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10 difference

X.68

6.8

26.8×26.8 718.24 square of the mean diameter ×40 length

28729.6,0

.0034

1149184

861888

+20. head diameter

26.8 mean diameter.

Ans. 97.68064 gallons.

2. What is the content of a cask of whiskey whose bung diameter is 35 inches, head diameter 25 inches, and length 40 inches, it being considerably curved, as seen by the diamters? Ans. 137.52864 gals.

3. What is the content in gallons, in a cask of large curvature, whose bung diameter is 36 inches, head diameter 24 inches, and length 40 inches? Ans. 142.76736 gals.

4. Required the content in gallons, of a cask of moderate curvature (.62) whose bung diameter is 30 inches, head 24 inches, and length 38 inches. Ans. 99.277+gals.

5. Required the content in gallons, of a cask of a small curvature (.6), whose bung diameter is 31 inches, head 26 inches, and length 36 inches. Ans. 102.9384.

A short method to find the content of a cask of a common curvature without the help of instruments.

RULE.

With any straight rod take the diagonal of the cask from the centre of the hung-hole correctly, and make a mark on the rod, which may be measured with a carpenter's rule; then multiply the cube of the diagonal in inches, by .00272 for the content in gallons. Or,

Divide the cube of the diagonal, by 368 for the answer required.

EXAMPLES.

1. Suppose a cask to measure diagonally 36 inches: required the content in gallons.

36 × 36 × 36.00272=126.9+gals.
Or, by rule 2d.

36 × 36 × 36÷÷368=126.782+gals.

2. Required the content of a cask of apple brandy, whose diagonal measure at the bung is 30 inches.

Ans. by rule 1st, 73.44gals.

by rule 2d, 73.369+gals.

ANNUITIES.

An annuity is a sum of money payable either yearly, half yearly, or quarterly, for a certain number of years, or for

ever.

When the debtor keeps the annuity in his own hands beyond the time of payment, it is said to be in arrears.

The sum of all the annuities for the time they have been forborne, together with the interest due upon each, is called the amcunt.

If an annuity is to be bought or paid all at once, at the beginning of the first year, the price which ought to be given for it is called the present worth.

To find the amount of an annuity at simple interest.

RULE.

1. Find the sum of the natural series of numbers, 1, 2, 3, &c. to the number of years less one.

2. Multiply this sum by one year's interest of the annuity at the given rate, and the product will be the whole interest due upon the annuity.

3. To this product add the product of the annuity and time, and the sum will be the amount sought.

ANNUITIES AT COMPOUND INTEREST.

CASE 1.

When the annuity, the time it continues, and the rate per cent. are given, to find the amount.

RULE.

1. Make 1 the first term (of a geometrical progression,) and the amount of $1, at the given rate per cent. the ratio. 2. Carry the series to as many terms as the number of years, and find its sum.

3. Multiply the sum thus found by the given annuity, and the product will be the amount sought.

Or, multiply the amount of $1 for one year, at the given rate per cent., into itself as many times as there are years given; from the product subtract one, then divide the remainder by the interest of $1 for one year, at its given rate per cent., and multiply the quotient by the annuity for the amount required.

If the payments be half yearly or quarterly, multiply the amount found as above, by the proper number in the following

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The construction of this Table is as follows: For half yearly payments take one from the ratio, and from the square root of the ratio; half the quotient of the first remainder divided by the latter, will be the tabular number.

For quarterly payments, use the 4th root as above, and take one quarter of the quotient.

Or, without the help of the table, find the amount of the whole number of years as before, then find the interest of that amount for the given parts of a year; add this interest to the former amount, and it will give the whole amount required.

EXAMPLES.

1. What will an annuity of 300 dollars amount to in four years, at 5 per cent., compound interest?

Ans. $1293 03 cents 7.5 mills.

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Or, 1.05×1.05 × 1.05 × 1.05=1.21550625
Subtract 1.

Interest of $1 for 1 year at 5 per cent.=.05).21550625

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Ans. by the second rule

$1293.03,7,5100

2. What will 50 dollars annual rent amount to, at 4 per cent., if it be forborne 7 years? Ans. $394.91+ 3. What will an annuity of 50 dollars per annum, payable yearly, amount to in 4 years, at 6 per cent. per annum; and what would be the respective amounts, if the payments were to be half yearly or quarterly?

Ans. Amount for yearly payments, $218.73+ do. for half yearly do.

221.96,3+

do. for quarterly do. 223.59,9+

CASE 2.

The annuity, time, and rate given, to find the present worth.

RULE.

1. Divide the annuity by the amount of one dollar for one year, at the given rate per cent., and the quotient will be the present worth of one year's annuity.

2. Divide the annuity by the square of the last divisor, (the amount of one dollar, &c.) and the quotient will be the present worth of two years.

3. In like manner, find the present worth of each year by itself, and the sum of all these will be the present value of the annuity sought.

Or, divide the annuity by the amount of one dollar €

one year, at the given rate, involved to the time, and subtract the quotient from the annuity; divide the remainder by the interest of one dollar, for one year, at the given rate; the quotient will be the present worth.

EXAMPLES.

1. What is the present worth of an annuity of 40 dollars, to continue three years, discounting at 5 per cent. per annum, compound interest?

Amt. of $1 one y. at 5 per cent.=1.05)40.(38.095-present worth for one year.

1.05 x 1.05 1.1025)40.(36.281 = two ys. 1.05 × 1.05 x 1.05-1.157625)40.(34.556-three y.

Ans. Whole present worth, $108.93,2

Second Method.

Annuity. 40.0000 annuity

1.05 x 1.05 1.05-1.157625)40.000000(34.5535 quot.

Divisor 1.05-1

=.05)5.4465

Ans.

$108.93

2. A, B, and C, each hold a tenement that they lease for 20 dollars each, for six years; A covenanted with his tenant to receive his rent yearly, B was to have his rent half yearly, and C's rent was to be paid quarterly; required the present value of each lease, computing discount at five per cent. per annum? Ans. Present worth of A's lease, $101.5138

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3. What is the present worth of a yearly rent of $300 for two years? Ans. $550.01,7.

NOTE.-When the payments are half yearly or quarterly, multiply the present worth so found by the proper number in the above table. Or, find the present worth for the whole years, as before, then find the present worth for the given parts of a year, and it will be the whole present worth.

4. Required the present worth of one dollar for one year at 5 per cent. Ans. $.9523+

CASE 3.

When the present worth, time, and rate are given, to find the annuity, rent, &c.

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