293. Map Questions. 1. When it is noon in Philadelphia, what time is it in Chicago? in Denver? in San Francisco? in New York? 2. When it is 9 A.M. in Chicago, what time is it in San Francisco? in Washington? in New Orleans? in Denver? in Seattle? in New York? 3. A telegram was sent from Washington at 2 P.M. and was received in San Francisco at 11 A.M. of the same day. Explain. 4. At 10 P.M. the people of Los Angeles, Cal., were reading the election returns of New York, which had been compiled at 11 P.M. of the same day. Explain. 5. The passengers on a west-bound train arrived in Sparks, Nev., at 6.05 A.M. and after a stop of 10 min. started on their journey at 5.15 A.M. Explain. 6. On leaving North Platte, Neb., the passengers on an east-bound train found that their watches were all 1 hr. behind time. Explain. 7. How many times must a person reset his watch in traveling from Boston to San Francisco, if he wishes to have correct time on the journey? 8. If the telegraph office in Chicago, Ill., closes at 6 P.M., what is the latest time a message can be sent from San Francisco in time to reach this office before it closes, allowing 30 min. for delays in transmission? 9. When it is noon by standard time in western Iowa, is it earlier or later than noon by local time? Name some place where it is 6 P.M. by standard time before it is 6 P.M. by local time. RATIO 294. 1. What is the ratio of 4 ft. to 6 ft.? Compare the ratio of 4 ft. to 6 ft. with the ratio of 2 ft. to 3 ft., and with the ratio of 8 ft. to 12 ft. 2. What effect upon the ratio of two quantities has (a) multiplying both terms by the same number? (b) dividing both terms by the same number? 3. Name two quantities whose ratio is . Name two other quantities having the same ratio. 4. Name numbers whose ratio is expressed by the fraction,,,, §. §. §. 5. Name whole numbers whose ratio is as to 2; as 3 to; as to 6; as 6 to 2. 6. Write ratios equivalent to the following, but with one or both terms a fraction: 1 to 2; 1 to 4; 2 to 3; 2 to 1; 3 to 7. 7. Supply the number in place of x: 4 X 5 = 10 The Compare fractions are stated as equivalent fractions. 8. The ratio between two numbers may be stated in the form of a fraction. In = § we have an equality of ratios. The equality between ratios is called a proportion. 2 4 6 X 5 10 X 12 X 10 9. Solve: 5 X 10 30 10. If 20 bbl. of flour cost flour can be bought for $120? much as $80.) 11. In a certain city the ratio of the number of schoolcensus children to the total population is 1 to 41. If the school census is 20,000, what is the population of the city? PART V POWERS AND ROOTS as a factor to give 9. What number is the 295. 3 x 3 = 9. 3 is used twice 9 is called the second power of 3. second power of 2? of 4? of 5? of 10? of 1? of 12? 2. The second power of a number is called its square, as the number of units in the area of a square surface is found by taking the second power of the number denoting the length of a side of the square. Give the 3. The square of 3 may be indicated thus: 32. Indicate the square of 4; of 5; of 1; of 10; of 12. value of each: 72, 82, 22, 62. The small figure written at the right and above indicates how many times the number is to be taken as a factor and is called the exponent of the number. 4. 3 x 3 x 3 = 27. 27 is the third power, or cube, of 3. What number is the cube of 1? of 2? of 4? of 5? of 6? of 10? of 12? 5. The cube of 3 may be indicated by an exponent, thus 33. Indicate the cube of 7; of 8; of 9. Give the value of each: 13, 23, 103, 123. 54 is read the fourth power of 5, or 5 to the fourth power; it means 5 × 5 × 5 × 5. Read and tell meaning of: 64, 35, 2o. 6. Find the volume of a cube whose edge is 5 in. Find the cube of 5. 7. Give the square of each of the numbers from 1 to 12. 8. Square 1, 1, 4, .5, 1.5, .04, 16, 21. 9. Find and memorize the cubes of 1, 2, 3, 4, 5, 6, 10, 12. The process of finding a power of a number is sometimes called involution. 10. A number that is the square of some integer or fraction is called a perfect square. Thus, 25 (5 × 5) and (x) are perfect squares. Is 24 a perfect square? 11. Square each: 20, 30, 40, 50, 60, 70, 100. 12. Is the square of 2 plus the square of 3 the same as the square of 5? 296. 1. Which is the more and how much, 202 + 52 or 252 ? 2. The square of any number composed of tens and units may be found thus: 400+2(100) + 25 = 202+2(20 x 5)+52 The square of 25 is seen to be the square of the tens, plus twice the product of the tens and the units, plus the square of the units. 3. Square as above: 23, 47, 105 (100 + 5). 5 20 5 100 255 4. The figure represents a square whose side is 25 units. The square whose side is 20 units contains 400 square units. The two rectangles 20 by 5 contain 100 square units each. The square is completed by the addition of the small square 5 by 5, containing 25 square units. The area of the square is (400 + 2(20 × 5) + 25), or 625 square units. 400 100 20 5. Construct a square whose side is 10+ 5 units. S 20 297. Roots. 1. Since 9 is the square of 3, 3 is the square root of 9; that is, it is one of the two equal factors of 9. What number is the square root of 4? of 25? of 64? of 36? of 49? of 16? of 144? of 100? of 81? of 121? of 1? 2. Since 27 is the cube of 3, 3 is the cube root of 27. What is meant by the cube root of a number? What number is the cube root of 1? of 125? of 8? of 1000? of 1728? 3. The sign (√) is called the radical, or root sign, and is placed over a number to show that its root is to be taken. The root to be taken is indicated by a small figure, called an index, written in the radical thus, 27, which is read the cube root of 27. The index 2 for square root is usually omitted. 4. Read and give the roots: √64, √64, √49, √100, 125, 81, √36, √144, 81. The process of finding the root of a number is sometimes called evolution. 298. Finding Roots by Factoring. Roots of perfect squares may be found by factoring. 1. Find the square root of 324. By factoring, 324 = 2 × 2 × 3 × 3 × 3 × 3. Arranging the factors into two like groups, 2. Find the cube root of 2744. Factor 2774. Group the factors into three like groups. The product of one of these groups is the cube root. 3. The square root of a fraction is the square root of its numerator over the square root of its denominator, thus: √= . |