Janet has n dollars, where n is an integer and 80 < n < 130, to buy either shirts, jackets, or coats for her clothing store. The wholesale cost of each shirt is $4, the wholesale cost of each jacket is $14, and the wholesale cost of each coat is $56. If she buys the greatest possible number of shirts, she will have $3 left over. If she buys the greatest possible number of jackets, she will have $9 left over. If she buys the greatest possible number of coats, she will have d dollars left over, where d is a positive integer.Quantity AdQuantity B51
Answer(s): C
We need n such that:n 3 (mod 4)n 9 (mod 14)80 < n < 130Work with the second congruence:n 9 (mod 14) possible values: 9, 23, 37, 51, 65, 79, 93, 107, 121, 135...Restrict to 80 < n < 130 possible values: 93, 107, 121Check against first congruence (n 3 mod 4):93 mod 4 = 1 not valid107 mod 4 = 3 valid121 mod 4 = 1 not validSo, n = 107Compute remainder when buying coatsCoat price = 56n = 107Greatest number of coats: floor(107 / 56) = 1 coatRemainder: 107 - 56 = 51So d = 51
k is an odd positive integer.Quantity AThe median of the first k even positive integersQuantity BThe median of the first 2k + 1 positive integers
The first k even integers are 2, 4, 6, ..., 2kSince k is odd, the middle one isThat equals:So Quantity A = k + 1First 2k+1 positive integers are 1, 2, 3, ..., 2k + 1Since there are 2k + 1 terms (odd count), the median is the middle one:the (k + 1)-th integer.That equals: k + 1
In the xy-plane, the graph of y + 4x = 2 has y-intercept k and x-intercept n.Quantity Ak - nQuantity B2
Answer(s): B
Find the y-intercept (k)Set x = 0 in the equation:y + 4(0) = 2y = 2k = 2Find the x-intercept (n)Set y = 0 in the equation:0 + 4x = 2Quantity A is
Quantity AQuantity B
Quantity A:Quantity B:
Quantity AThe range of the ages of the 45 oldest studentsQuantity B12
Answer(s): A
We need to find the age range of the 45 oldest students. Let's start from the oldest age group and count upward:Ages 25-29: 11 studentsAges 20-24: 20 studentsAges 15-19: 43 studentsTo reach 45 oldest students:11 (25-29) + 20 (20-24) = 31We still need 14 more students; they must come from the 15-19 age group.Oldest students: in the 25-29 group oldest possible age = 29 Youngest students: from 15-19 groupBut only the oldest 14 students in this group are included. So their youngest possible age is 15, but more likely closer to 19. To be conservative (worst-case range), we assume the youngest of those 14 is 15.Then the range = 29 - 15 = 14
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