58. Using equations to solve problems. 1. How many degrees are there in the sum of the angles of any triangle? 2. If told the sum of two angles of a triangle how may the third angle be found? 3. If told one of the acute angles of a right triangle how may the other acute angle be found? 4. If told one of the equal angles of an isosceles triangle how may the other two angles be found? 5. If told a person's age 5 years ago how can you find his age now? How can you find how old he will be 15 years from now? 6. If told the length and the width of a rectangle how can you find its area? Its perimeter? Its diagonal? 7. What is the next integer greater than n? The next smaller than n? Give five consecutive integers having 7 for the first; having 7 for the middle one; having n for the first; having n for the last; having n for the middle one. Exercise 64 1. Twice a certain number is 10 more than the number. Find the number. 2. If 24 is taken from 3 times a certain number the result is 4 more than the number. Find the number. 3. The length of a rectangle is 2 inches more than its width. Its perimeter is 28 inches. Find its length and its width. 4. The length of a certain rectangle is 3 in. more than twice its width. Its perimeter is 6 in. more than 2 times its length. How long is it? 5. One of two children is 3 times as old as the other. In 2 years he will be just twice as old as the other. How old is each? 6. In 6 years a certain boy will be just twice as old as he was 4 years ago. How old is he? 7. The sum of 3 consecutive integers is 24. What are they? 12. Three times ZX is the supplement of 24°. How many degrees in <X? 13. Find two complementary angles whose difference is 20°. 14. Find two supplementary angles whose difference is 36°. 15. How many degrees in the angle which is double its complement? 16. How many degrees in the angle which is 10° more than of its supplement? 17. Find the number of degrees in each angle of the triangle of Figure 72. FIG. 73 -1560 18. Find the number of degrees in each angle of the triangle of Figure 73. 19. The vertical angle of an isosceles triangle is 24° less than the sum of the two equal angles. Find each angle of the triangle. 20. One of the acute angles of a right triangle is double the other. How many degrees in each angle of the triangle? 21. The supplement of a certain angle is 3 times its complement. How many degrees in the angle? 22. Write the equation stating that the supplement of a certain angle is double its complement. Solve it. 23. In Figure 74 find the value of x if C is a right angle. ZA is 3x and LMBN is 2x. 24. How many sides has a polygon the sum of whose interior angles is 540°? HINT. The formula for the sum of the interior angles of a polygon of n sides is S=(n-2)180°. B N M FIG. 74 25. How many sides has the polygon the sum of whose interior angles is 1800°? 26. The circumference of a circle is 6 feet longer than its radius. Find its radius. 27. If the radius of a certain circle were increased 2 inches the circumference would then be 36 inches. What is the radius of the circle? 28. The diameter of one circle is 3 times the diameter of a second circle. The circumference of the first is 16 feet more than the circumference of the second. Find the diameter of each. 59. Equations involving fractions. The pupil will need to make use of the following rule for multiplication of fractions which he has learned: To multiply a fraction by an integer multiply the numerator or divide the denominator by the integer. For example, 3x=4; 3x=3; 6x=5. What is the smallest number by which can be multiplied so that the product is an integer? What is the product? Answer the same questions concerning ; ; ; 25; 1000; Give a number by which can be multiplied so that the product is an integer, and give the product. Do the same х 5 m a a+b. 5+x for ; ; y m 5 a+b; 7 2 What is the smallest number by which both and can be multiplied so that the two products are integers? What are the products? Answer the same questions concerning 5 5 4 and; and } ; ; and }; 2 3x 5 and a. How are the numbers 5 found? 2x and 2; 1x and x; a 3 9 by which you multiplied We shall need to perform such multiplications in solving equations containing fractions. SOLUTION. To get an equation that contains no fractions we multiply both members of = 12 by 3. This gives x = х 3 = 36. Multiplying both members of an equation by a number so as to get an equation that does not contain fractions is called clearing the equation of fractions. |