Mrs. Johnson, wholoves to give parties, invited all her neighbors to a barbecue

Question

Mrs. Johnson, wholoves to give parties, invited all her neighbors to a barbecue lastnight.   The highlight of Mrs. Johnson’s parties isalways the delicious fruit punch. At her party last week, shemade a punch with 7-ounce cans of lemonade and 6-ounce cansof grape juice, using 10 cans in all. Last night Mrs.Johnson, using 3 more cans of grape juice than last week and8-ounce cans of orange juice instead of lemonade, made an ounceless punch than last week. She used whole number of cans oforange juice. How many? I need to know how many cans ofOrange juice used. Thanks for helping I got Two cans of OJ for theanswer is this what you got.

Explanation / Answer

Let x be the total number of lemonade cans and y the number ofgrape cans, in the first week. Let t be the number of ounces, alsoin the first week.Since each lemonade can contains 7 ounces,and each one of grape, 6 ounces, then 7x + 6y = t. The total number of cans is 10. Since x and y are integersthere are only 10 possibilities, each of which has associated atotal number of ounces. x      y      t 1      9      61 2      8      62 3      7      63 4      6      64 5      5      65 6      4      66 7      3      67 8      2      68 9      1      69 The next week, instead of x cans of lemonade, z cans of orangejuice were used, each one with 8 ounces. And by the conditionsgiven the following holds 6(y +3) + 8z = t -1. The difference of the new total number of ounces(t-1) and the new number of grape juice ounces6(y+3) must be a multiple of 8, so there exist an integernumber of orange juice cans. Here is the list of cases y +3       onces_grape         t-1                12               72                     60 11               66                     61 10               60                    62 9                 54                      63 8                  48                     64 7                  42                     65 6                  36                     66 5                  30                     67 4                  24                     68 The case in green is the only one in which the difference(t-1) - onces_grape is a multiple of 8. 64 -48 = 16. 16 = 8z z= 2 So there are 2 cans of orange juice. So there are 2 cans of orange juice.

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