The whole interest involved is therefore equal to the interest on $1, for 1200+1800-3000 months; and the same interest would accrue on the $600 in 3000-600-5 months.

Hence 5 months is the credit to be allowed on the sum $600.0

EXERCISES.

1. C is indebted to D $900; of which $200 will be due in 6 months; $300 more in 9 m.; and the remainder in 12 months. What would be the proper time for the payment of the whole at once! Ans. 9 months.

2. A merchant bought goods amounting to $5000; of which he was to pay $3000 in hand, and the remainder in 6 months. It is since agreed that the whole shall be paid at one time; what is the proper credit to be allowed? Ans. 2 months.

3. A certain sum of money was to be paid as follows, viz : of it in 2 years, of it in 3 years, and the rest in 4 years and 6 months. The debtor proposing to pay the whole at the same time, it is required to find the proper term of credit.

Ans. 3 years.

4. A engaged to pay to B, $200 on the 1st day of January; $300 on the 15th of April; and $400 on the 20th of August. They now agree to make but one payment of the whole, and wish to know on what day that payment will be equitably due.

The $200 was due on the 1st of January; when the $300 was entitled to a credit of 3 m. 15 da., to the 15th of April ; and the $400, to a credit of 7 m. 20 da., to the 20th of August. $300X(3 m. 15 da.)=$300 × 105 da. =31500;

and $400X(7 m. 20 da.)=$400 × 230 da. —92000. Then (3150092000)÷(200+300+400)=123500-900= 137 da.

We thus find that the sum of the payments will claim a credit of 137 days, to be reckoned from the 1st of January. Disregarding the of a day, and allowing 30 days to a month, we shall find that this credit will extend to the 17th of May. 5. On the 5th of September, 1846, a merchant bought goods amounting to $8000; of which $4000 was to be paid in 4 months, $2000 in 6 m.; and the remainder in 8 m. It was afterwards agreed that one payment might be made of the whole; what was the proper day of payment?

Ans. February 17th, 1847.

6. On the 10th of January, 1848, A bought of B, 100 acres of land, at $24 per acre, to be paid in three equal instalments, on the 20th of October, 1848, the 1st of June, and 30th of December, 1849. If the whole be converted into one payment, on what day should that payment be made?

Ans. May 25th, 1849.

COMPOUND INTEREST.

281. Simple Interest is interest on a given principal only; (§ 265). Compound Interest is interest on both principal and interest, when the latter remains unpaid after it hus become due.

The interest is compounded annually, half yearly, or quarterly, &c., according to the time at which it becomes due.

Compound interest is not sanctioned, by law, on money lent, or debts contracted in ordinary commercial transactions.

§ 282. To calculate Compound Interest.-Make the amount, at simple interest, for the first year, or period when the interest becomes due, the principal for the second; the amount for the second, the principal for the third; and so on. From the last amount subtract the original principal; the remainder will be the compound interest.

EXERCISES.

1. What is the compound interest on $200, for 3 years, at 6 per cent., allowing interest to be due annually?

Ans. $38.203'. 2. What is the compound interest on $1000, for 2 years, at 8 per cent., allowing interest to be due half yearly?

Ans. $169.858'. 3. What would $500 amount to in 5 years, at 6 per cent. interest, if the interest be compounded annually?

Ans. $669.112.

EXERCISES ON CHAPTER XI.

$2500.183.

Philadelphia, June 1st, 1845.

1. On the 1st day of January, 1846, I promise to pay to William Kind, the sum of Two thousand, five hundred dollars, 18 cents, with interest; for value received.

Simon Thankful.

This note was endorsed as follows:

January 1st, 1846, received $1000.

October 10th, 1846, received $35.25.
August 16th, 1847, received $200.

The balance on the note was not paid until the 1st of Jan

uary, 1848. What amount was then to be paid?

Ans. $1541.40'.

2. A farmer bought of a merchant, goods amounting to $175.12, on a credit of 12 months; but paid the debt in 2 months and 10 days. What sum should have been discounted from the debt, allowing the rate of interest to be 8 per cent? Ans. $10.596'.

3. A debt of $3000.75 will be due in 2 years, 7 months and 18 days, without interest. What sum in hand would be an equivalent for the debt, money being at 7 per cent. interest? Ans. $2533.775'.

4. A held a note against B for $473.50, due April 3d, 1846; and B held a note against A for $500.621, due June 10th, 1846; no interest accruing in either case, until the note is due. Settlement was had May 5th, 1846; what was then the balance between A and B, allowing money to be worth 6 per Ans. A owed B $21.61'.

cent.?

5. A merchant bought 43 cut. 3 qr. of sugar, at $5.25 per cwt., which he immediately sold at $7 per cwt., on 6 months credit. Taking the purchaser's note for the amount, he gets the note discounted in bank, at 6 per cent.; what profit did the merchant make? Ans. $67.22'.

6. Wishing to raise the sum of $3760.50, I design, for the purpose, to put a note in bank for 4 months. For what principal must the note be drawn,-interest being at 8 per cent.? Ans. $3866.042'.

7. A promissory note for $350.75 was at interest from the 4th of July, 1844, to the 19th of January, 1847, when it had amounted to $404.238, at what rate was the interest computed?

8. In what time will $400 produce the terest, at 6 per cent., that would acrue on 7 months and 25 days, at 7 per cent. ?

Ans. per cent. same amount of in$375.183, in 5 years, Ans. 6.185' years.

9. A owes B $5000; of which $1200 is to be paid in 9 months, $3000 in 1 year and 3 months, and the remainder in 2 years. In what time might the whole sum be paid at once, without injustice to either? Ans. 15 months.

10. A rice plantation was to be paid for as follows, namely; of the purchase money in hand; of it in 12 months; and the remainder in 1 year and 9 months. The parties have since agreed that the whole shall be paid at one time; when should the payment be made? Ans. In 12 months. for 2 years, at 6 per such a manner as to What profit did he Ans. $5.508.

11. A money dealer borrowed $1000 cent. interest; and loaned the same in compound the interest every 6 months. make in the 2 years, by this proceeding?

CHAPTER XII.

POWERS AND ROOTS.-INVOLUTION.-EVOLUTION.-APPLICATION OF SQUARE AND CUBE ROOT

§ 283. The first power of a number is the number itself. Thus, the first power of 3 is 3.

$284. The second power, or square, of a number, is the product of that number multiplied into itself.

Thus, the second power or square of 3, is 3X3=9.

What is the second power, or square, of 4? Of 5?
What is the second power, or square, of 8 ? Of 11

Of 7? Of 10?
Of 12 Of 20?

$285. The second root, or square root, of a number, is that number which, multiplied into itself, produces the given number.

Of 64 ? Of 100?
Of 81? Of 144?

Thus the square root of 36 is 6, because 6×6=36. What is the square root of 4? Of 25? Of 49? What is the square root of 9? Of 16? Of 36? § 286. The third power, or cube, of a number, is the product of that number multiplied into its second power, or square.

Thus the third power, or cube, of 3, is 3×3×3=27.

What is the third power, or cube, of 2? Of 4? Of 5? Of 7? Of 10?

§ 287. The third root, or cube root, of a number, is that number which, being multiplied into its second power, or square, produces the given number.

Thus, the cube root of 27 is 3, because 3×3×3=27.

What is the cube root of 8? Of 64? Of 125? Of 216? Of 1000?

$288. The 4th power of a number is the product of that number multiplied into its 3d power, or cube. Thus, the 4th power of 2, is 2×2×2×2=16.

What is meant by the 5th power of a number? By the 6th power?

The 4th root of a number is that number which, being multiplied into its 3d power, or cube, produces the given number thus, 2 is the 4th root of 16.

What is meant by the 5th root of a number?

By the 6th root?

From the preceding it is plain, that,

When one number is any power of another, the latter is the corresponding root of the former. Thus 9 is the square of 3; then 3 is the square root of 9.

Powers and Roots of Unity.

§ 289. Any power or root whatever, of unity, is unity; since any number of 1s, multiplied together, produce 1.

Thus, 1X1=1; 1X1X1=1; and so on.

Powers and Roots of Fractions.

§ 290. A power or root of a fraction is found by taking the power or root of the numerator and denominator, separately. Thus, the square of is }; and the square root of is therefore. So the cube of

What is the square of ?

What is the square root of?
What is the cube of ?

is.

What is the cube root of?

Of ?
Of 22? Of 125? Of %?

1000

Powers and Roots of Mixed Numbers.

§ 291. A power or root of a mixed number may be found by reducing to an improper fraction, and taking the power or root of the numerator and denominator, separately.

Thus, the square root of 51= the square root of

=}=2}.

What is the square root of 24? Of 201? Of 11f ? Of lff? What is the cube root of 33? Of 219? Of 19?

Perfect and Imperfect Powers.

§ 292. A perfect power, of any order, is a number which has an exact root of the corresponding order. An imperfect power has no exact root of the corresponding order. Thus,

A perfect square, or a square number, is any number, integral or fractional, which has an exact square root; and a cube number is one which has an exact cube root.

Name all the square numbers, in succession, from unity to the square of 12. Name several cube numbers, beginning with unity.

Name three fractions which are perfect squares. Name three which are perfect cubes.

An imperfect power is also called a Surd; and its root is called an irrational number, because its ratio to unity cannot be exactly determined.