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? 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?

Explanation / Answer

First time: x = the number of cans of lemonade used to make thefruit punch y = the number of cans of grape juice used to make thefruit punch x + y = 10 z = the amount of fruit punch made, measured inounces z = 7x + 6y x + y = 10 => y = 10 - x => z = 7x + 6(10 - x) => z = 7x + 60 - 6x => z = x + 60 Second time: w = the number of cans of orange juice used to make thefruit punch y + 3 = the number of cans ofgrape juice used to make the fruit punch w + (y + 3) = 10 z - 1 = the amount offruit punch made, measured in ounces z - 1 = 8w + 6(y + 3) w + (y + 3) = 10 => w = 7 - y => z - 1 = 8(7 - y) + 6y + 18 => z - 1 = 56 - 8y + 6y + 18 => z - 1 = 74 - 2y => z = 75 - 2y z = x + 60,    y = 10 - x => x + 60 = 75 - 2(10 - x) => x + 60 = 75 - 20 + 2x => x = 60 - 75 + 20 => x = 5 y = 10 - x = 10 - 5 = 5 w = 7 - y = 7 - 5 = 2 z = x + 60 = 5 + 60 = 65 Check: x + y = 5 + 5 = 10 z = 7x + 6y = 7(5) + 6(5) = 35 + 30 = 65 z - 1 = 65 - 1 = 64 8w + 6(y + 3) = 8(2) + 6(5 +3) = 16 + 48 = 64

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