ANNUITIES.

Q. What is meant by an Annuity?

A. It is a sum of money, payable every year, for a certain number of years, or for life, or for an indefinite term of time. Q. When is an annuity said to be Contingent?

A. When it depends on some circumstance or contingency, as on the life of the person.

Q. When is an annuity said to be in Reversion?

A. When it does not commence, until a certain number of years have elapsed.

Q. When is an annuity said to be in Arrears?

A. When the debtor keeps the annuity in his own hands, and does not pay it at the stipulated time.

Q. What is meant by the Present worth of an annuity? A. When the annuity is bought off or paid all at once, at the beginning of the first year, the price, which is paid for it, is called the present worth; and is such a sum as, if put to interest, would exactly pay the annuity as it became due. Q. What is the Amount of an annuity?

A. It is the sum of all the annuities for the time they have been forborne, together with the interest due on each.

CASE FIRST.

Q. What is the first Case?

A. It is to find the amount of an annuity, at simple interest. Q. What is the RULE in this case?

A. Find the interest of the given annuity for one year, then for 2, 3, 4, &c. up to the number of years, less, by one, and their sum will be the whole interest; then multiply the given annuity by the number of years, and add the product to the whole interest, and the sum will be the amount sought.

Q. Why do you not find the interest for the whole number of years, instead of the number less, by one?

A. Because one year elapses before the annuity becomes due, therefore no interest is due on the first year.

Q. There is another RULE for finding the amount of an annuity at simple interest; what is that?

A. Add together the series of numbers, 1, 2, 3, 4, &c. up to the number, less, by one, than the given number of years: then multiply the sum by the interest of the annuity for one year, and the product will be the whole interest. Multiply the annuity by the number of years, and add this product to the whole interest, and the sum will be the amount.

EXAMPLES.

1. What is the amount of an annuity of 200 dollars, for 5 years, with interest at 6 per cent?

Operation by Rule 1.

Interest of 200 dollars for 1 year-12

66

200 66

$120 amount of interest.

The $200×5=1000+120=1120 dollars, answer.

By RULE 2. 1+2+3+4=10×12=$120 amount of in

terest.

Then $200×5+120=1120 dollars answer, as before.

2. A gentleman let his house for 10 years, at a rent of 250 dollars a year; what is the whole amount, computing interest at 6 per cent?

Operation. 1+2+3+4+5+6+7+8+9-45 sum of the

series.

Interest of $250 for 1 year is $15×45=$675 whole in

terest.

Then $250×10-2500+675-3175 dollars. Ans.

3. If I lease a farm for 5 years, at a rent of 300 dollars a year, what will it amount to, with interest at 5 per cent per annum? Ans. 1650 dollars. 4. What will an annuity of 150 dollars amount to in 6 years, at 7 per cent?' Ans. 1057 dollars, 50 cts. 5. If I hire a man for 12 months, at 25 dollars a month, and agree to pay him monthly, but retain the money in my own hands, till the end of the year, how much must I then pay him, allowing money to be worth 1 per cent a month?

Ans. $324,75. 6. If a pension of 33 dollars be forborne 12 years, what will it amount to, at 7 per cent? Ans. 548 dollars, 46 cents.

CASE SECOND.

Q. What is the Second Case?

A. It is when the present worth of the annuity is required. Q. What is the RULE ?

A. By the rule of Discount find the present worth of each year, separately, discounting from the time it falls due; and the sum of all these present worths will be the answer.

N*

EXAMPLES.

Say as 106

1. What will be the present worth of an annuity of 100 dollars, to continue 4 years, discounting 6 per cent? 100 94,339 present worth of 1 year.

100: : 89,285 66
100: 84,745

66

124

100: 80,645

66

66

4 66

Ans. $349,01,4.

2. If a house be leased for 5 years, at 250 dollars a year, what amount will pay the whole rent, at the commencement of the term, discounting 6 per cent? Ans. $1064,846 m.

3. What is the present worth of $400 per annum, to continue 4 years, discounting 6 per cent? Ans. $1396,06.

4. If a store be let for 3 years, at 100 dollars a year, how much ready money will pay the whole rent, at 6 per cent discount? Ans. $268,37.

5. A gentleman leased an estate for 5 years, at 300 dollars a year; at the expiration of which he leased it again, to the same person, for 5 years more, at 250 a year; as the rent on the first lease is still in arrear, it is required to know what amount will pay the 10 years rent, reckoning interest and discount at per cent? Ans. $2744,846 m. 6. What is the present worth of an annuity of 150 dollars a year, payable half yearly, and to continue 3 years, discounting 6 per cent? Ans. $408,121 m. 7. What amount must be given for an annuity of $500, for six years; 4 years of which are in arrear, reckoning interest on the arrear at 6 per cent, and discounting, for the present worth, 7 per cent? Ans. $3085,885 m.

LOSS AND GAIN.

Q. What is Loss and Gain?

A. It is a rule by which merchants and traders find their profit and loss in buying and selling goods. It also teaches them to fix the prices of their goods, so as to gain or lose so much per cent.

Q. What RULE is most applicable to answering questions in

Loss and Gain?

A. The Rule of Three Direct.

CASE FIRST.

Q. What is the first case in Loss and Gain?

A. It is, when an article is sold at a certain profit or loss, to find what that profit or loss is per cent, or how much on 100 dollars.

Q. What is the RULE in this case?

:

A. First, find the whole gain or loss by subtraction, then by the Rule of Three, say, as the price it cost is to the whole gain or loss: so is 100: to the gain or loss per cent; or, which is the same, annex two ciphers to the whole gain or loss, and divide it by the price it cost, and the quotient will be the gain or loss per cent.

EXAMPLES.

1. If I buy cloth at 85 cents a yard, and sell it again at 90 cents, what is my gain per cent ?

Operation. From the selling price,

90 cents

Deduct the purchase price, 85 cents

5 cts. gain per yard. Ans. 515 or $5,88.+

Then say, 85:5:: 100:515. 2. If I buy cloth at 90 cents a yard, and am forced to sell it at 85 cents, what do I lose per cent? Ans. 55 or $5,55+ 3. A merchant imported 12 bales of cloth, each containing 10 pieces, which, together with the charges of shipping, amounted to $360; he paid duties here, 92 cents a piece; freight $12; and portage $1,25; what does it stand him in, per piece, and what will he gain per cent, if he sell it at $4,43 cents the piece? Ans. 10 per cent. Will stand him in $4,03 per piece. 4. Bought a farm for 5000 dollars, and hired the money at 6 per cent to pay for it; at the end of 2 years, I sold it for 6000 dollars; what do I gain per cent, after paying the interest? Ans. 8 per cent.

5. A merchant bought 4 pieces of satin, each containing 36 yards, at 45 dollars a piece, and retailed two pieces of it at $1,87 a yard, and the other two pieces at $1,50 a yard; what was his whole gain, and how much per cent?

Ans. Whole gain $63. Gain per cent 35 dollars.

6. Bought 50 bank shares, including 6 months interest on them, for 55 dollars a share, and sold them again, at 54 dollars a share, charging 6 months interest to the selling price; did I gain or lose, and how much per cent?

Ans. I gained 13 or $1,127 m. per cent.

7. If I buy a lot of goods, amounting to 750 dollars, on 4 months credit, and sell the same for 800 dollars in cash, what is my gain on the purchase, and how much per cent, allowing money to be worth 6 per cent a year ?*

Ans. My neat gain is $65. My gain per cent is $8,84+. 8. A merchant bought 60 yards of broadcloth at 4 dollars a yard, and 38 yards of cassimere at 2 dollars a yard, and sold them together for 44 dollars a yard; what was his whole gain, and how much per cent?

Ans. His whole gain $51,50. His gain per cent $14,11.

CASE SECOND.

Q. What is the Second Case in Loss and Gain?

A. It is to find how a commodity must be sold to gain or lose so much per cent.

Q. What is the RULE in this case?

A. By the Rule of Three, say, as 100 is to the purchase price so is 100 with the gain per cent added, or loss per cent subtracted to the selling price.

EXAMPLES.

1. A merchant bought a quantity of silks for 90 cents a yard; how must he sell them a yard to gain 13 per cent? Operation. 100+13=113

Ans. $1,02.

Then say, as 100 : 90 :: 113 : 1,0215. 2. If I buy 100 barrels of flour for 750 dollars, how must I sell it a barrel, to gain 12 per cent? Ans. $8,434 cts.

3. A merchant bought in New-York, 4 pipes of Madeira wine, containing 125 gall. each, at $2,50 a gall; paid freight and other expenses, amounting to $5,25 a pipe; by accident 15 gall. leaked out; how must he sell it a gallon to gain 20 per cent on the whole purchase? Ans. $3,144+m.

4. Bought a bale of broadcloths, containing 14 pieces, each piece contained 25 yards, at $4,50 a yard; but, proving to be damaged, have 8 per cent deduction allowed me; what does it stand me in per yard, and at what price must I sell it per yard, to gain 12 per cent on the purchase?

It stands me in $4,14 a yard. Ans. I must sell it at $4,63,6% mills.

*All discounts in this Rule are made by the Merchants and Bankers Rule.