15. A person receives £859 3s. 3d. 3.2544 qrs. for £875 5s. 6d., due at a certain time to come, allowing 41 per cent. discount-in what time should the debt have been discharged without any rebate?

Ans. 5 months. 16. I have received £70 11s. 9.1764d+. for a debt of £75, allowing the person per cent. for prompt payment-when would the debt have been payable without the rebate? Ans. 15 months.

EQUATION OF PAYMENTS.

To find the equated time for the payment of a sum of money due at several times.

RULE.-Find the present worth of

each payment for its respective time,

thus,

Add all the present worths together;

tr+1

then, s-p=0.

d

and -=e.
pr

1. D owes E £200, whereof £40 is to be paid at 3 months, £60 at 6 months, and £100 at 9 months-at what time may the whole debt be paid, rebate at 5 per cent.?

then 200-39.5061+58.5365+96·3855=5·5719, and

5.5719

194-4281 X 0.5

=57315= 6 months, 26 days. Ans.

2. D owes F £800, whereof £200 is to be paid in 3 months, £200 at 4 months, and £400 at 6 months; but they agreeing to make but one payment of the whole, at the rate of 5 per cent. rebate-find the true equated time. Ans. 4 months, 22 days.

3. E owes F £1200, which is thus to be paid; £200 down, £500 in 10 months, and the rest in 20 months; but they agreeing to have one payment of the whole, rebate at 3 per cent.-find the true equated time. Ans. 1 year, 11 days.

COMPOUND INTEREST.

The letters made use of in Compound Interest are,

A, the amount.

P, the principal.

T, the time.

R, the amount of £1 for a year, at any given rate, which is thus found:

As 100 105:: 1:1·05. As 100; 105·5;: 1; 1·055.

A TABLE OF THE AMOUNT OF £1 FOR ONE YEAR.

A TABLE SHOWING THE AMOUNT OF £1 FOR ANY NUMBER OF YEARS UNDER 31, AT 5 AND 6 PER CENT. PER ANNUM.

The above Table is thus made: As 100 105: 1: 1.05, for the first year; then, As 100 105 105 11025, second year, &c.

I. When p, t, r, are given to find a.

RULE. pxrt=a.

1. What will £225 amount to in 3 years' time, at 5 per cent. per annum? 1.05×1.05 × 1.05=1.157625, then 1.157625x225 £260 98. 3d. 3 qrs. Ans. 2. What will £200 amount to in 4 years, at 5 per cent. per annum? Ans. £243 2.025s. 3. What will £450 amount to in 5 years, at 4 per cent. per annum? Ans. £547 9s. 10d. 2.0538368 qrs. 4. What will £500 amount to in 4 years, at 5 per cent. per annum? Ans. £619 8s. 2d. 3.8323 qrs.

II. When a, r, t, are given to find p.

α

RULE.=p.

5. What principal, being put to interest, will amount to £260 9s. 3d. 3 qrs. in 3 years, at 5 per cent. per annum?

1.05×1.05×1·05=1·157625

260.465625

1.157625

= =£225 Ans.

6. What principal, put to interest, will amount to £243 2.025s. in

4 years, at 5 per cent. per annum?

7. What principal will amount to £547 9s. 10d.

5 years, at 4 per cent. per annum ?

8. What principal will amount to £619 8s. 2d.

years, at 5 per cent. per annum?

III. When p, a, t, are given to find r

Ans. £200. 2.0538368 qrs. in Ans. £150.

3.8323 qrs. in 4 Ans. £500.

a

RULE. =rt.
Р

which being extracted by the rule of extraction (the time given to the question showing the power) will give r.

9. At what rate per cent. will £225 amount to £260 9s. 3d. 3 qrs.

in 3 years?

260.465625

225

1.157625, the cube root of which is (it being the third power) 1.05 5 per cent. Ans.

10. At what rate per cent. will £200 amount to £243 2.0258. in 4 years? Ans. 5 per cent. 11. At what rate per cent. will £450 amount to £547 9s. 10d. 2.0538368 qrs. in 5 years?

Ans. 4 per cent. 12. At what rate per cent. will £500 amount to £619 8s. 2d. 3.8323 qrs. in 4 years? Ans. 5 per cent.

IV. When p, a, r, are given to find t.

which being continually divided by r, till nothing RULE.rt.remains, the number of those divisions will be

a

Р

equal to t.

13. In what time will £225 amount to £260 9s. 3d. 3 qrs. at 5 per cent?

=1·1025

1.05

-=1.05

1.05

260.465625
1.157625
1.1025
-=1.157625
225
1.105
1.05
=1; the number of divisions being 3 times sought.

14. In what time will £200 amount to £243 2.025s., at 5 per cent? Ans. 4 years. 15. In what time will £450 amount to £547 9s. 10d. 2·0538368 qrs., at 4 per cent? Ans. 5 years. 16. In what time will £500 amount to £619 8s. 2d. 3-8323 qrs., at 5 per cent?

Ans. 4 years.

ANNUITIES OR PENSIONS IN ARREARS.
Ku represents the annuity, pension, or yearly rent; a, r, t, as before.

A TABLE SHOWING THE AMOUNT OF £1 ANNUITY FOR ANY NUM-
BER OF YEARS UNDER 31, AT 5 AND 6 PER CENT. PER ANNUM.

The above table is made thus: take the first year's amount, which is £1, multiply it by 105, to which add 1, gives 2:05- the second year's amount: this being multiplied by 1·05, and 1 added, gives 3·1525= the third year's amount.

Multiply the amount of £1 for the number of years, and at the rate per cent. given in the question, by the annuity, pension, &c., and it will give the answer.

17. What will the annuity of £50 per annum, payable yearly, amount to in 4 years, at 5 per cent. ?

then

60.77531250-50

1.05-1

1.05×1.05×1.05×1·05×50=60·77531250, £215 10s. 1d. 2 qrs, Ans.; or,

by the table thus, 4·31012×50=£215 10s. 1d. 1.76 qrs.

18. What will a pension of £45 per annum, payable yearly, amount to in 5 years, at 5 per cent. ? Ans. £248 13s. 3.27 qrs.

19. If a a salary of £40 per annum, to be paid yearly, be forborne 6 years, at 6 per cent.-what is the amount? Ans. £279 3.057d.+ 20. If an annuity of £75 per annum, payable yearly, be omitted to be paid for 10 years, at 6 per cent.-what is the amount?

Ans. £988 11s. 2d. 1.228+qrs..

II. When a, r, t, are given to find u. RULE.

21. What annuity, being forborne 10s. 1d. 2 qrs., at 5 per cent. ?

years, will amount to £215

22. What pension, being forborne 5 years, will amount to £248 13s. 3.27 qrs., at 5 per cent.?

Ans. £45.

23. What salary, being omitted to be paid 6 years, will amount to £279 3.058d. at 6 per cent.?

Ans. £40. 24. If the payment of an annuity, being forborne 10 years, amount to £988 11s. 2d. 1.228 qrs., at 6 per cent.-what is the annuity?

III. When u, a, r, are given to find t.

ar+u-a

RULE.

Ans. £75.

I which being continually divided by r, till rt nothing remains, the number of those divisions will be equal to t.

25. In what time will £50 per annum amount to £215 10s. 1d. 2 qrs., at 5 per cent. for non-payment?

215.50625x1.05+50-215.50625
50

=1.21550625,

which being continually divided by r, the number of those divisions will be 4 years.

26. In what time will £45 per annum amount to £248 13s. 3.27 qrs. allowing 5 per cent. for forbearance of payment? Ans. 5 years.

27. In what time will £40 per annum amount to £279 3.058d., at 6 per cent.? Ans. 6 years. 28. In what time will £75 per annum amount to £988 11s. 2d. 1.228 qrs., allowing 6 per cent. for forbearance of payment ?

Ans. 10 years.

PRESENT WORTH OF ANNUITIES, PENSIONS, &c.

A TABLE SHOWING THE PRESENT WORTH OF £1 ANNUITY FOR ANY NUMBER OF YEARS UNDER 31, REBATE AT 5 AND 6 PER CENT.

The above table is thus made: divide £1 by 1.05=95238, the present worth of the first year, which÷1·05=90703, added to the first year's present worth 1.85941, the second year's present worth; then 907031·05, and the quotient added to 1.85941= 2-72324, third year's present worth, &c.

น

I. When u, t, r, are given to find p. RULE. u- rt

Multiply the present worth of £1 annuity for the time and rate per cent. given by the annuity, pension, &c., it will give the answer. 29. What is the present worth of an annuity of £30 per annum, to continue 7 years, at 6 per cent. ?

10.0483

1.06-1

30. What is the present worth of a pension of £40 per annum, for 8 years, at 5 per cent.? Ans. £258 10s. 6d. 1.45+qrs. 31. What is the present worth of a salary of £35, to continue 7 years, at 6 per cent. ?

32. What is the yearly rent of £50, money, at 5 per cent.?

II. When p, t, r, are given to find u.

Ans. £195 7s. 8d. for 5 years, worth in ready Ans. £216 9s. 5d. 2.08 qrs.