of A's note, and charge the remaining 60% to Farrel, and credit Profit and Loss with full amount.

3. Salaries are credited to the partners who earn them and charged to Profit and Loss account the same as interest.

5. A and B formed a partnership on January 1, 1905. A is to conduct the business and receive 65% of the gain, and B, whose time is not employed, is to receive 35% of the gain. On January 1, A invests $4000; on May 3, $1200, and withdraws $1600 on June 15. On January 1, B invests $3800; on March 23, $1800; and withdraws $860 on July 1. Each partner is to receive 8% interest on the amounts he invests, and pays 8% interest on all amounts which he withdraws; the interest is to be adjusted on the basis of each receiving of the gain. The partnership was dissolved on August 25, 1905, with a cash resource of $12000. They owed $450 to C, which B assumes. How much of the cash should each receive?

NOTES.-1. Find the time by counting exact days.

2. Since the interest is not adjusted on the same basis as other profits, it will be necessary to make up the Profit and Loss account with the interest only, and find remainder of gain or loss from the resources and liabilities.

6. A, B and C commenced business on June 1, 1905, with resources as follows: A invested cash $4000, Merchandise $1800, Fixtures $250. B invested Store $4800, Cash $2750, Personal Accounts $800. C invested Cash $6000, Notes $1200, Accrued Interest $120. The firm assumed a mortgage of $2500 on the store, and A's note of $1450. A was to share 30% of the gain or loss; B, 35%; C, 35%. B was allowed $800 per year for keeping the books. On June 1, 1906, A invested $1500; B withdrew $1150; C invested $2000. On January 1, 1907, each partner withdrew $1000. June 1, 1907, the partners agreed upon a dissolution, the ledger showing:

Find the net gain or loss not counting interest on the partners' accounts. What was each partner's net worth at closing, if interest was allowed at 9% ?

7. J. W. White and H. Murray associating together, purchased a flouring mill for $8400, in which White holds a twothirds interest and Murray one-third. During the year White paid out on account of the mill $1548.26, and received $4862.48. Murray paid out $956 and received $2686.40. A settlement is now made, the mill having just been sold for $9000; $4500 received, in cash and the balance a note at 60 days which both agree that White may take to apply on his account at 20% discount; and the $4500 is then properly divided between them; make the division.

INVOLUTION

=

478. A Power is the product obtained by multiplying a number by itself, or using it as a factor. Thus, 9 3 X 3 is the second power of 3; 27 = 3 × 3 × 3 is the third power of 3.

479. The Exponent of a power is the number denoting how many times the factor is repeated. The exponent is usually a small figure, placed at the right and a little above the factor. Thus, 32 signifies that 3 is to be raised to the second power; 33 signifies that 3 is to be raised to the third power, etc.

480. The Square of a number is the second power of the number.

481. The Cube of a number is the third power of the number. 482. Involution is the process of raising any number to a given power.

From the preceding we have the following:

To Raise Any Number to a Given Power

a. Multiply the number by itself until it has been used as often as there are units in the exponent of the power.

1. Find the second power of 18.

2. What is the third power of 54?
3. What is the second power of 4.36?

4. Find the fourth power of 75.

5. What is the sixth power of 1.12?

6. What is the second power of 4.86?

7. What is the fifth power of 4?

8. Find the third power of .3 to three places.

9. What is the third power of 3?

10. What is the fifth power of 1.04? 11. Raise 1.05 to the sixth power. 12. What is the eighth power of ? 13. What is the second power of 43? 14. Expand the expression 65.

15. What is the second power of 5?

16. What part of 83 is 26.

17. What is the difference between 5° and 46?

18. Expand 35 X 24?

19. Express with a single index, 473 × 475

476.

20. How many acres are in a square lot, each side of which is 135 rods?

21. What is the sixth power of .01?

22. What is the fourth power of .03? 23. What is the fifth power of 1.05? 24. What is the third power of .001?

25. What is the second power of .0044?

26. Each side of a room is 12 feet long. How many square yards of carpet will be required to cover the floor?

27. A box in cubical form is 6 feet long on any inner side. How many cubic inches will it contain?

28. From 2.125 subtract 13.

29. From the fifth power of take the fourth power of 4.

EVOLUTION

483. A Root is one of the factors which are repeated duce a power.

484. Evolution is the process of finding or extracting the root of a power. Evolution is the exact reverse of Involution.

485. The Radical Sign is a character V placed over the number considered as the power to indicate that the root is to be extracted.

486. The Index of the root is a small figure placed above the radical sign to indicate what root is to be extracted. Thus indicates that the third or cube root of 64 is to be extracted. The square root is indicated by the radical sign alone without any index.

487. A Perfect Power is a number whose root can be exactly extracted. Thus 64 is a perfect power whose third or cube root

488. An Improper Power is a number whose root cannot be exactly extracted. Thus 10 whose square root is 3.1622+.

The only roots that are of much practical use are the Square and Cube roots.

SQUARE ROOT

489. The Square Root of a number is one of the two equal factors that produce the number. Thus the square root of 25 is 5.

By trial and inspection we find that the square of any number has twice as many or one less than twice as many figures as the number. Hence the square root of a number will contain as many figures as one-half the number of figures in the power and one figure more for an odd figure in the power.

1. Find the square root of 576 sq. ft.

EXPLANATION.-Pointing off the power into periods of two places each by the principle just laid down we see that there will be two figures in the root. Now it is required to construct a square which shall contain 576 sq. ft. We can see that the root of the left period is 2 and since there will be two figures in the root, this must be 2 tens or 20 ft. Construct a square which shall be 20 feet on each side as in the diagram. The surface of this square is 20 20 400 sq. ft. But subtracting this 400 ft. (expressed as 4 in hundreds place in the solution) we still have 176 sq. ft. Our square must be increased to absorb this 176 sq. ft. This can be done by adding to two sides, and still retain the figure as a square. These two additions will together be 40 ft. long, and this explains why we "double the root already found," in the solution. Now the length of our additions being 40 ft. the width will be as much as the length is contained times in the area. 176 40 = = 4 ft. and this gives the second figure of the root. But our square is not yet complete. We still require a corner which we find is 4 ft. square. This corner added to the additions on the two sides gives the length of the three additions as 44 ft.