c. The quantity of each ingredient when the mixture is limited. 464. Given the quantity of each ingredient and the price of each to find the average price. 1. A grocer mixed sugars at different prices, as follows: 25 lbs. at 6 cents, 40 lbs. at 62 cents, 50 lbs. at 8 cents, 15 lbs. at 4 cents, 20 lbs. at 9 cents. At what price must he sell the mixture? Suggestion. - Find the cost of 25 lbs. at 6 cents. Find the cost of 40 lbs. at 6 cents. Find the cost of 15 lbs. at 4 cents. Find the cost of 20 lbs. at 9 cents. Then the cost of 150 lbs. was 1050 cents. 2. Find the average price of the following mixture : 10 lbs. of tea, at 70 cents, 20 lbs. at 80 cents, 35 lbs. at $1.10? 3. At what price per pound can a mixture of 20 lbs. of butter at 40 cents, 10 lbs. at 30 cents, 5 lbs. at 15 cents, and 2 lbs. at 10 cents, be sold so as to neither gain nor lose by the operation? 4. Find the average price of 100 lbs. of flour at 31 cents, 30 lbs. of meal at of a cent, and 15 lbs. of bran at of a cent? Ans. 21 cts. nearly. 465. Given the prices of the several quantities, and the average price, to find the ratio of the mixture. 1. In what ratio must sugars at 8 cents, 10 cents, 12 cents, and 15 cents a pound, be mixed to make the mixture worth 11 cents a pound? Av. price, 11 cents. OPERATION. On 1 lb. worth 8 cts., but sold at 11 cts., there is a gain of 3 cts. Hence, to gain only 1 cent of a pound must be taken. On 1 lb. worth 10 cts., but sold at 11 cts., there is a gain of 1 ct. Hence, to gain only 1 cent 1 pound must be taken. On 1 lb. worth 12 cts., but sold at 11 cts., there is a loss of 1 ct. Hence, to lose only 1 cent 1 pound must be taken. On 1 lb. worth 15 cts., but sold at 11 cts., Gain, 1 cent. Gain, 1 cent. Loss, 1 cent. Loss, there is a loss of 4 cts. Hence, to lose only 1 cent. 1 cent Thus, by taking of a lb. must be taken. of a pound at 8 cents, and 1 pound at 10 cents, there is a gain of 2 cents. But, by taking 1 pound at 12 cents, and of a pound at 15 cents, there is a loss of 2 cents. Now, any integers in the ratio of 1, 1, 1 and 1, will represent the relative number of pounds of each ingredient to be taken to make a mixture worth 11 cents a pound. We may take 4 lbs. at 8 cents, 1 lb. at 10 cents, and 3 pounds at 15 cents. Proof. -4 lbs. at 8 cents are worth 32 cents. Then, 9 pounds are worth......... 99 cents. And 1 pound is worth 11 cents, the average price. lb. at 12 cents, NOTE-This class of problems admits of a variety of solutions, each solution giving a different result, but all the results satisfying the conditions of the problem. 466. The principle involved in ART. 265 underlies ARTS. 467 and 468. 467. To find the quantity of each ingredient when the quantity of one is limited. 1. A grocer mixed 72 pounds of sugar at 12 cents, with sugars at 4, 5 and 9 cents a pound, so as to sell the mixture at 8 cents. How many pounds of each did he take? Suggestion. - By ART. 465 we find the ratio of the mixture to be as follows: 3 lbs. at 4 cts., 4 lbs. at 5 cts., 12 lbs. at 9 cents., and 3 lbs. at 12 cents. But, as there are 24 times 3 lbs. at 12 cts., there must be in the mixture 24 × 3, or 72 lbs. at 4 cts. ; 24 × 4, or 96 lbs. at 5 cts. ; 24 × 12, or 288 lbs. at 9 cts. NOTE. — Let the pupils find five other answers, and prove them. 468. To find the quantity of each ingredient when the quantity of the mixture is limited. 1. What are the respective quantities of sugar at 4, 8, 10 and 12 cents a pound, in a mixture of 200 pounds, worth 9 cents a pound? Suggestion.- Find, by ART. 465, the integral ratios. Then, by ART. 440, divide 200 in the ratio of these numbers. 469. A large class of problems may be solved by a judicious application of the principles involved in ARTS. 465, 466, 467, 468. They are, however, of little practical importance. MISCELLANEOUS PROBLEMS. 2. A man bought 200 bushels of grain at 90 cents, 150 at 70 cents, and 250 at $1. He wishes to pur chase a sufficient number of bushels at 50 cents, to enable him to sell the lot at an average price of 75 cents a bushel, and realize a profit of 25 %. How many bushels must he buy? Ans. 1750. 3. A speculator bought land at an average price per acre, of $75. His investment amounted to $18750. If the land cost him $60, $70, $80 and $85 an acre, how many acres at each of these prices did he buy? 4. At what price per bushel must a mixture of wheat consisting of 45 bushels worth 95 cts., 30 bushels worth $1, and 25 bushels worth $1.05, be sold to realize a profit of 5 per cent? POWERS AND ROOTS. 470. Finding any power of a number is called invo lution. 471. The first power of a number is the number itself. 472. The Second Power of a number is the Product found by taking the number twice as a factor. 473. The Third Power of a number is the product found by taking the number three times as a factor 474. Any Power of a number is the product found by taking that number as many times as a factor as there are units in the index of the power. 475. The index of a power is a figure placed at the right and a little above the number. Thus: 22 means the second power of 2, or 2 X 2. 33 means the third power of 3, or 3 × 3 × 3. 4' means the fourth power of 4, or 4 X 4 X 4 X 4. Numbers. 1, 2, 3, 4, 5, 6, 7, 8, 9. 81. Hence, the square, or second power of any number consisting of one figure, cannot consist of more than two figures. Numbers. 10, 20, 30, 40, 50, 60, 70, 80, 99. Squares. 100, 400, 900, 1600, 2500, 3600, 4900, 6400, 9801. Hence, the square of a number consisting of two figures cannot consist of more than four figures. 476. To involve, or raise a number to any power consists in taking the number a given number of times as a factor. 477. The second power is called the square. 478. The third power is called the cube. EXERCISES. 1. Find the first five powers of 2; of 3; of 4. 2. Find the first five powers of 5; of 6; of 7. 3. Find the third power of 8; the fourth power of 9. 4. Find the fifth power of 10; the sixth power of 11. Find the powers indicated below: 5. (12)3, (13)*, (14)3, (15)*, (16)2, (17)3, (18)2, (19)2, (20)2. 6. Find the square of 21, 23, 25, 27, 29. 7. Find the cube of 22, 24, 26, 28, 30. |