« ΠροηγούμενηΣυνέχεια »
11. Suppose a freehold estate of D140 per annum, to com mence three years hence, is to be sold; what is it worth, allow ing the purchaser 7 per cent. ? Thus, 1.07x1.07x1.07X.07= .08575301; then D140-08575301=1632.59.5D. Ans.
12. What is an estate of D260 per annum, to continue for ever, worth in present money, allowing 6. per cent. to the pur chaser ?
D4333.33.3. Ans. T find the present worth of a freehold estate, or an annuity to
continue for ever at compound interest.
As the rate per cent. is to D100, so is the yearly rent to the value required.
13. What is the worth of a freehold estate of D40 per annum, allowing 5 per cent. to the purchaser ?
Thus, D5 : 100 :: 40 : 800. Ans. 14. What is the amount of an annuity of D180 for 9 years at 5 per cent. ?*
Ans. D1984.781520. 15. What will an annuity of D200 amount to in 5 years, to be paid by half-yearly payments at 6 per cent.? Thus, 5.637093 X 200=1127.4186 X1.014781=1144.08.2D. Ans.
16. Required the amount of an annuity of D150 for 10 years at 5 per cent. ?
Ans. 1886.68.3. 17. If a salary of D100 per annum, to be paid yearly, be forborne 5 yrs. at 6 per cent., what is the amount ? Ans. D563.70.9.
18. What sum of ready money will buy an annuity of D300, to continue 10 years, at 6 per cent.
10 years=7.36008 X 300=2208.024D. Ans. 19. What salary, to continue 10 years, will D2208.024 buy ? This example is the reverse of the one above, consequently D2208.024 -7.36008=300. Ans.
20. If the annual rent of a house, which is D150, remain in arrears for 3 years at 6 per cent., what will be the amount due for that time?
Ans. D477.54. Note.--From the nature of an annuity, as explained in the proof of the rule in annuities at simple interest, there is due one year's interest less than the number of years the annuity has been continued. By the rule and examples in compound interest, the amount of D1 at the given rate is equal to that power of the amount for one year, which is indicated by the number of years. This amount is obtained for one less than the numbe: of years by forming the geometrical series as directed in the rule, or beginning with unity; thus, in question 1 (annuities) the series
See preceding rules.
is 1, 1.06, 1.062, 1.063, and the last term is the amount of DI for 1 less than 4, the number of years. The sum of this series is the amount at compound interest, of an annuity of Di for 4 years. The amount of any other annuity, for the same time and rate, will be as much greater or less as the annuity is greater or less than 11; that is, the amount of the annuity of D1 must be multiplied by the annuity to obtain its amount, &c.
What is an annuity? How will you find the amount of an annuity at simple interest ? How will you find the amount of an annuity at compound interest ? What is the rule ? What will you do when the payments are half-yearly, or quarterly? How is table 1st calculated ? How will you find the present worth of an annuity? What is the rule? How is table 2d calculated ? When quarterly payments are required, what is to be done? How is table 3 constructed? What can you say of annuities in reversion ? What is the rule ?
What name is given to annuities which continue for ever ? What is the rule for perpetuities? What more can you say of annuities?
DUODECIMALS, OR CROSS MULTIPLICATION.
DUODECIMALS, that is, numbering by 12, are fractions of a foot, or of an inch, or parts of an inch, having 12 for their denominator. It is a rule in general use with artificers in casting up the contents of their work, and an excellent and useful rule for that purpose. Considering a foot as the measuring unit, a prime is the 12th part of a foot, a second the 12th part of a
It is to be observed, that in measures of length, inches are primes, but in superficial measure they are seconds. In both, a prime is a of a foot; but la of a square foot is a parallelogram, a foot long and an inch broad. The 12th part of this is a square inch, which is 1ha of a square foot. This method of multiplying crosswise is not confined to twelves, but may be greatly, extended, for any number, whether its inferior denominations decrease from the integer in the same ratio or not, may be multiplied crosswise. Thus, pounds multiplied by pounds are pounds, pounds multiplied by shillings are shillings, &c.; shillings multiplied by shillings are twentieths of a shilling; shillings multiplied by pence are twentieths of a penny, pence multiplied by pence are 240ths of a penny, &c. The word duodecimal is derived from the Latin word duodecim, signifying twelve. The denominations are12 fourths Illi make
1 third 12 thirds
1 second ! 12 seconds
1 inch I. 12 inches
ADDITION OF DUODECIMALS.
Add as in Compound Addition, carrying 1 for each 12 to the next denomination.
3. Five floors in a certain building contain each 1295 feet, 9 inches, 8''; how many feet in all ? Ans. 6479ft., Oin., 4",
4. Several boards measure as follows, namely: 27st., 3in.; 25ft., 1lin. ; 23ft., 10in.; 20ft., 9in.; 20ft., 6in.; and 18ft., 5 in.; what number of feet in all ?
Ans. 136ft., 8in..
SUBTRACTION OF DUODECIMALS.
RULE. Work as in Compound Subtraction, borrowing 12, &c. when necessary. (5) ft. in.
ft. in. From 176 1 2 6 10 3786 10 4 6 7 Take 97 10 1 7 11
6 9 Rem. 78 3 0 10 11
7. From a board measuring 41ft. 7in., cut 19ft. 10in.; wha' remains ?
Ans. 21ft. 9in.
MULTIPLICATION OF DUODECIMALS.
1. UNDER the multiplicand write the corresponding denominations of the multiplier.
2. Multiply each term of the multiplicand, beginning at the lowest, by the highest denomination in the multiplier, and write the result of each under its respective term, observing, in duodecimals, to carry a unit for every 12 from each lower denomination to its next superior, and for other numbers accordingly.
3. In the same manner, multiply all the multiplicand by the primes, or seconds, denomination in the multiplier, and set the result of each term one place removed to the right of those in the multiplicand.
4. Do the same with the seconds in the multiplier, setting the result of each term two places to the right of those in the multiplicand.
5. Proceed in like manner with all the rest of the denominations, and their sum will be the answer required.
Note.--If there are no feet in the multiplier, supply their place with a cipher, and in like manner supply the place of any other denomination between the highest and lowest. Feet x by feet give feet. Inches x by inches give seconds. Feet x by inches give inches. Inches x by seconds give thirds. Feet X by seconds give sec'ds. Seconds x by sec. give fourths.
Note.-Let it be remembered, that though the feet obtained by the rule are square feet, the inches are not square inches, but the twelfth part of a square foot, 14 8. Multiply 21 feet by 21 feet. Thus :
2-hin. or, 2} or, 2.5 decimally.
Ans. feet 6-3 ft. 61 6.25 So that 3 is not 3 inches, but 36 inches, or or a square foot, . 12X3=, &c.
9. Multiply 10 feet 6 inches by 4 feet 6 inches ? Thus :
4-6 in. 4-6 4.5
67-11-6=product of the feet in the muliplier.
do, the primes.
ft. 75-5-347-6 Ans. 11. Multiply 9 ft. 7 in. by 3. ft. 6 in. Ans. 33 ft. 6 in. 6'. 12. Multiply 3 ft. 11 in. by 9 ft. 5 in. 36 ft. 10 in. 7'. 13. Multiply 8 ft. 6 in. 9'' by 7 ft. 3 in. 8'. 62 ft. 6 in. 7", gu.
14. In a load of wood, 8 feet 4 Thus : 8 ft. 4 in. length. inches long, 2 feet 6 inches high,
2 6 x height. and 3 feet 3 inches wide, how many solid feet ?
Cub.p. od. 67 8 6". Ans. 15. How many solid feet are there in a stick of timber, 70 feet long, 15 inches thick, and 18 inches wide ? Thus : 70 ft. x 15 in.=1050 X 18 in.=18900"144=131, and 36'' rem., hen 36:12=3 in.
Ans. 1314 ft. 16. How many cubic feet of wood in a load 6 st. 7 in. long, 3 st. 5 in. high, and 3 ft. 8 in. wide ?, Ans. 82 ft. 5' 8" 411.
When the feet of the multiplier exceed twelve. RULE.—Multiply by the feet of the multiplier as in compound multiplication, and take parts for the inches &c.