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4. If 1.47cwt. of sugar be worth 4.5L. how much is 1.7lb. worth? Ans. 11.1d. 5. If 1.6cwt. of sugar sell for 3L. 12.76s. what is the value of 3 hogsheads, each 11cwt. 3qrs. 10.12lb.? Ans. 80L. 15s. 3d. 3.36qrs. 6. What is the value of 3 pieces of cloth, each containing 21.5yds. at 12.3s. per yard?

Ans. 39L. 13s. 4.2d. 7. If 1 pint of wine cost 1.2s. what cost 12.5 hogsheads? Ans. 378L. 8. If 19 yards of linen cost 25.75dols. what will 435.5 yards come to? Ans. 590.217dols. +Or, 590dols. 21 cts. 7m. 9. What will a merchant gain by buying 436 yards of linen, at 8.5s. per yard, and selling it at 10.75s. per yard? Ans. 49L. 1s. 10. A grocer bought 7.6cwt of sugar, at 40.1s. per cwt. and retailed it out at 4.5d. per Ib. he gain or lose, and how much?

Whether did

Ans. He gained 14s. 5d. 1.12qrs. 11. A bought 3cwt. 1.5qr. of cloves, at 2.75s. per lb. which he afterwards sold for 60L. 11s. 6d. How much did he gain by the transaction?

Ans. 8L. 12s. 12. If 1 yard of ribbon sell for 4.5 cents, how much will 345 yards bring?

Ans. 15.525dols. Or $15.52

INVERSE PROPORTION.

1. How long will 3 men be in performing a piece. of work which will occupy 5 men 40.5 days?

Ans. 67.5. 2. How many men can do as much work in .4 of a month, as 16 can do in 1.5 month?

Ans. 60.

3. How much silk .75 of a yard wide, will line 25.5 yards of cloth that is 5qrs. wide? Ans. 42.5yds.

4. If a board be .75 of a foot broad, what length must it be to measure 12 square feet? Ans. 16 feet. 5. A had 40.7 yards of linen, for which B gave him 25.6 ells of Holland, at 4.5s. per ell. How much was the linen per yard? Ans. 2s. 9d. 3.84qrs.

THE DOUBLE RULE OF THREE,

IN DECIMALS.

Gestions this rule are wrought as in whole numbere, placing the points agreeably to former directions.

EXAMPLES.

1. If 3 men receive 8.9L. for 19.5 days labour, how much must 20 men have for 100.25 days?

men 3 : 20

Ans. 305L. Os. 8.2d.

days 19.5: 100.25 days: 89L.: 305L. Os. 8.2d.

2. If 2 persons receive 4.625s. for 1 day's labour, how much should 4 persons have for 10.5 days? Ans. 4L. 17s. 1 d. 3. If the interest of 76.5L. for 9.5 months, be 15.24L. what sum will gain 6L. in 12.75 months?

Ans. 22L. 8s. 9 d. 4. How many men will reap 417.6 acres in 12 days, if 5 men reap 52.2 acres in 6 days? Ans. 20 men. 5. If a cellar 22.5 feet long, 17.3 feet wide, and 10.25 feet deep, be dug in 2.5 days, by 6 men, working 12.3 hours a day; how many days of 8.2 hours, should 9 men take to dig another, measuring 45 feet long, 34.6 wide, and 12.3 deep? Ans. 12 days.

INVOLUTION,

OR THE RAISING OF POWERS.

A power is the product arising from multiplying any given number into itself continually a certain number of times; thus,

2x2 4 the second power or square of 2. 2×2×2= 8 the third power or cube of 2. 2×2×2×2-16 the fourth power of 2, &c. The number denoting the power is called the index or exponent of that power.

If two or more powers of the same number are multiplied together, their product is that power whose index is the sum of the exponents of the factors; thus,

2×2-4 the square of 2; 4×4-16-4th power of 2; and 16x16-256-8th power of 2, &c.

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636 2161296 7776 46656 279936 1679616j 10077696 749 343 2401 16807 117649 823543 5764801 40353607 864 512 4096 32768 262144 2097152 16777216 134217728

981 729 6561 59049 531441 4782969 43046721 387420489

EXAMPLES.

1. What is the square of 22?

2. What is the cube or third power of 4?

4x4x4−-64.

3. What is the fifth power of 7?

Ans. 484.

Ans. 64.

Ans. 16807.

4. What is the cube or third power of 35?

Ans. 42875.
Ans. 81

5. What is the fourth power of ?
6. What is the cube or third power of .13?

7. What is the sixth power of 5.03?

256

Ans. .002197

Ans. 16196.005304479729

EVOLUTION,

OR THE EXTRACTING OF ROOTS.

The root of a number, or power, is such a number, as being multiplied into itself à certain number of times,

will produce that power. Thus 2 is the square root of 4, because 2x24; and 4 is the cube root of 64, because 4x4x4 64, and so on.

THE SQUARE ROOT.

The square of a number is the product arising from that number multiplied into itself."

Extraction of the square root is the finding of such a number as being multiplied by itself will produce the number proposed.

RULE!

1. Separate the given number into periods of two figures, each, beginning at the units place.

2. Find the greatest square contained in the left hand period, and set its root on the right of the given number: subtract said square from the left hand period, and to the remainder bring down the next period for a dividual.

3. Double the root for a divisor, and try how often this divisor (with the figure used in the trial thereto annexed) is contained in the dividual: set the number of times in the root; then, multiply and subtract as in division, and bring down the next period to the remainder for a new dividual.

4. Double the ascertained root for a new divisor, and proceed as before, till all the periods are brought down. Note. If, when all the periods are brought down, there be a remainder, annex ciphers to the given number, for decimals, and proceed till the root is obtained with a sufficient degree of exactness.

Observe that the decimal periods are to be pointed off from the decimal point toward the right hand; and that there must be as many whole number figures in the root, as there are periods of whole numbers, and as many decimal figures as there are periods of decimals.

PROOF.

Square the root, adding in the remainder, (if any,) and the result will equal the given number.

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Ans. 327.

Ans. 672.

2. What is the square root of 106929? 3. What is the square root of 451584? 4. What is the square root of 36372961? Ans. 6031. 5. What is the square root of 7596796?

Ans. 2756.228+

6. What is the square root of 3271.4007?

Ans. 57.19+

7. What is the square root of 4.372594?

Ans. 2.091+
Ans. 3.24

8. What is the square root of 10.4976?
9. What is the square root of .00032754?

Ans. .01809+
Ans. 3.1622+

10. What is the square root of 10? To extract the Square Root of a Vulgar Fraction.

RULE.

Reduce the fraction to its lowest terms, then extract the square root of the numerator for a new numerator, and the square root of the denominator for a new denominator.

Note. If the fraction be a surd, that is, one whose root can never be exactly found, reduce it to a decimal, and extract the root therefrom.

EXAMPLES.

1. What is the square root of $6 ? 2. What is the square root of 2794? 3. What is the square root of $18?

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