1.3 Problem-Solving Procedures 27 Notice in the proportion that tablespoons and gallons are placed in the same relative positions. Often, the unknown quantity is replaced with an x. The proportion may be written as follows and solved using cross-multiplication. x x x x x 16 2 3.5 16(3.5) 2 56 2 56 2 2 2 28 = = = = = Thus, 28 tbsp of salt must be used to make 3.5 gal of a brine solution. This answer seems reasonable, since we would expect to get an answer greater than 16 tbsp. b) To answer this question, we use the same procedure discussed in part (a). This time, we will use the information that a 12-lb turkey requires 16 tbsp of salt. The pounds may be placed in either the top or bottom of the fraction, as long as they are placed in the same relative position. ⎧ ⎨ ⎩ = 12 lb 16 tbsp 20 lb ? tbsp Now replace the question mark with an x and solve the proportion. x x x x x 12 16 20 12( ) 16(20) 12 320 12 12 320 12 26.67 = = = = ≈ Thus, about 26.67 tbsp of salt are needed to make enough brine solution for a 20-lb turkey. This answer is reasonable because we would expect the answer to be more than the 16 tbsp of salt required for a 12-lb turkey. 7 Cross-multiplication Divide both sides by 2 to solve for x. Given ratio d Other information given d Item to be found Cross-multiplication Divide both sides by 12 to solve for x. Now try Exercise 45 Most of the problems solved so far have been practical ones. Many people, however, enjoy solving brainteasers. One example of such a puzzle follows. A Chinese myth says that in about 2200 B.C. , a divine tortoise emerged from the Yellow River. On his back was a special diagram of numbers from which all mathematics was derived. The Chinese called this diagram Lo Shu. The Lo Shu diagram is the first known magic square. Arab traders brought the Chinese magic square to Europe during the Middle Ages, when the plague was killing millions of people. Magic squares were considered strong talismans against evil, and possession of a magic square was thought to ensure health and wealth. For more magic square problems, see Exercises 51–54 on page 33. RECREATIONAL MATH East Meets West: Magic Squares Example 7 Magic Squares A magic square is a square array of distinct numbers such that the numbers in all rows, columns, and diagonals have the same sum. Use the digits 1, 2, 3, 4, 5, 6, 7, 8, and 9 to construct a magic square. Solution The first step is to create a figure with nine cells, as in Fig. 1.2(a). We must place the nine numbers in the cells so that the same sum is obtained in each row, column, and diagonal. Common sense tells us that 7, 8, and 9 cannot be in the same row, column, or diagonal. We need some small and large numbers in the same row, column, and diagonal. To see a relationship, we list the numbers in order: 1 , 2, 3, 4, 5 , 6, 7, 8, 9
RkJQdWJsaXNoZXIy NjM5ODQ=