19. Two opposite angles are denoted respectively by 3x+12 and 5x-8. Find the angles. Suggestion: Make a sketch. Form the equation. Add 8 to both members. Subtract 3x from both. 20. John has 6 marbles more than Henry. Together they have 26 marbles. How many marbles has John? 21. A man who paid $8000 for his house wishes to sell it with a profit of $500. How much must he ask for the house, if he pays a commission of 3% of the selling price to a realestate agent for making the sale? 22. A boy had 3 times as much money as his brother. Having spent 22 cents he had as much as his brother. How much money did he have? Solve the following equations and check each result: 23. 6x-6=126-5x. Suggestion: Add 5x to both sides; then add 6 to both sides. 30. Ellen is 4 years younger than Mary. The sum of their ages is 20 years. How old is each? 159. Use of the multiplication axiom in solving fractional equations. The solution of equations containing fractions can often be simplified by using the multiplication axiom stated in §88. Thus by multiply ing every term of the equation t t 16 by the least Dividing both members of this equation by 8, t= 30. This result may be checked by substituting 15 for t in the original equation. This may be done as follows: 160. What every pupil should know and be able to do. At the end of this chapter the pupil is expected to be able: 1. To change per cents to decimal fractions and to common fractions. 2. To solve problems in percentage and interest by means of formulas. 3. To make graphs of equations of the form ax = b, a and b being whole numbers, decimal or common fractions. 4. To solve equations of the forms w+5=20; 5w+3=18; 2x-40-90; 8a-10=6a+20; 9x 5x 2 3 = 28. 5. The following principles and formulas should be known: a. If equal numbers are added to equal numbers, the sums are equal. (Addition axiom.) b. If equal numbers are subtracted from equal numbers, the remainders are equal. (Subtraction axiom.) 161. Typical problems and exercises. The following represent types of problems every pupil should be able to solve: 1. The winner of a 100-mile automobile race finished in 95 minutes, 18 seconds. How fast did he travel? 2. A man saved 20% of his income of $3500. How much did he save? 3. In a school of 480 pupils, 260 are girls. What per cent are girls? 4. In a test a pupil makes a score of 16 out of a possible score of 22. What is his grade? 5. Find the number which increased by 6% of itself gives 424. 6. Find the number 5% of which is 75. 7. Express as a common fraction, and as a decimal fraction, each of the following: 5%, 10%, 121%, 20%, 30%, 45%, 50%, 75%. 8. One of two complementary angles is 48° smaller than the other. Find the two angles. 9. Which is the better bargain so far as price reduction is concerned, a $55 suit at $38, or a $65 suit at $48? 10. Find the interest on $3850 at 61% for 2 years. 17. Make a graph of the equation p=4.6b. 18. Write a paper on one of the following topics: a. The value of the algebraic formula. b. The graph of a formula. c. The use of mathematics in business. d. The equation as a tool for solving problems. CHAPTER IX SUPPLEMENTARY ARITHMETICAL PRACTICE EXERCISES IN THE FUNDAMENTAL 162. Why practice is needed. In the first six grades you spent much time training yourself to become accurate and quick in arithmetical work. In the junior high school you must continue practice in arithmetic. Ordinarily this is supplied by the regular classroom work. Whenever you feel that more training is needed, you should practice working the exercises in this chapter, and thereby supplement the regular class work. Like the athlete who practices daily to keep physi cally fit, you must keep up practice in number work to preserve your skill in arithmetic. Lack of ability to perform arithmetical operations with ease and accuracy interferes seriously with all classroom work. When working the exercises place a sheet of paper under one line at a time and write your results on it. 246 |