the method of finding the mean time for the payment of several debts, due at different times.

1. In how many months will $1 gain as much as 5 dollars will gain in 6 months ?

2. In how many months will $1 gain as much as $40 will gain in 15 months? Ans. 600.

3. In how many months will the use of $5 be worth as much as the use of $1 for 40 months?

4. Borrowed of a friend $1 for 20 months; afterwards lent my friend $4; how long ought he to keep it to become indemnified for the use of the $1?

5. I have three notes against a man; one of $12, due in 3 months; one of $9, due in 5 months; and the other of $6, due in 10 months; the man wishes to pay the whole at once; in what time ought he to pay it?

$12 for 3 months is the same as $1 for 36 months, and $9 for 5 months is the same as $1 for 45 months, and $6 for 10 months is the same as $1 for 60 months.

27

141

He might, therefore, have $1 141 months, and he may keep 27 dollars part as long; that is, 4: 6+ days, Answer.

= 5 months

Hence, To find the mean time for several payments,-RULE: -Multiply each sum by its time of payment, and divide the sum of the products by the sum of the payments, and the quotient will be the answer.

Note. This rule is founded on the supposition, that what is gained by keeping a debt a certain time after it is due, is the same as what is lost by paying it an equal time before it is due; but, in the first case, the gain is evidently equal to the interest on the debt for the given time, while, in the second case, the loss is only equal to the discount of the debt for that time, which is always less than the interest; therefore, the rule is not exactly true. The error, however, is so trifling, in most questions that occur in business, as scarce to merit notice.

6. A merchant has owing him $300, to be paid as follows: $50 in 2 months, $100 in 5 months, and the rest in 8 months; and it is agreed to make one payment of the whole in what time ought that payment to be?

:

Ans. 6 months. 7. A owes B $136, to be paid in 10 months; $ 96, to be paid in 7 months; and $260, to be paid in 4 months: what is the equated time for the payment of the whole ?

Ans. 6 months, 7 days +. 8. A owes B $600, of which $200 is to be paid at the present time, 200 in 4 months, and 200 in 8 months; what is the equated time for the payment of the whole ?

Ans. 4 months. 9. A owes B $300, to be paid as follows: in 3 months, in 4 months, and the rest in 6 months: what is the equated time? Ans. 44 months.

RATIO;

OR

THE RELATION OF NUMBERS.

T 93. 1. What part of 1 gallon is 3 quarts? 1 gallon is 4 quarts, and 3 quarts is of 4 quarts. Ans. of a gallon.

2. What part of 3 quarts is 1 gallon? 1 gallon, being 4 quarts, is of 3 quarts; that is, 4 quarts is 1 time 3 quarts and of another time.

Ans. 1.

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3. What part of 5 bushels is 12 bushels?

Finding what part one number is of another is the same as finding what is called the ratio, or relation of one number to another; thus, the question, What part of 5 bushels is 12 bushels? is the same as What is the ratio of 5 bushels to 12 bushels? The Answer is 12 22.

Ratio, therefore, may be defined, the number of times one number is contained in another; or, the number of times one quantity is contained in another quantity of the same kind.

4. What part of 8 yards is 13 yards? or, What is the ratio of 8 yards to 13 yards?

13 yards is 3 of 8 yards, expressing the division fractionally. If now we perform the division, we have for the ratio 1; that is, 13 yards is 1 time 8 yards, and of another time.

We have seen, (TT 15, sign,) that division may be expressed fractionally. So also the ratio of one number to another, or the part one number is of another, may be expressed fractionally, to do which, make the number which is called the part, whether it be the larger or the smaller number, the numerator of a fraction, under which write the other number for a denominator. When the question is, What is the ratio, &c.? the number last named is the part; consequently it must be made the numerator of the fraction, and the number first named the denominator.

5. What part of 12 dollars is 11 dollars? or, 11 dollars is what part of 12 dollars? 11 is the number which expresses the part. To put this question in the other form, viz. What is the ratio, &c.? let that number, which expresses the part, be the number last named; thus, What is the ratio of 12 dollars to 11 dollars? Ans. H

6. What part of 1 £. is 2 s. 6 d. ? or, What is the ratio of 1 £. to 2 s. 6 d. ?

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1 £.

240 pence, and 2 s. 6 d. 30 pence; hence, , is the Answer.

7. What part of 13 s. 6 d. is 1 £. 10 s. ? or, What is the ratio of 13 s. 6 d. to 1 £. 10 s.?

8. What is the ratio of 3 to 5?

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of 5 to 3?

of 15 to 90? of 160 to 84 ?

Ans. 20.

of

of 90 to

of 615 to

Ans. to the last, §.

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