You received a $150 gift certificate to a clothing store. The store sells T-shirts for $17 and dress shirts for $26. You want to spend no more than the amount of the gift certificate. You want to leave at most $8 of the gift certificate unspent. You need at least two dress shirts. What are all of the possible combinations of T-shirts and dress shirts you could buy?

Question

You received a ​$150 gift certificate to a clothing store. The store sells​ T-shirts for ​$17 and dress shirts for ​$26. You want to spend no more than the amount of the gift certificate. You want to leave at most ​$8 of the gift certificate unspent. You need at least  two dress shirts. What are all of the possible combinations of​ T-shirts and dress shirts you could​ buy?

Bot GPT 3.5 · Answer

Let's solve this problem step by step: 1) We know that we want to spend no more than the amount of the gift certificate, which is $150. Therefore, we can form a basic equation: 17T + 26D ≤ 150, where T represents the number of T-shirts and D represents the number of dress shirts. 2) We also want to leave at most $8 of the gift certificate unspent. This means that 17T + 26D ≥ 142. 3) Since we need at least two dress shirts, the minimum value for D is 2. 4) Now we can start determining the combinations: a) Let's start by plugging in D = 2 into the equation: 17T + 26(2) ≤ 150 Simplifying this, we get: 17T + 52 ≤ 150 Subtracting 52 from both sides: 17T ≤ 98 Dividing by 17: T ≤ 5.764 Since the number of T-shirts must be a whole number, the maximum value for T in this case is 5. Therefore, we have the first combination: T-shirt = 5, Dress shirt = 2 b) Plugging in D = 3 into the equation: 17T + 26(3) ≤ 150 Simplifying this, we get: 17T + 78 ≤ 150 Subtracting 78 from both sides: 17T ≤ 72 Dividing by 17: T ≤ 4.235 The maximum value for T in this case is 4. So, we have the second combination: T-shirt = 4, Dress shirt = 3 c) Plugging in D = 4 into the equation: 17T + 26(4) ≤ 150 Simplifying this, we get: 17T + 104 ≤ 150 Subtracting 104 from both sides: 17T ≤ 46 Dividing by 17: T ≤ 2.764 The maximum value for T in this case is 2. So, we have the third combination: T-shirt = 2, Dress shirt = 4 d) Plugging in D = 5 into the equation: 17T + 26(5) ≤ 150 Simplifying this, we get: 17T + 130 ≤ 150 Subtracting 130 from both sides: 17T ≤ 20 Dividing by 17: T ≤ 1.176 The maximum value for T in this case is 1. So, we have the fourth combination: T-shirt = 1, Dress shirt = 5 Therefore, all possible combinations of T-shirts and dress shirts that can be bought are: 1) T-shirt = 5, Dress shirt = 2 2) T-shirt = 4, Dress shirt = 3 3) T-shirt = 2, Dress shirt = 4 4) T-shirt = 1, Dress shirt = 5