Question

ifede can use a limited number of spells to defeat trolls and goblins. it takes the same number of spells to defeat each troll, and it takes 3 spells to defeat each goblin. let ( t ) represent the number of trolls and ( g ) represent the number of goblins that ifede can defeat without running out of spells. 5t + 3g leq 85 according to the inequality, how many spells does it take to defeat each troll, and what is the maximum number of spells that ifede can use? it takes (\boxed{quad}) spells to defeat each troll, and ifede can use a maximum of (\boxed{quad}) spells.

Explanation:

Step1: Understand the problem

We know that the number of trolls and goblins is the same, so \( T = G \). The inequality is \( 5T + 3G \leq 85 \). Substitute \( G \) with \( T \) in the inequality.

Step2: Substitute and solve

Substitute \( G = T \) into \( 5T + 3G \leq 85 \), we get \( 5T + 3T \leq 85 \), which simplifies to \( 8T \leq 85 \). Then \( T \leq \frac{85}{8}=10.625 \). Since the number of trolls (and goblins) must be an integer, the maximum value of \( T \) (and \( G \)) is 10. Now, we need to find the number of spells per troll. Let \( x \) be the number of spells per troll (and per goblin, but the problem says "it takes the same number of spells to defeat each troll, and it takes 3 spells to defeat each goblin" – wait, no, re - reading: "It takes the same number of spells to defeat each troll, and it takes 3 spells to defeat each goblin." Wait, the inequality is \( 5T+3G\leq85 \), and \( T = G \) (same number of trolls and goblins). Wait, maybe I misread. Wait, the problem says: "It takes the same number of spells to defeat each troll, and it takes 3 spells to defeat each goblin." Wait, no, the inequality is \( 5T + 3G\leq85 \), and we need to find the number of spells per troll (let's call it \( x \)), so \( T=x \) (no, \( T \) is the number of trolls, \( x \) is spells per troll). Wait, no, let's re - express. Let \( x \) be the number of spells per troll (so \( 5T \) means 5 spells per troll times \( T \) trolls), and 3 spells per goblin times \( G \) goblins. And \( T = G \) (same number of trolls and goblins). So substitute \( G=T \) into \( 5T + 3T\leq85\Rightarrow8T\leq85\Rightarrow T\leq10.625 \). So maximum \( T = G = 10 \). Now, substitute \( T = 10 \) and \( G = 10 \) into \( 5T+3G \): \( 5\times10 + 3\times10=50 + 30 = 80\leq85 \). Now, we need to find \( x \) (spells per troll) such that \( 5x+3x\leq85 \) (since \( T = G \), let's say \( T = G = n \), then \( 5n+3n=(5 + 3)n=8n\leq85\Rightarrow n\leq10.625 \), \( n = 10 \)). Wait, no, the coefficient of \( T \) is 5, which is the number of spells per troll? Wait, the inequality is \( 5T+3G\leq85 \), so 5 is the number of spells per troll, 3 is the number of spells per goblin. Wait, the problem says "how many spells does it take to defeat each troll" – so 5 is the number of spells per troll? Wait, no, the inequality is \( 5T+3G\leq85 \), where \( T \) is the number of trolls, \( G \) is the number of goblins. So 5 spells per troll, 3 spells per goblin. And we need to find the number of spells per troll (5? No, that can't be. Wait, maybe the problem is: "It takes \( x \) spells to defeat each troll, and 3 spells to defeat each goblin. The inequality is \( xT+3G\leq85 \), and \( T = G \) (same number of trolls and goblins). We need to find \( x \) such that the maximum number of spells can be used. Wait, no, the original inequality is \( 5T+3G\leq85 \), so 5 is the number of spells per troll. Wait, the problem is asking "how many spells does it take to defeat each troll" – so the coefficient of \( T \) in the inequality is the number of spells per troll. Wait, that must be it. Wait, the inequality is \( 5T+3G\leq85 \), where 5 is spells per troll, 3 is spells per goblin. And \( T = G \) (same number of trolls and goblins). So we found that the maximum number of trolls (and goblins) is 10. Now, substitute \( T = 10 \) and \( G = 10 \) into \( 5T+3G \): \( 5\times10+3\times10 = 80 \). But we need to check the inequality \( 5T+3G\leq85 \). Now, if we want to find the number of spells per troll, let's assume that we use the maximum number of spells (85). Wait, no, the problem says "how many spells does it take to defeat each troll" – wait, maybe the inequality is \( 5T+3G\leq85 \), and \( T = G \), and we need to find 5 (the coefficient of \( T \))? No, that doesn't make sense. Wait, maybe I made a mistake. Let's start over.

Let \( x \) be the number of spells per troll (so the term for trolls is \( xT \)) and 3 spells per goblin (term \( 3G \)). The inequality is \( xT+3G\leq85 \), and \( T = G \) (same number of trolls and goblins). We need to find \( x \). Wait, but the inequality is given as \( 5T+3G\leq85 \), so \( x = 5 \)? No, that can't be. Wait, the problem says "how many spells does it take to defeat each troll" – so the answer is 5? Wait, no, let's re - read the problem:

"Ifede can use a limited number of spells to defeat trolls and goblins. It takes the same number of spells to defeat each troll, and it takes 3 spells to defeat each goblin. Let \( T \) represent the number of trolls and \( G \) represent the number of goblins that Ifede can defeat without running out of spells. \( 5T + 3G\leq85 \)

According to the inequality, how many spells does it take to defeat each troll, and what is the maximum number of spells that Ifede can use?

It takes \(\boxed{5}\) spells to defeat each troll, and Ifede can use a maximum of \(\boxed{85}\) spells. Wait, no, the second part: "It takes \(\underline{}\) spells to defeat each troll, and Ifede can use a maximum of \(\underline{}\) spells."

Wait, from the inequality \( 5T+3G \), the coefficient of \( T \) is 5, so it takes 5 spells to defeat each troll. To find the maximum number of spells, we need to consider the maximum \( T \) and \( G \) such that \( 5T + 3G\leq85 \) and \( T = G \) (same number of trolls and goblins). As we calculated before, \( T = G = 10 \), then \( 5\times10+3\times10 = 80 \), but if we take \( T = 10 \) and \( G = 10 \), the total spells used is 80. But the inequality is \( 5T+3G\leq85 \), the maximum value of \( 5T + 3G \) is 85 when \( 5T+3G = 85 \) and \( T = G \). Wait, solving \( 5T+3T=85\Rightarrow8T = 85\Rightarrow T=\frac{85}{8}=10.625 \), not an integer. So the closest integer values are \( T = 10 \), \( G = 10 \) (total spells 80) or \( T = 11 \), \( G = 11 \) (total spells \( 5\times11+3\times11=88>85 \), which is not allowed). So the maximum number of spells used is when \( T = 10 \), \( G = 10 \), total spells \( 5\times10 + 3\times10=80 \). But the problem says "what is the maximum number of spells that Ifede can use?" Wait, maybe the problem has a different interpretation. Maybe \( T \) is the number of spells per troll, and \( G \) is the number of spells per goblin? No, the variable definitions say "Let \( T \) represent the number of trolls and \( G \) represent the number of goblins".

Wait, going back to the problem's first blank: "It takes \(\underline{}\) spells to defeat each troll" – from the inequality \( 5T \), so 5 spells per troll. The second blank: "Ifede can use a maximum of \(\underline{}\) spells" – to find the maximum, we need to find the maximum \( 5T+3G \) subject to \( T = G \) (same number of trolls and goblins). As \( T = G \), \( 5T+3T = 8T\leq85\Rightarrow T\leq10.625 \), so \( T = 10 \), \( G = 10 \), then \( 5\times10+3\times10 = 80 \). But if we don't assume \( T = G \), and just find the maximum of \( 5T+3G \) (which is 85 when \( 5T+3G = 85 \)). But the problem says "it takes the same number of spells to defeat each troll, and it takes 3 spells to defeat each goblin" – so the number of trolls and goblins is the same? The problem statement: "It takes the same number of spells to defeat each troll, and it takes 3 spells to defeat each goblin. Let \( T \) represent the number of trolls and \( G \) represent the number of goblins that Ifede can defeat without running out of spells. \( 5T + 3G\leq85 \)"

So, the first blank: the coefficient of \( T \) is 5, so it takes 5 spells to defeat each troll.

The second blank: to find the maximum number of spells, we need to find the maximum value of \( 5T+3G \) such that \( T \) and \( G \) are non - negative integers and \( 5T+3G\leq85 \). To maximize \( 5T + 3G \), we can set \( 5T+3G = 85 \). We want to find non - negative integers \( T \) and \( G \) that satisfy this. Let's solve for \( T \): \( T=\frac{85 - 3G}{5}\). We want \( 85-3G \) to be divisible by 5. \( 85\) is divisible by 5, so \( 3G \) must be divisible by 5, so \( G \) must be a multiple of 5. Let \( G = 5k \), then \( T=\frac{85 - 15k}{5}=17 - 3k \). We want \( T\geq0 \) and \( G\geq0 \), so \( 17-3k\geq0\Rightarrow k\leq\frac{17}{3}\approx5.666 \). So \( k = 5 \), \( G = 25 \), \( T=17 - 15 = 2 \); \( k = 4 \), \( G = 20 \), \( T=17 - 12 = 5 \); \( k = 3 \), \( G = 15 \), \( T=17 - 9 = 8 \); \( k = 2 \), \( G = 10 \), \( T=17 - 6 = 11 \); \( k = 1 \), \( G = 5 \), \( T=17 - 3 = 14 \); \( k = 0 \), \( G = 0 \), \( T = 17 \). The maximum value of \( 5T+3G \) is 85 (when \( T = 17 \), \( G = 0 \) or \( T = 2 \), \( G = 25 \) etc.). But the problem says "it takes the same number of trolls and goblins" – wait, the original problem says "It takes the same number of spells to defeat each troll, and it takes 3 spells to defeat each goblin" – no, "It takes the same number of spells to defeat each troll, and it takes 3 spells to defeat each goblin" – no, the key part I missed: "It takes the same number of trolls and goblins" – no, "It takes the same number of spells to defeat each troll, and it takes 3 spells to defeat each goblin" – no, re - reading the original problem:

"Ifede can use a limited number of spells to defeat trolls and goblins. It takes the same number of spells to defeat each troll, and it takes 3 spells to defeat each goblin. Let \( T \) represent the number of trolls and \( G \) represent the number of goblins that Ifede can defeat without running out of spells. \( 5T + 3G\leq85 \)

According to the inequality, how many spells does it take to defeat each troll, and what is the maximum number of spells that Ifede can use?

It takes \(\underline{}\) spells to defeat each troll, and Ifede can use a maximum of \(\underline{}\) spells."

From the inequality \( 5T+3G \), the coefficient of \( T \) is 5, so it takes 5 spells to defeat each troll. To find the maximum number of spells, we consider the inequality \( 5T+3G\leq85 \). The maximum value of \( 5T + 3G \) is 85 (when \( 5T+3G = 85 \)).

Answer:

It takes \(\boxed{5}\) spells to defeat each troll, and Ifede can use a maximum of \(\boxed{85}\) spells.