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7 years.

{} {1",

Note-If the pay-Shalf-yearly then use

$4 a, žr, and 2 t. ments are made { quarterly } 28a, 1 r, and 4 t.

2 a - 2u xt III. Given, u, a, and t, to findr.-RULE.

u XtXt-uxt

EXAMPLE 20. If an annuity of $250 per annum, amounts to $2065 in 7 years : Required the rate per cent ? Ans. 6 per cent.

40–4X1uX2t Note-If the pay- Shalf-yearly

ju X 2tX2t-lux2t ments are made 2 quarterly

8a-8X1uX 4t

40 X 40 X 4t1uX4t IV. Given, u, a, and r, to find t. 2

2 a
-1=
RULE.

=x: then

EXAMPLE. 21. If a pension of 8250 per annum, being unpaid a certain time, amounts to $2065, at 6 per cent: What time has the payment been delayed ?

Ans. ments are made quarterly S fr, will give 4t.

PRESENT WORTH OF ANNUITIES. Here p, represents the present worth ; u, t, and r, as before. I. Given,ʻu, t, and r, to find p.--RUB. txt, matXr+2t

:Xu=p. 2t Xr+2

EXAMPLE. 22. What is a pension of $250 per annum worth in ready money, at 6 per cent. for 7 years ? Ans. $1454.225.

Note-Respecting half-yearly and quarterly the same as I. II. Given, p, t, and r, to find u.

txr-11 RULE.

: x2p=u. txtxr-xr+2t

EXAMPLE. 23. What annuity is that which for 7 years continuance, at per cent. produces $1454.225, present worth? Ans. $250, Note-If the pay. Shalf-yearly Str, 2t, and X by 4 p. ments are made {quarterly, {tr, 4t, and X by 8 p. III. Given, u, p, and t, to findr.-RULE. Xt---X2

2p xttu Xt--utx Xt

EXAMPLE 24. If an annuity of $250 per annum, to continue 7 years, produce $1454.225 for the present worth, what is the rate per cent ?

Ans.

6 per cent.

Note-If the pay- Shalf-yearly 2 | 4 u, 2t, will give } r.

{ ments are made quarterly #u, 4t, will give fr.

2p

=t.

r

u

+X+Xr--+Xr+2t:Xu=p.

a

IV. Given, u, p, and r, to find t.

2 RULE.

xxx -15x :12p

+

uXr

EXAMPLE. 25. Required the time that $250 per annum, may be purchased for $1454.225 at 6 per cent.

Ans.

7 years. Note-If the pay-Shalf-yearly Szu, fr, will give 2t. ments are made quarterly SE fu, &r, will give 4t.

ANNUITIES, &c. TAKEN IN REVERSION. I. To find the present worth of an annuity, &c. taken in reversion.

RULE 1.-Find the present worth of the yearly sum at the

Thus, given rate, and for the time of its continuance,

2tXr+2 2. Change pinto a, and find what principal being put to interest will amount to Thus, a, at the same rate, and for the time to

Xr+1 come before the annuity, &c. commences,

EXAMPLE. 26. What is the present worth of 35l. per ann. to continue 12 years ; but is not to commerce till the end of 5 years, allowing 10 per cent. to the purchaser ?

Ans. 1971. 58. 5d. 1.792 qr. II. To find the yearly income of an annuity, &c. in reversion.

RULE 1. Find the amount of the present worth at the given rate, and Thus, pXtXr+p=a. for the time before the reversion,

2.--Change a into p, and And what annuity being sold

tXr+1 will produce p, at the same

Thus,

:12p=4. rate, and for the time of its tXtXr--tXr+2t continuance,

EXAMPLE, 27. A person having an annuity left him for 12 years, which does not commence till the end of 5 years, sold it for 1971. 58. 5d. 1.792qr. allowing 10 per cent to the purchaser : What was the yearly income?

Ans. 351. REBATE OR DISCOUNT. Here s, represents the sum to be discounted : Do the pres. ent wortii, t, and r, es before.

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I. Given, s, t, and r, to find p.-RULE.

tXr+1 EXAMPLES. 28. What is the present worth of $600. due 3 years hence, at 5 per cent per annum?

Ans. $508.4745. 29. What is the present worth of $357.50 to be paid 9 months hence, at 5 per cent per annum? Ans. S344.5783. II. Given, p, t, and r, to find s. RULE. PXtXr+p=s.

EXAMPLES. 30. If the present worth of a sum of money due 9 months, hence, allowing 6 per cent, be $508.4745, what was the sun first due ?

Ans. $600. 31. A person paid $344.5783 for a debt due 9 months hence he being allowed 5 per cent for the discount, how much was the debt?

Ans. $357.50.

Sop III. Given s, p, and t, to find r. -RULE.

txe EXAMPLES. 32. At what rate per cent, will $600 payable 3 years hence, produce $508.4745 for present payment ? Ans. 6 per cent.

33. At" what rate per cent, will $357į payable 9 months hence, produce the present payment of $344.5783. Ans. 5

S-P IV. Given s, p, and r, to find t. -RULE.

riXP EXAMPLES. 31. The present worth of $600 due for a certain time to come, is $508.4745 at 6 per cent, in what time should the sum have peen paid without any rebate ? Ans.

3 years. 35. I have received $344.5783 for a debt of $3571 allowing the person 5 per cent for prompt payment, I desire to know when the debt would have been payable without the discount ?

Ans. 9 months. EQUATION OF PAYMENTS. I. 7 o find the equated time for the payment of a sum of money

due at several times. Rule 1 Find the present worth of

Thus, tXr+1=P each payment for its respective time

2. Add all the present worths together, and call that sum P, then will s-p=d the rebate.

d 3. And

=e, the true equated time. PXr

per cent.

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a

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EXAMPLES. 36 M. owes N. $200, whereof $40 is to be paid at 3 months, 860 at 6 months, and $100 at 9 months; at what time

may the whole debt be paid together, rebate being made at 5 per cent ? Ans. 57315 years=6 months, 26 days.

37. P. owes Q. $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 discount: the true equated time is demanded ?

Ans. 4 months 22 days. 38. R. owes S. $1200, which is to be paid as follows: $200 down, $500 at the end of 10 months, and the rest at the end of 20 months ; but they agreeing to have one payment of the whole, rebate at 3 per cent; the true equated time is demanded ?

Ans. 1 year 11 days.

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COMPOUND INTEREST. Compound Interest is that which arises from a principal increased by its interest, as the interest becomes due.

The letters here made use of, are, a, the amount. p, the principal, hence, amp, the interest. t, the time.

r, the ratio, or amount of $1, or £. for 1 year at any given rate, which is thus found :

: 105 : :1: 1.05=r, at 5 per cent. As 100 : 106 :: 1 : 1.06=r, at 6 per cent.

100 : 107 :: 1 : 1.07=r, at 7 per cent. I. Given, p, t, and r, to find a. Rule. pXr*

EXAMPLES. 1. What will $200 amount to in 4 years, at 5 per cent per annum ?

Ans. $243.10125. 2. What will $480 amount to in 5 years, at 6 per cent per annum ?

Ans. $642.348288. II. Given, a, r, and t, to find p.- RULE.

$ 100

:

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EXAMPLES. 3. What principal being put to interest will amount to $243.10125 in 4 years, at 5 per cent ?

Ans. $200. 4. What principal being put to interest will amount to $642.348288 in 5 years at 6 per cent ?

6

Ans. $480. III. Given, p, a, and r, to find t. -Rule. “Ert which

P being continually divided by r, till nothing remains, the numa. ber of those divisions will be =t.

EXAMPLES. 5. In what time will 8200 amount to $243. 10125 at 5 per cent?

6. In what time will $480 amount to $642.348288 at 6 per cent?

Ans. 4 years.

Ans. 5 years.

a

.

.

a

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IV. Given, p, a, and t, to find r. - Rule. -=rt: then yrt=r.

P

EXAMPLES. 7. At what rate per cent will $200 amount to $243.10125 in 4 years ?

Ans. 5 per cent. 8. At what rate per cent will $480 amount to $642.348288 in 5 years?

Ans 6 per cent. ANNUITIES, OR PENSIONS, IN ARREARS. Here u represents the annuity, pension, &c, a, r, t, as before

uX_u I. Given, u, t, and r, to find a.- -Rule.

EXAMPLES. 9. What will an annuity of $50 per annum, payable yearly, amount to in 4 years, at 5 per cent ? Ans. $215.50625.

10. What will an annuity of $75 per annum, payable yearly, amount to in 6 years, at 6 per cent ? Ans. $523.14885.

aXra II. Given, a, r, and t, to find u.- -Rule.

r_1

EXAMPLES. 11. What annuity being forborne 4 years, will amount to $215.50625 at 5 per cent?

Ans. $50. 12. What salary being omitted to be paid 6 years, will amount to $523.14885 at 6 per cent ?

Ans. $75.

aXr+1-a III. Given u, a, and r, to find t. -Rule. which being continually divided by r, till nothing remains, the number of those divisions will bet

EXAMPLES. 13. In what time will $50 per annum amount to $215.50625 at 5 per cent ?

Ans. 4 years. 14. In what time will 875 per annum amount to $523.14885 allowing 6 per cent for forbearance of payment ? Ans. 6 years. PRESENT WORTH OF ANNUITIES, PENSIONS, &c.

u.

t

u

u

I. Given, u, t, and r, to find p.Rule.

=r=1=p. , f

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