OPERATION. $.80 × 40 = $32.00 $.70 × 25= 17.50 $1.50 × 15 = 22.50 80 )$72.00 bushels, is worth $72, and 1 $7280 = $.90, Ans. RULE. Divide the entire cost or value of the ingredients by the sum of the simples. 2. A student received the following returns from an examination: arithmetic 98%; history 95%; geography 100%; grammar 100%; drawing 82%. What was his average per cent? SOLUTION.-98% + 95% + 100% + 100% + 82% = 475 % ÷ 5 = 95%, Ans. 3. If a grocer mixes 8 lb. of tea worth $.60 a lb. with 6 lb. at $.70 a lb., 2 lb. at $1.10, and 4 lb. at $1.20, what is 1 lb. of the mixture worth? Ans. $.80. 4. A grocer mixed 10 pounds of sugar at 4 cents with 12 pounds at 4 cents and 16 pounds at 5 cents, and sold the mixture at 6 cents per pound. Did he gain or lose by the sale, and how much? Ans. He gained 46 cents. 5. On a certain day the thermometer registered the following averages: from 6 to 9 A.M., 64°; from 9 to 12, 74°; from 12 to 3 P.M., 84°; and from 3 to 6, 70°. What was the mean temperature of the day? 6. A drover bought 84 sheep at $5 a head, 86 at $4.75, and 130 at $5.50. At what average price per head must he sell them to gain 20% ? 7. How fine is a mixture of 5 pwt. of gold 16 carats fine, 2 pwt. 18 carats fine, 6 pwt. 20 carats fine, and 1 pwt. pure gold (24 carats)? Ans. 184 carats fine. INVOLUTION. 418. A Power is the product arising from multiplying a number by itself, or repeating it several times as a factor. Thus, in 2 x 2 x 2 = 8, the product, 8, is a power of 2. 419. The Exponent of a power is the number denoting how many times the factor is used to produce the power, and is written above and a little to the right of the factor. Thus, 2 × 2 × 2 is written 23, in which 3 is the exponent. Exponents likewise give names to the powers, as will be seen in the following illustrations: 3 3 x 3 = 31 = - 3, the first power of 3; 9, the second power of 3; 27, the third power of 3. The sum of the exponents of two powers of the same number is equal to the exponent of the product of those powers. Thus, 22 × 23 = 25; for 22 = 2 × 2, and 28 = 2 × 2 × 2; hence, 22 × 23 = 2 × 2 × 2 × 2 × 2 = 25. 420. The Square of a number is its second power. The Cube of a number is its third power. 421. Involution is the process of raising a number to a given power. A Perfect Power is a number that can be exactly produced by the involution of some number as a root. Thus, 25 and 32 are perfect powers, since 25 = 5 × 5, and 32 = 2 × 2 × 2 × 2 × 2. EXAMPLES. 422. To find any power of a number. 1. What is the cube of 15? OPERATION. 15 × 15 × 15 = 3375, Ans. or cube of 15, since 15 has been RULE. Multiply the number by itself as many times, less 1, as there are units in the exponent of the required power. 215 2 5 SOLUTION. We multiply 15 by 15, and the product by 15, and obtain 3375, which is the 3d power taken 3 times as a factor. OPERATION. 2 × 2 × 2 5x5x5 = Ans. 625. Ans. 18225. Ans. 373248. Ans. 331776. Ans. 373.248. Ans. 1.26247696. Ans. .0000248832. Ans. 1.00040004. 8 53 125 SOLUTION. We multiply × × by finding the product of the numerators 2 × 2 × 2 and of the denominators 5 × 5 × 5, or by cubing both the numerator and denominator. RULE. A common fraction may be raised to any power, by raising each of its terms, separately, to the required power. 11. What is the square of & ? 12. What is the square of &? 13. What is the cube of 13? 14. Raise 24 to the 2d power. Ans. 2744 Ans. 127. Ans. 612 423. To find the square of a number in terms of its tens and units. 23= 23 69 = 1. Find the square of 23 in terms of its tens and units. SOLUTION.-23=20+3. Multiplying 20 + 3 by 3 and indicating the operation, we have 20 × 3 and 3 x3 or 32. Multiplying 20x 3 by 20, we have 20 × 20 or 202 and 20 × 3. Adding the partial products, the result is 202 + 2 times 20 × 3 + 32 which is equal to 529. = OPERATION. 20+ 3 20 x 3 +32 46 =202+20 x 3 529 = 202 + 2(20 × 3)+32 RULE. ·To find the square of a number consisting of tens and units, to the square of the tens add twice the product of the tens by the units and the square of the units. When a number is separated into any two parts, its square is always equal to the square of the first part + twice the product of the first by the second + the square of the second part. Thus, 23 = 12 + 11 and 232 = 122 + 2(12 × 11) + 112 = 529. In the same way find the square of: 5. 42. 8. 71. 6. 57. 9. 79. 7. 65. 10. 83. 2. 13. 3. 25. 4. 39. 23= 23= 69 46 = 424. To find the cube of a number in terms of its tens and units. = 14. Find the cube of 23 in terms of its tens and units. 529= 23 = = = 11. 101. 12. 119. 13. 235. OPERATION. 20+ 3 20 x 332 1587 202 + 20 x 3 202+2(20 × 3) + 32 20+3 SOLUTION. The cube of 23 = 23 × 23 × 23 or 232 × 23. We proceed to find the square as before which is 202 + 2 (20 × 3) + 32. Multiplying this first by 3 and then by 20 and adding these partial products, the result is 203 + 3 (202 × 3) + 3 (20 × 32)+ 33. RULE. To find the cube of a number consisting of tens and units, to the cube of the tens add three times the product of the square of the tens by the units, three times the product of the tens by the square of the units, and the cube of the units. The cube of a number divided into any two parts is equal to the cube of the first part 3 times the square of the first part by the second part, + 3 times the first part by the square of the second part + the cube of the second part. Thus, 23 = 12 + 11, and 233 = 123 + 3 (122 × 11) +3 (12 × 112) + 113 = 12167. In the same way find the cube of 15. 17. 16. 25. 17. 34. 18. 49. 19. 67. 20. 78. 21. 89. 22. 95. 23. 125. 24. 135. 25. 225. 26. 319. APPLICATIONS OF INVOLUTION. 425. The following principles of physics afford application for the rules of involution: - PRINCIPLES.-I. The intensity of light varies inversely as the square of the distance from the source of illumination. II. The intensity of sound varies inversely as the square of the distance. III. The heating effect of a small radiant mass upon a distant object, varies inversely as the square of the distance. IV. The force of attraction or repulsion exerted between two magnetic poles is inversely proportional to the square of the distance between them. V. Gravitation varies inversely as the square of the distance between the centers of gravity. |