481. 1. What is the ratio of 3 to 6? 5 to 10? 7 to 21 ? 2. What is the ratio of $3 to $10? 12 lb. to 6. lb.? 27 bush. to 9 bush.? 3. What is the ratio of 7 to 35? 24 to 48? 13 to 39 ? 4. If the antecedent be 20, and the consequent 15, what is the ratio ? 5. What is the ratio when the antecedent is 45, and the consequent 25.? 6. What is the ratio of to ? to a? to ? Fractions should be reduced to similar fractions. They will then have the ratio of their numerators. 7. What is the ratio of 54 to 31? 7} to 61? 93 to 51? 8. What ratio will the work of 12 men sustain to that of 8 men? 9. What will be the ratio of 8 yd. to 24 yd.? 6 yd. to 9 yd. ?" at w 10. When the antecedent is 3 and the ratio }, what is the consequent? 11. When the consequent is 8 and the ratio , what is the antecedent? 12. When the antecedent is į and the ratio \, what is the consequent? 13. What number has to 3 the ratio of 5 to 6? 16. If two numbers have the relation of 6 to 8, and the first is 12, what is the other ? 17. What number has to 12 the ratio of 8 to 9 ? 18. If two. numbers have the relation of 10 to 15, and the antecedent is 40, what is the consequent? TV PROPORTION 482. 1. What two numbers have the same relation to each other as 3 to 6? As 2 to 8? As 7 to 21 ? 2. What two numbers have the same ratio as 5 to 15? 6 to 30? 12 to 48? 121 to 25? 24 to 41? 121 to 50 ? 3. What number has the same relation to 6 that 3 has to 9? 4. What number has the same relation to 5 that 7 has to 14? 5. What number has the same relation to that 4 has to 8? 6. To what number has 5 the same relation that 3 has to 9? 7. To what number has 21 the same relation that 7 has to 21 ?, 8. 24 is to 7 as 12 is to what number? 9. 12 is to 5 as what number is to 15 ? 10. If the cost of 9 yards of cloth is $5, how will the cost of 18 yards compare with that suim? 11. If 10 men can earn $30 per day, what ratio will the earnings of 15 men bear to that sum ? 12. Write two equal ratios; multiply the first and last terms together; multiply the second and third terms together. How do the products compare? 13. Write two other equal ratios; multiply as before. How do the products compare? DEFINITIONS. 483. A Proportion is an equality of ratios. The double colon (::) may be regarded as the extremities of the sign of equality (=). It is written between the ratios. A proportion must have four terms, viz: two antecedents, and two consequents. Any four numbers that can be formed into a proportion are called proportionals. 485, The Antecedents of a proportion are the antecedents of the ratios, or the first and third terms. Thus, in the proportion 5:10:37:14, 5 and 7 are the untecedents. 486. The Consequents of a proportion are the consequents of the ratios, or the second and fourth terms. Thus, in the proportion 5:10 :: 7:14, the consequents are 10 and 14. 487, The Extremes of a proportion are the first and fourth terms. Thus, in the proportion 7:8::14:16, 7 and 16 are the extremes, 488. The Means of a proportion are the second and third terms. Thus, in the proportion 7:8::14:16, 8 and 14 are the means. 489. PRINCIPLES. -1. The product of the extremes is equal to the product of the means. 2. The product of the extremes divided by either mean gives the other mean. 3. The product of the means divided by either extreme gives the other extreme, EXERCISES.. Find the term that is wanting in the following: 1. 18 : 24 :: 6 :( ). 9. ( ) : 14 :: 16 : 35. 17. 5 men : 7 men :: 8.1 :( ). . SIMPLE PROPORTION. 490. A Simple Ratio is a ratio between any two numbers. Thus, 6:8, $10:$8, 5 lb. 6 oz.: 7 lb. 3 oz., are simple ratios. 491. A Simple Proportion is an equality between two simple ratios. 492. A Direct Proportion is one in which each term increases or diminishes, as the one on which it depends increases or diminishes. Thus, proportions involving quantity and cost, men and work done, etc., are direct proportions, for as the quantity increases or diminishes, the cost increases or diminishes, and as the number of men increases or diminishes, the amount of work done will increase or diminish. 493. An Inverse Proportion is one in which each term increases as the term upon which it depends diminishes, or diminishes as it increases. Thus, in the problem, "If 6 men can mow a field of grass in 9 days, how long will it take 9 men to mow it,” as the number of men increases, the number of days required to do the work decreases, and the proportion is an inverse proportion. WRITTEN EXERCISES. PROCESS. 494. 1. If 8 yd. of silk cost $24, what will 15 yd. cost? ANALYSIS.--It is eviyd. yd. dent that 8 yd. have the (1) 8 : 15 :: 24 : ( ) same relation to 15 yd. that the cost of 8 yd. has yd. yd. $ to the cost of 15 yd. Hence (2) 15:8:: ( ) : 24 we have the proportion, The term wanting (1) 15724 = $45 8 yd. : 15 yd. :: $24, the The term wanting (2) 15X24 = $45 cost of 8 yd. : the cost of 15 yd., or 15 yd. : 8 yd. :: the cost of 15 yd. : $24, the cost of 8 yd. To find the cost of 15 yards, the term wanting, we divide the product of the means by the extreme, as in (1); or the product of the extremes by the mean, as in (2). 2. If 5 men can cut a quantity of wood in 18 days, in how many days could 12 men do the same work? ANALYSIS It is evident men, men days. days. that exactly in proportion as (1) 5:12 :: 0 :18 the nunber of men is increased, the number of days required to men, men. days. days. do the work is diminished, and (2) 12:5 :: 18 : () therefore 5 men : 12 men :: the Term wanting = 1875 = 74 da. days it will require 12 men to do the work : 18 the number of days required for 5 men to do the work. Or, 12 men : 5 men :: :18 days, the number of days it requires 5 men to do the work : the number of days 12 men require to do the work. PROCESS. |