E. 5. What is the prefent worth of an annuity of 2461, to continue 30 years, at 5 per cent. per annum? In table 5, against 30 years at 5 per cent. is

15,372451
246

Answer 3781,622946 = 37817. 125, 5d. the prefent worth.

E. 6. What is the annuity which 2467. will purchase, to continue 30 years, reckoning 5 per cent? In table 6, against 30 years, at 5 per cent. is ,0650514

246

3903084 2602056 1301028

Anf. 16,0036444 167. os. oåd. the purchased annuity per annum.

If the amount of any fum be fought, for a number of days, which are not in the first table, and years which are not in the second; divide the given number of days or years into two fuch numbers as are in the table; then multiply the amount pertaining to each into each other, the product will be the amount for the time required.

E. 7. What will 5237. amount to, in 194 days, at 5 per cent. per

annum ?

In table 1, againft 190 days, under 5 per cent. is
And against 4 days, at the fame rate, is

The product is the amount of 17. for 194 days, viz.
Which multiply by the principal fum, viz.

The product is the answer

1,0257228
1,0005348
1,0262714
523

536,7399840 = (5361. 14s. 9 d.

E. 8. What is the amount of 150l. in 81 years, at 5 per cent? In table 2, against 40 years, under 5 per cent. is 7,0399887 And against 41 years, at 5 per cent. is

7,3919881

The product is the amount of 17. for 81 years, viz. 52,0395126
Which multiply by the principal fum, viz.

150

The other tables of compound intereft, cannot be extended in

Note.

this manner.

Questions

Questions for exercife, to fhew the extenfive ufe of the Tables.

Queft. 1. A perfon having 12 years to run in a leafe of an eftate of 60%. per annum, for 40 years, would know what present money he must pay, in order to complete the leafe by adding 28 years thereto, computeing at 5 per cent, compound intereft? By table 5, the prefent value of 17. per annum, at 5 per cent. for 40 years, is 17,1590862

By the fame table, the value of 17. per } 8,8632516

annum, at that rate, for 12 years to run,

Anfwer £.497,7500760

Queft. 2. Which is the most advantageous, a term of 15 years in an eftate of 100l. per annum ; or the reverfion of fuch an eftate for ever, after the expiration of the faid 15 years, computing at the rate of 5 per cent. per annum, compound intereft?

A Freehold eftate of 100l. per annum, at 5 per cent. is worth

·£. 2000

In table 5, the prefent value of the fame eftate, at the fame rate for 15 years, is

The difference is

1037,965

962,035 val. of rever.

Hence it appears that the first term of 15 years is better than the reverfion for ever afterwards, by 75,930=751. 18s. 7d, the anfwer.

Queft. 3. What annuity, to continue 14 years, may be purchased with 1000l, due at the end of 5 years; the annuity to commence prefently, at 5 per cent? By table 3, the prefent worth of 1000/, due 5 years hence, at 5 per cent. may be found equal to 783,5262; and by table 6, it may be found that the annuity which 783,5262 will purchase for 14 years, at the rate of 5 per cent, is 79,1518-791, 35, old. per annum, the answer.

Queft. 4. For a lease of certain profits for 7 years, A offers to pay 150l. gratuity, and 300l. per annum; B offers 400/. gratuity, and 250/. per annum; C bids 650l. gratuity, purchase without any yearly rent; query, which is the beft offer, and what is the difference, computing at 5 per cent? By table 5, the prefent worth of 300l, per annum, for 7 years, at 4 per cent. is

To which add

The value of A's offer

1800,6164
150
1950,6164

The prefent worth of 250%. per annum, for 7 years 1500,5136

Add

The value of B's offer

400

1900,5136

The

The prefent worth of 200l. per annnm, for 7 years 1200,4109

650,

Add

Hence it appears that A's is the best offer; and that rejecting the decimals, he bids 50l. more than B, 100l. more than C, and 150l. more than D.

Queft. 5. What annuity is fufficient to pay off a debt of 50 millions, in 30 years, at 41. per cent. compound intereft?

In table 6, against 30 years, under 4 per cent. is Which multiply by the debt

The annuity fought

,0578301 50000000

So that fuppofing the national debt to be 50 millions, the intereft at 4 per cent. would be

-£.2891505

2000000

Then it would require a finking fund of 28915057. per annum to clear the whole debt in 30 years.

Queft. 6. A fon previous to his marriage, is minded to have 50l. a year freehold fettled on his family; and to have immediate possession of it, offers his father in lieu an anuuity for his life, valued at 12 years purchase, discounting at 4 per cent. thereon; whereas he is content the eftate fhould be valued at a difcount of 3 per cent. and confequently will be worth 33 years purchase; pray what had the father for his life?

First 33,3X50 1666,6 = = 16667. 135. 4d. nearly the value of the annuity. Then per table 6, 17. for 12 years, at 4 per cent. will purchafe,1065522 per annum.

I

'.* 1666,6 ×,1065522=177,587=1777• 11s. 83d. the answer,

LXIV. Concerning Divifors.

T being often neceffary in arithmetical calculations, to find fuch multipliers, or numbers, which may be divided by any number of given divifors, without any remainder, or remainders; by which means many pleasant questions, not reducible to any other rule in common arithmetic, may be folved.

To find the leaft number that can be divided by any number of divifors, with a remainder.

RULE. Multiply all the prime numbers, and the root of fuch as are fquare or cube numbers, continually; the product will be the number required.

Note. A prime number is fuch as hath no measure but itself and unity, and confequently cannot be produced by the multiplication of two or more integers; as, 1, 2, 3, 5, 7, 11, &c. are prime numbers.

Compofite numbers are fuch as are divifible by fome numbers befides unity; as 8 is divisible by 4 and 2, &c.

A number

A number that will divide several numbers exactly, is called a common measure, as 3 is a common measure to 12 and 15.

EXAMPLE I. Required the three leaft numbers, which divided by 20, fhall leave 19 for a remainder; but if divided by 19, fhall leave 18; if divided by 18, fhall leave 17; and fo on, always leaving one lefs than the divifor, to unity?

First 1, 2, 3, 5, 7, 11, 13, 17, and 19, are prime numbers:

IX 2 X 3 X 2 X 5 X 7 X 2 X 3 X 11 X 13 X 2 X 17 X 19 = 232792560, the leaft number that can be divided by the given divifors without a remainder.

Then the number 232792560—1—232792559, the first number.
And 232792560 X 2 — -1=465585119, the fecond number.
Alfo 232792560 X 3-1698377679, the third number.
And after this manner may the other numbers be found.

E. 2.

What is the leaft number that can be divided by the nine digits, without a remainder?

The given divifors are 1, 2, 3 4, 5, 6, 7, 8, 9.

Now √ 42; 6 may be cancelled, being composed of 2 X 3; and 3,

5

3

and 7 are prime numbers; and √ 8 = 2. Alfo√93.

Then per rule, 1 X 2 X 3 X 2 X 5 X 7 X 2 X 3 = 2520, the number required.

E. 3. Required the leaft number which being divided by 7, 6, 5, 4,3 and 2, fhall leave 6, 5, 4, 3, 2 and 1 refpectively ?

Firft the divifors are 7, 6, 5, 4, 3, 2.

Now 42, and 3, 5 and 7 are prime numbers, 6 may be cancelled being a compofite number.

Then per rule, 2X 3X2X5X7=420, the leaft number that can be divided by the given divifors without a remainder. Therefore 420. I 419, the number required.

E. 4. John the gardener counting fome apples into a basket, found that when he counted them in by two at a time, three at a time, and four at a time, there remained one; but when he counted them in by five at a time, there remained none; quere, the number of apples?

Firft 2, 3 and 4, are the divifors; now 2 and 3 are prime numbers, and 142.

Then per rule, 2X 3X212; then 12+1=13, which divided by 2, 3 and 4, leaves 1, according to the question; but divided by 5 will leave 3, which is 2 fhort of 5. . To twice 12 add 1, and the fum will be 25, the number fought.

E. 5. A country girl to torn did go,
Some walnuts there to fell;
A gentleman fhe chanc'd to meet,
And thus is her befell:
My pretty maid, fays be to fhe,
What number have you here?
I can't tell, Sir, fays he to him,
But this I'll make appear;

And

every

I told them o'er, e'er I came out,
By fix's, fives, four's,three's,two's,
time I numbered them,`
One remained overplus:
I told them o'er by feven's at last,
And there were no remains;
If you can find the number out,
Pray take them for your pains.

First, the leaft number that can be divided by 1, 2, 3, 4, 5, 6, without a remainder, will (per rule) be 1X 2X 3X2X5=60. Then 60+i =61, which divided by 2, 3, 4, 5, 6, will leave 1 according to the question; but divided by 7, will leave 5; .. 60 X 5+1=301, the leaft number which admits of the conditions of the question. Then to find the next least number which admits of the fame conditions, by proceeding as above we shall find to be 60X12+1=721. Alfo 721-301 =420, the common difference of all the numbers answering the conditions of the queftion. Therefore 30:, 721, 1141, 1561, 1981, &c. ad infinitum, will anfwer the conditions of this question.

THIS

LXV. DUODECIMALS;

OR,

CROSS MULTIPLICATION.

HIS rule is called duodecimals, because the unit, or integer, is divided into 12 equal parts; and hence this way of computation is chiefly used amongst workmen in cafting up the contents of fuperficial and folid works, the lineary dimensions being generally taken in feet, inches, and parts.

RULE. 1. Under the multiplicand, write the correfpondent denominations of the multiplier.

2. Multiply each term in the multiplicand, beginning with the lowest, by the feet in the multiplier; placing each refult under its refpective term, remembering to carry an unit for every 12 from each lower denomination to its next fuperior.

3. Work in the fame manner with the inches and parts, fetting the refult, of each term one place more to the right-hand; having thus finished multiplication, the fam of all will be the product required.

Note. In multiplying feet, inches, and parts; if feet be multiplied by feet, the product is feet; and feet multiplied by inches, the product is inches; aud parts multiplied by feet, the product is parts; parts multiplied by inches, the product is feconds; and parts multiplied by parts, the product is thirds.

EXAMPLE