7. The length of a rectangular garden is 50 yards and its width is 40 yards. Around it is a path containing 475 square yards. Find the width of the path. 8. The height of a mirror is two feet greater than its width. The glass cost $1 per square foot, and the frame cost 50 per linear foot inside measure. The cost of the mirror was $34. Find the dimensions of the mirror. 9. A man buys a farm for $6400. The number of dollars an acre cost is 25% of the number of acres bought. Find the cost of one acre. 10. "Some bees were sitting on a tree. At one time the square root of half their number flew away, at another time of the whole flew away. There were then 2 bees left. How many bees were there?" (Taken from the Bija Ganita, the second chapter of a Hindu work on astronomy.) HINT. Let 2x2 the number of bees. = Example 1. A farmer bought a number of cattle for $480. If he had bought 4 less for the same money they would have cost $6 apiece more. How many did he buy? x = the number of cattle bought. Let the number of dollars each would have cost had he bought 4 less. number of cattle were sold for $480. If 4 more were sold for the same money, the selling price would have been $6 apiece less. How many were sold?" the answer would be 16. EXERCISE 108 1. Two numbers differ by 2 and the sum of their reciprocals is 23. Find them. 2. A number of men were paid $112 for doing a piece of work. If two more men had been employed, each man would have received $1 less. How many men were employed? 3. The circumferences of two wheels differ by 1 foot, and the smaller wheel makes 40 revolutions more than the larger in going a distance of 1 mile. Find the circumference of each. 4. A man sold a mule for $16 at a loss of as many per cent as the mule cost him dollars. Find the cost. 5. A crew can row 20 miles down a stream and back in 6 hours. The rate down the stream is 3 miles per hour more than the rate up. Find the rate down, the rate up, and the rate in still water. 6. A and B working together do a piece of work in 71⁄2 days. B alone takes 8 days longer than A alone to do the work. Find A's and B's time. He 7. A man bought a number of cattle for $2000. sold all but 5 at an advance of $5 per head for $2025. How many cattle did he buy? 8. The sum of a number and its reciprocal is 21. Find the number. 9. A number consists of two digits of which the first exceeds the second by unity, and the number itself exceeds the sum of the squares of its digits by 4. Find the number. 10. A sum of money amounts in 1 year to $262.50. The rate per cent of interest is of the principal. Find the principal. CHAPTER XII RADICALS. THEORY OF EXPONENTS. RADICAL 129. The second of the four fundamental rules, namely subtraction, gave rise to the introduction of zero and the negative number. To illustrate, take the problem: What number added to b gives a for the sum? The answer to this question is the root of the equation If b equals a + c, c being a positive number, then a − b = a − (a + c) = − c. Division gave rise to the introduction of the fraction. To illustrate, take the problem: What is the fifth part of 22? The answer is the root of the equation 5x = 22. 130. The four fundamental rules, when applied to integers, give in every instance results known as rational numbers. The general type of a rational number is m and n being positive integers. m n 131. Evolution gave rise to a new kind of number known as the irrational or surd number. What is the square root of 10? The answer is, the number which being multiplied by itself gives 10 for product. Algebraically the square root of 10 is defined as a root of the equation x2 = 10. All irrational numbers are not expressible by a finite number of figures. Their values, however, can be expressed to any desired degree of approximation. Thus, √103.16228 correct to five decimal figures. In fact, 3.16228 differs from V10 by less than the hundredthousandth part of one-fourth of a unit. Indicated roots of positive rational quantities which cannot be expressed by a finite number of terms are called irrational or surd quantities. Inasmuch as irrational numbers occur in investigations, it is well to know the rules by which they are combined. √4 × √9 = 2 × 3 = 6, Since a and b being real positive numbers, and n a positive integer.* *Real numbers include both rational and irrational. |