EVOLUTION is the reverse of Involution, and teaches to find the roots of any given powers.

The root is a number whose continual multiplication into itself produces the power, which is denominated the 2d, 3d, 4th, &c., power, according to the number of times which the root is multiplied into itself. Thus, 4 is the square root of 16, because 4 × 4 = 16; and 3 is the cube root of 27, because 3 × 3 × 3 =27; and so on.

Although there is no number of which we cannot find any power exactly, yet there are many numbers of which precise roots can never be determined; but, by the help of decimals, we can approximate towards the root to any assigned degree of

exactness.

The roots which approximate are called surd roots; and those which are perfectly accurate are called rational roots.

Roots are sometimes denoted by writing the character ✔ before the power, with the index of the root over it; thus, the 3d root of 36 is expressed 36, and the second root of 36 is ✔ 36, the index 2 being omitted when the square root is designed. If the power be expressed by several numbers with the sign + or between them, a line is drawn from the top of the sign over all the parts of it; thus, the 3d root of 42+22 is √42+22, and the second root of 59-17 is 59-17, &c.

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Sometimes roots are designated like powers with fractional indices. Thus the square root of 15 is 15, the cube root of 21 is 21, and the 4th root of 37-20 is 37- 20a, &c.

It sometimes will happen that one root is involved in another, thus:

125—5+ √19+6, or 161–147.

√ √ 178+7 33—8+ √84 — 5 — √/87 + 16.

or lost a certain per cent., to ascertain what would be gained or lost at some other price.

The eight following examples will illustrate the above problems.

EXAMPLES.

1. If I buy cloth at $4 per yard, and sell it at $5 per yard, what per cent. do I gain? Ans. 25 per cent.

OPERATION BY PROPORTION.

$5 $4 = $1; $4 : 81: : $100: $25, that is, 25 per cent.

By analysis. If $4 gain $1, it is evident that $1 will gain of $1 = $0.25; and $100 will gain 100 times $0.25= $25, that is, 25 per cent., answer as before.

2. When cloth is purchased at $5 per yard, and sold at $4 per yard, what per cent. is lost? Ans. 20 per cent.

OPERATION BY PROPORTION.

$5 $4 $1. 85: 81: $100: $20, that is, 20 per cent.

By analysis. If $1 be lost on $5, it is certain that on $1 there will be lost of $1 - $0.20; and if on $1 there be lost $0.20, on $100 there will be lost 100 times $0.20 = $20, that is, 20 per cent., answer as before; therefore, when we wish to know what per cent. is gained or lost, either in purchasing or disposing of goods, we adopt the following

RULE.

As the price of the goods is to the gain or loss, so is $100 to the per cent. gained or lost.

3. If I buy cloth at $4 per yard, for how much must I sell it to gain 25 per cent. ? Ans. $5.

OPERATION BY PROPORTION.

$100+$25 $125; $100: $125 :: $4: $5 Ans.

By analysis. If for $100 I receive $125, it is evident that

for $1 I shall receive only

I shall have 4 times $1.25

of $125 $1.25, and for $4 $ 5, answer as before.

4. If I buy cloth at $5 per yard, for what must I dispose of it per yard to lose 20 per cent. ?

Ans. $4.

By analysis. If $80 are to be received for $100, it is certain that for $1 there will be paid only 10% 8.0

of a dollar =

=

$0.80, and for $5 I can receive only 5 times $0.80 $4, answer as before; therefore, to ascertain at what price to sell an article to gain or lose a certain per cent. we adopt the following

RULE.

As $100 is to $100 with the profit added or loss subtracted, so is the price given to the price required.

5. If I sell cloth at $5 per yard, and thereby make 25 per cent., what was the prime cost of the goods?

OPERATION BY PROPORTION.

$100+$25

- $125; $125 : $100 :: $5 : $4 Ans.

By analysis. As $125 are received for $100, it is evident that for $1 there will be received only 12%

$0.80

of a dollar

=

0; and for $5, 5 times $0.80 = $4.00 Ans. 6. If I dispose of cloth at $4 per yard, and by so doing lose 20 per cent., required the prime cost of the goods. Ans. $5.

OPERATION BY PROPORTION.

$100 - $20 $80; $80: 100 :: $4: $5 Ans.

=

By analysis. As $100 are received for $80, it is certain that for 100 of a dollar = = $1.25 there would be received only $1; therefore, for $4 which I receive I should have had 4 times $1.25 = $5, answer as before; therefore, when we wish to ascertain the price of an article, when we know what per cent. is gained or lost, we adopt the following

RULE. -As $100 with the gain per cent. added or loss per cent. subtracted is to $100, so is the price to the prime cost.

7. If I sell cloth at $5 per yard, and thereby gain 25 per cent., what would be my gain if I were to obtain $7 per yard? Ans. 75 per cent.

OPERATION BY PROPORTION.

=

$100 + $25 $125; $5 : $7:: $125: $175;
$175

$100

=

$75, that is, 75 per cent.

By analysis. As $5 amounts to $125, it is evident that $7 will amount to of $125 = $175; and if $7 amount to $175 it is certain that $175 - $100 $75 are gained on $100, that is, 75 per cent.

=

8. If I purchase cloth at $7 per yard, and thereby gain 75 per cent., do I gain or lose if I sell the same at $3 per yard? Ans. lose 25 per cent.

OPERATION BY PROPORTION.

=

$100 + $75 $175; 87: $3 :: $175: $75.

$100

=

$75 $25; the loss is therefore 25 per cent.

By analysis. If $7 give $ 175, $3 will give of $175 = $75. Therefore, for $100 there are received only $75; therefore there is $100 - $75 = $25, or 25 per cent. loss, answer.

If, therefore, goods be sold at a certain price, and there be gained or lost a certain per cent., and we wish to ascertain what would be gained or lost per cent. at some other price, we deduce the following

RULE.

- As the first price is to the proposed price, so is $ 100 with the profit per cent. added or the loss per cent. subtracted to the gain or loss per cent. at the assumed price.

NOTE. If the answer exceeds $100, the excess is the gain per cent. ; but if it be less than $100, the deficiency is the loss per cent.

EXAMPLES TO EXERCISE THE PRECEDING RULES.

9. Sold broadcloth at $ 6.124 per yard, and by so doing lost. 121 per cent. What was the original cost per yard?

Ans. $7.00.

871
100°

By analysis. If 12 per cent. be lost, 87 per cent. will remain. It is now required to find of what number $6.12 is This is done by multiplying $6.12 by 100, and dividing by 871, and it produces the answer, $7.00.

10. Bought cloth at $7.00 per yard, and sold it at $6.12. What per cent. did I lose? Ans. 12 per cent. 11. Bought cloth at $7.00 per yard, and sold it for 121⁄2 per cent. less than what it cost. What did I receive?

Ans. $6.123. 12. Bought cloth at $3.60 per yard. For how much must

I sell it to gain 12 per cent.?

13. Sold cloth at $4.05 per yard, and by so

121 per cent. What did it cost?

14. Bargained for cheese at $8.50 per cwt. must I sell it to gain 10 per cent. ?

15. Sold cheese at $9.35 per cwt.

What did I give for it?

cent.

Ans. $4.05.
doing I gained
Ans. $3.60.
For how much

Ans. $9.35 per cwt. and gained 10 per cent. Ans. $8.50 per cwt.

16. Sold cloth at $1.25 per yard, and by so doing lost 15 per For what should I have sold it to gain 12 per cent. ? Ans. $1.64,7 per yard. 17. Sold cloth at $1.25 per yard, and lost 15 per cent. What

cent.

per

per cent. should I have gained had I sold it for $1.64,7 yard? Ans. 12 per cent. 18. Sold cloth for $1.64,7 per yard, and gained 12 per For what should I have sold it to lose 15 per cent.? Ans. $1.25 per yard. 19. Sold cloth for $1.64,7 per yard, and gained 12 per cent, What should I have lost had I sold it for $1.25 per yard? Ans. 15 per cent. 20. A buys corn for $0.90 per bushel, and sells it for $1.20. B buys for $1.12, and sells for $1.50. Who gains the most per cent. ? Ans. both gain alike. 21. If I buy cotton cloth at 13 cents per yard, on 8 months' credit, and sell it again at 12 cents cash, do I gain or lose, and how much per cent.? Ans. lose 4 per cent. 22. If 24 yards of cloth are sold at $2.50 per yard, and there is 7 per cent. loss in the sale, what is the prime cost of the whole ? Ans. 864.86,439.

23. Bought 24 yards of cloth for $64.86,432. For what must I sell it per yard to lost 7 per cent.? Ans. $2.50. 24. Bought a certain quantity of cloth for $ 64.86,434, and by selling it at $2.50 per yard, I lost 7 per cent. How many yards were bought? Ans. 24 yards.

25. Bought 24 yards of cloth for $ 64.86,433, and sold it at $2.50 per yard; what per cent. is lost? Ans. 7 per cent. 26. If 27 cwt. of sugar be sold at $12.50 per cwt., and there is gained 17 per cent., what was the first cost?

Ans. $10.68,382. 27. Sold a horse for $75, and by so doing I lost 25 per cent.; whereas, I ought to have gained 30 per cent. much was he sold under his real value?

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How Ans. $55.00.

28. Bought a horse which was worth 30 per cent. more than gave for him; but having been injured, I sold him for 25 per cent. less than what he cost, and thereby lost $55 on his original value. What was received for the horse? Ans. $75.00. 29. Bought molasses at 42 cents per gallon, but not proving so good as I expected, I am willing to lose 5 per cent. what must I sell it per gallon?

Ans. $0.39,9.

For

30. Bought a hogshead of molasses for $112, but 15 gallons having leaked out, I am willing to lose 5 per cent. on the cost. For how much per gallon must I sell it? Ans. $2.21,63.

31. Bought a hogshead of molasses for $112, but a number of gallons having leaked out, I sell the remainder for $2.21,63