6. What will be the insurance of a ship and cargo, valued at $5643, at 1 per cent.?

per cent. ?

at per cent.?

at per cent.?

Ans. at

at per cent. ? Note. Consult fi 82, ex. 11.

17. A man having compromised with cents on a dollar, what must he pay on

at

per cent. $42'322. his creditors at 621 a debt of $137'46 ? Ans. $85'92.

8. What is the value of $800 United States Bank stock, at 112 per cent.? Ans. $900. 9. What is the value of $56075 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. 1 year 9 months,

14. The interest on a certain note, for 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. $19'76. 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 124 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. $1'265 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.

T 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 $9 for 5 months is the same as $6 for 10 months is the same as 27

$1 for 36 months, and $1 for 45 months, and $1 for 60 months.

141

He might, therefore, have $1 141 months, and he may keep 27 dollars part as long; that is, 45 montlis days, Answer.

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. 4 months.

RATIO;

OR

THE RELATION OF NUMBERS.

93. 1. What part of 1 gallon is 3 quarts? 1 gallon is 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.

4 quarts, and 3 quarts is of 4 quarts.

Ans.

1

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 = 23.

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 1 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, (¶ 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 frac tionally, 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 numcrator 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. ?

1 £. 240 pence, and 2 s. 6 d. 30 pence; hence, 20 , 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?

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, f.

PROPORTION;

OR

THE RULE OF THREE.

94. 1. If a piece of cloth, 4 yards long, cost 12 dollars, what will be the cost of a piece of the same cloth 7 yards long?

Had this piece contained twice the number of yards of the first piece, it is evident the price would have been twice as much; had it contained 3 times the number of yards, the price would have been 3 times as much; or had it contained only half the number of yards, the price would have been only half as much; that is, the cost of 7 yards wni be such part of 12 dollars as 7 yards is part of 4 yards. 7 yards is of 4 yards; consequently, the price of 7 yards must be of the price of 4 yards, or 4 of 12 dollars.of 12 dollars, that is, 12 x 84 21 dollars, Answer.

2. If a horse travel 30 miles in 6 hours, how many miles will he travel in 11 hours, at that rate?

11 hours is of 6 hours, that is, 11 hours is 1 time 6 hours, and of another time; consequently, he will travel, in 11 hours, 1 time 30 miles, and of another time, that is, the ratio between the distances will be equal to the ratio between the times.

11 —

of 30 miles, that is, 30 X 339 55 miles. If, then, no error has been committed, 55 miles must be of 30 miles. This is actually the case; for v.

Ans. 55 miles. Quantities which have the same ratio between them are said to be proportional. Thus, these four quantities,

hours. hours. miles. miles.

6, 11, 30, 55,

written in this order, being such, that the second contains the first as many times as the fourth contains the third, that is, the ratio between the third and fourth being equal to the ratio between the first and second, form what is called a proportion.

It follows, therefore, that proportion is a combination of two equal ratios. Ratio exists between two numbers; but proportion requires at least three.