COMPOUND INTEREST.

442. If P represent the principal, r the rate of interest, n the number of years, A the amount, then :

1+r will represent the amount of $1 for 1 year. (1+r)2 will represent the amount of $1 for 2 years. (1+r) will represent the amount of $1 for 3 years. (1+r)" will represent the amount of $1 for n years. P× (1+r)" will represent the amount of $P for n years. Therefore, A PX (1+r)";

and

log Alog P+n × log (1+r).

By means of this formula, all questions in compound interest can be solved.

(1) A deposits $15 in a savings bank, and draws it out at the end of 10 years with interest at 4% per annum, compounded annually. How much does he receive? In the formula A = P× (1+r)", P= $15, r = 0.04, n = 10. log 15 = 1.1761

log 1.0410 = 0.1700

1.3461=log 22.19.

That is, the amount drawn out is $22.19.

(2) In how many years will a sum of money double itself at 4%, compound interest?

Here n is required; and if P be considered 1, A will be 2.

Therefore, the number of years required is 17.7.

EXERCISE XCI.

1. A deposits $60 in a savings bank, and draws it out at the end of 8 yrs., with 4% compound interest. What does he receive?

2. What will $100 amount to in 7 yrs., interest at 8% per annum, compounded semi-annually?

NOTE. Interest for 7 yrs. at 8% per annum, compounded semiannually, is the same as interest for 14 yrs. at 4%, compounded annually.

3. In how many years will a sum of money double itself at 6%, compounded annually?

4. In how many years will a sum of money treble itself at 6%, compounded annually?

5. In how many years will $87 amount to $99 at 3%, compounded annually?

6. In how many years will $100 amount to $175 at 4%, compounded annually?

7. At what rate per cent will a sum of money double itself in 12 yrs., compound interest?

NOTE. Consider 1+r the quantity sought, and its logarithm will be of log 2.

8. At what rate will a sum of money treble itself in 15 yrs. at compound interest?

9. At what rate will $80 at compound interest amount to $110 in 8 yrs.?

10. What sum must be invested at 5%, compound interest to amount to $1200 in 7 yrs.?

11. What sum must be invested at 4%, compound interest, to amount to $2000 in 10 yrs.? To amount to $5000

in 8 yrs.?

12. If A puts $100 a year into a savings bank that pays 4% per annum, compound interest, what will he have in the bank at the end of 10 years?

13. What will be the amount in the last problem if the bank pays 4% per annum?

14. What should be paid to-day for an annuity of $500 a year, for 12 years, if money is worth 34%, compound interest?

NOTE. First find the sum of the payments and interest at the end of the 12 yrs., and then the present worth of that sum.

15. What should be paid to-day for an annuity of $300 a year, for 10 years, if money is worth 4%, compound

interest?

16. What should be paid to-day for the assurance that 5 yrs. hence I shall begin to receive $500 a year, for 8 yrs., if money is worth 41%, compound interest? NOTE. Find what should be paid, if paid all at once, 5 yrs. hence; then find the present worth of that sum.

17. If interest is reckoned at 6%,

what sum of money must be paid annually, beginning a year hence, to clear off a debt of $10,000 in 5 equal payments?

18. If interest is reckoned at 6%, what is the amount of

each of 12 equal semi-annual payments, the first to

be paid 6 mos. hence, required to clear off a debt of $24,000?

CHAPTER XXV.

MISCELLANEOUS PROBLEMS.

1. Make six different numbers with the digits 1, 2, 3, and find their sum.

2. Make six different numbers with the digits 2, 3, 5, and find, by logarithms, their continued product.

3. Make six different numbers with the digits 8, 7, 3, and find, by logarithms, their continued product.

4. Find, by logarithms, the missing term in each of the following proportions:

(i.) 7.13 3.57 4.18: ? (iii.) 7.37:? :: 86.1: 43.7. (ii.) 5.89 76.3:?: 38.7. (iv.) ?: 69.7 3.79: 29.4. 5. Find, by logarithms, the values of .08; 2734; 21.97*. 73.6

6. Find, by logarithms, the values of 9.71; 7.935.

NOTE. In solving the following problems use logarithms whenever they can be used with advantage.

7. What is the horizontal distance between two points, when the air-line distance is 1534 ft., and the difference of level 34 ft.?

8. Find the horizontal distance when the road distance is 1 mile, and the rise 347 ft.

9. If the road distance is half a mile, and the horizontal distance 2513 ft., find the difference of level.

10. The diagonal of a rectangular floor is 34.6 ft., and the width is 17.8 ft. Find the length of the floor.

11. The height of a tower on a river's bank is 55 ft., the length of a line from the top to the opposite bank is 78 ft. Find the breadth of the river.

12. The number of seamen at Portsmouth is 800, at Charlestown 404, and at Brooklyn 756. A ship is commissioned whose complement is 490 seamen. Determine the number to be drafted from each place in order to obtain a proportionate number from each.

13. Show, without division, that 36,432 contains 8, 9, 11 as factors.

14. Find the smallest multiplier that will make 47,250 a perfect cube.

15. Find the proper fraction which, when reduced to a continued fraction, has for quotients 1, 3, 5, 7, 2, 4. 16. If the meter is equal to 1.09362 yds. find a series of four fractions that will express more and more nearly the true ratio of the meter to the yard.

17. Find the square factors contained in 33,075.

18. The top of St. Peter's, Rome, 18 T of a mile above the ground, and that of St. Paul's, London, is 17 of a mile. By how many feet does the height of St. Peter's exceed that of St. Paul's?

19. How many days elapsed between the annular eclipse of May 15, 1836, and that of March 15, 1858?

20. In a gale, a flag-staff 60 ft. high snaps 28.8 ft. from the bottom; and, not being wholly broken off, the top touches the ground. If the ground is level, how far is the top from the bottom?

21. Seventeen trees are standing in a line, 20 yds. apart from each other; a person walks from the first to the second and back, then to the third and back, and so on to the end. How far does he walk?

22. A level reach in a canal is 14 mi. long and 48 ft. broad. At one end is a lock 80 ft. long, 12 ft. broad, and with a fall of 8 ft. 6 in. How many barges can

pass through the lock before the water in the canal is lowered 1 in.?