Or, which is the same thing:

Take the difference of the tabular numbers for these two periods, and multiply by the number of dollars in the annuity.

EXAMPLES.

1. What is the present worth of an annuity of $200, to be continued 5 years, but not to commence till 2 years hence, interest being 6 per cent.?

By our table, we find the present worth of $1 for 7 years, at 6 per cent., to be $5.582381; the same for 2 years, is $1.833393; the difference is $3.748988, which, multiplied by 200, gives $749.798 for the present worth.

2. A father leaves to his son a rent of $310 per annum for 8 years, and the reversion of the same rent to his daughter for 14 years thereafter. What is the present worth of the legacy of each, at 6 per cent.?

$1925 036140=present worth of son's.

12.041582=tabular number 8+14=22 years. 6.209794 tabular number for 8

5.831788

310

58317880

17495364

years.

$1807 854280-present worth of daughter's.

3. What is the present worth of a reversion of $100

a year, to commence in 4 years, and to continue for 10 years, interest being at 6 per cent.? Ans. $582 988.

4. What is the present worth of a reversion of $800 a year, to continue 7 years, but not to commence until the end of 8 years, interest being 4 per cent.?

Ans. $3508 514.

When the annuity is to continue for ever, it is obvious that its present worth will be that sum whose interest for 1 year is equal to the annuity; therefore, to find the present worth of an annuity to continue for ever, we must divide the annuity by the interest of $1 for 1 year, at the given rate per cent.

5. How much must be paid at present for the title to an annuity of $1000, to commence in 7 years, and to continue for ever, interest at 6 per cent.?

Dividing $1000 by $0.06, we get, for the present worth, if entered upon immediately, $16666 663. From table under Compound Discount, we find the present worth of $1 for 7 years, at 6 per cent., to be $0.665057; this, multiplied by 16666.663, gives $11084-283 for the present worth of $16666 663, which is evidently the same as the present worth of the annuity.

6. What is the present worth of a reversion of $100 a year, to commence in 4 years, and to continue for ever, interest being 6 per cent.?

Dividing $100 by $0.06, we get 1666 for the number of dollars in the present worth, if entered upon immediately.

From the table under Compound Discount, we find the present worth of $1 for 4 years, at 6 per cent., to be $0.792094, which must be multiplied by 16663. But, since 1666 is of 1000, we may multiply by 5, divide

by 3, and remove the decimal point three places to the right, as in the following

Operation.

$0.792074

5

3)3960470

Ans. $1320-15

71. The following tables, which have been computed by the aid of logarithms, are added more for curiosity than for any view to their utility.

This table gives the time required for a given principal to double itself, at compound interest, the interest being compounded yearly.

The following table gives the time required for a

given principal to double itself at compound interest, the interest being compounded half-yearly.

This table gives the time required for a given principal to double itself at compound interest, the interest being compounded quarter-yearly.

The following table gives the time required for a given principal to double itself at compound interest, the interest being compounded every instant.

The following table gives the amount of $1, or £1, for any number of years up to 30, for 5 and 6 per cent., compound interest, the interest being compounded every

If we compute the instantaneous compound interest at 6.76587 per cent., it will, at the end of the year, be equal to the simple interest at 7 per cent.

In the same way, the instantaneous compound interest, at 5.8269 per cent., is the same as simple interest at 6 per cent.

[For some curious results in regard to Instantaneous Compound Interest, see an article which I prepared for the American Journal of Science, Vol. 47, No. 1.]