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37. If 85 tons of coal are required to run 6 engines 17 hours a day for a certain number of days, how many tons will be required to run 25 engines 12 hours a day for the same number of days?
38. If 500 lbs. of wool worth 424 a pound are given for 75 yds. of cloth 18 yds. wide, how much wool worth 364 a pound should be given for 27 yds. that is 1} yds. wide ?
39. If it costs $ 135.00 to carry 855 pounds 64 miles, what will it cost to carry 1288 pounds 15} miles ? 40. If 200 rods of wall can be built by 25 men
in 9} days of 10 hours each, how many rods can be built by 12 men in 1 day of 12 hours ?
41. If it costs a certain family $ 700 a year to live in Brownville, and the cost of living is twice as great in Chicago as in Brownville, what will it cost to live 8 months in Chicago?
42. How many men will be required, working 12 hours a day for 250 days, to dig a ditch 750 ft. long, 4 ft. wide, and 3 ft. deep, if it requires 27 men, working 13 hours a day for 62 days, to dig, a ditch 403 ft. long, 3 ft. wide, and 3 ft. deep ?
43. If 9 men working 12 hours a day for 7 days can make 7 cases of boots, in how many days of 11 hours each can 3 men and 4 boys (one boy's work being equal to of the work of a man) make 33 cases of the same kind of boots ?
44. Wishing to find the number of bricks in a wall 6 rods long, 4 feet high, and 13 inches thick, I found that a part of the wall 63 feet long, 2 feet high, and 13 inches thick, contained 330 bricks. How many bricks did the whole wall contain ?
45. Wishing to find the weight of a block of marble 5 feet long, 2 feet wide, and 1} feet thick, I weighed a smaller block 6 inches long, 4 inches wide, and 2 inches thick, and found it weighed 4 lb. 5 oz. What was the weight of the larger block ?
46. The weight of a cubical block of granite measuring 2 feet on each edge is 1352 pounds. What is the weight of a cubical block measuring 4 feet on each edge ?
655. ILLUSTRATIVE EXAMPLE. A and B associated themselves together in business for one year. A invested $ 500 and B $ 700, agreeing to share their gains or losses in proportion to their investments. They gained $ 2700. What was each person's share ?
Explanation. The whole invest$ 2700 x 5
ment was $ 1200, of which A put in $1125. A's gain. 12
and B . A should then have
of $2700, or $1125, and B should $ 2700 x 7 $ 1575. B's gain.
have 1 of $ 2700, or $ 1575. 12
Ans. A's, $ 1125; B's, $ 1575. Each person's share of the gain can be found by proportion, thus : 1200 : $ 500 = $ 2700 : X.
$ 1125 A's gain. $ 1200 : $ 700 = $ 2700 : X.
$ 1575 B's gain. 656. The gains or losses of a partnership are shared according to the agreement or contract of the partners.
657. In the following examples, when no agreement or contract is mentioned, divide the gains or losses in proportion to the capital invested by each partner and the time it is employed.
NOTE. Some simple examples in partnership have already been given. See page 113, Example m, and page 114, Example 196.
658. Examples for the Slate. 47. Two men, A and B, formed a partnership, A putting in $ 5000, and B $3000. They gained $3000. What was the share of each ?
48. Blood and Searle shipped coal from Philadelphia to New York. Blood had on board 450 tons, and Searle 900 tons. It became necessary to throw overboard 250 tons. What was the loss to each person?
49. A bankrupt owed to M $ 900, to N $ 350, and to O and P $500 each ; his whole property was sold for $ 1584.80, of which $ 158.48 was used to pay the expenses of the sale. What was each person's share of the remainder ?
50. Of a store valued at $ 90000, A owned 1 fourth, B owned 1 third, and C the rest. The store was insured for sof its value, and was entirely consumed by fire. What was the loss to each owner ?
51. Divide $ 1500 among three persons so that their shares shall be in the proportion of 3, 4, and 5.
52. Hinds, Bascom, and Ladd traded in company. Hinds put in $ 2500 for 10 months, Bascom $2300 for 11 months, and Ladd conducted the business, which was considered equal to $ 2000 in trade for 12 months. They gained $ 1486. What should each receive ?
53. X and Y received $857.50 for grading a road. X furnished 5 hands for 20 days, and 6 others for 15 days; Y furnished 10 hands for 12 days, and 9 others for 20 days. What was the share of each contractor ?
54. Rand and Parker engaged in trade. Rand had in trade $ 1000 from January 1 till April 1, when he withdrew $550; July 1 he added $ 700. Parker had in trade $ 3000 from Feb. 1 to Oct. 1, when he added $ 300; Nov. 1 he withdrew $ 900. The net gain during the year was $ 3500. What was the share of each ?
659. Questions for Review. What is a RATIO ? Name the terms of a ratio. What is a simple ratio ? a compound ratio ?
What is a PROPORTION ? Which are the means of a proportion ? the extremes ? What is a mean proportional between two numbers ? When are four quantities inversely proportional ? When four general quantities form a proportion, what two products are equal ? How do you find a missing extreme ? a missing mean? How do you solve examples by simple proportion ? When is compound proportion used ?
How are the gains or losses of a PARTNERSHIP shared ?
POWERS AND ROOTS.
INVOLUTION. 660. Name some products made by using 3's only as factors. Ans. 9, which equals 3 x 3 ; 27, which equals 3 x 3 x 3; 81, which equals 3 x 3 x 3 x 3.
A product made by using only equal factors is a power.
661. A power made by using two equal factors is a second power.
A power made by using three equal factors is a third power. A power made by using four equal factors is a fourth power; and so on. What is the second power of 2? the third ? the fourth ? the fifth ? 662. The process of forming powers is involution.
NOTE. The process of forming any power of a number is sometimes called raising the number to that power; the process of forming the second power is called squaring the number; the process of forming the third power is called cubing the number.
663. The second power of 3 is indicated thus, 32 ; the expression is read, “The second power of three.” The third power of 3 is indicated thus, 38; the expression is read, “ The third power of three "; and so on.
NOTE. The names square and cube are often used for “second power' and “third power,” because the contents of a square is found by raising to the second power the number of units in one of its sides, and the contents of a cube by raising to the third power the number of units in one of its edges.
.664. The small figure above and at the right, which shows to what power a number is raised, is the index or exponent of the power.
665. Oral Exercises. a. Name the squares of the numbers from 1 to 10 inclusive. b. What is the cube of 1 ? of 2 ? of 3? of 4? of 5 ?
c. What is the fourth power of 2 ? of 3? the fifth power of 2 ?
d. What is the square of } ? of } ? of } ? of 0.5 ? e. What is the cube of 1? of } ? of 3 ? of 0.22
666. Examples for the Slate. 1. Find and commit to memory the cubes of the integers from 1 to 10.
Find the powers indicated below. (2.) 17 (6.) (11) (10.) 11
(14.) (164) (3.) 282
(7.) 10.22 (11.) (3) (15.) 2.4. (4.) (1) (8.) 153. (12.) 0.55.
(16.) 0.24 (5.) 0.17 (9.) 0.123.
(13.) 1.92 (17.) 0.16%
667. Name one of the two equal factors of 4; of 9; of 25; of 36; of 64; of 81.
Name one of the three equal factors of 8; of 27 ; of 125 ; of 216.
One of the equal factors which produce a number is the root of that number.
668. One of the two equal factors of a number is its second or square root. One of the three equal factors of a number is its third or cube root. One of the four equal factors of a number is its fourth root, and so on.
669. The process of finding the root of a number is evolution.
NOTE. The process of finding the root of a number is sometimes called extracting the root.