Question

emma mixed up 6 lb of trail mix. she put the trail mix in bags that are \\(\frac{2}{3}\\) lb each. which equation represents the situation? a \\(\frac{2}{3}(6)=x\\) b \\(x - \frac{2}{3}=6\\) c \\(\frac{2}{3}x = 6\\) d \\(\frac{3}{2}x = 6\\)

Explanation:

Step1: Understand the problem

We have a total of 6 lb of trail mix. Each bag holds $\frac{2}{3}$ lb? Wait, no, looking at the options, let's re - evaluate. Wait, maybe the per - bag amount is $\frac{2}{3}$? Wait, no, the options have $\frac{2}{3}x = 6$? Wait, the problem is: A person has 6 lb of trail mix, puts it into bags that are $\frac{2}{3}$ lb each. Let $x$ be the number of bags. Then the total weight of the bags (number of bags times weight per bag) should equal the total weight of the trail mix. So, weight per bag $\times$ number of bags = total weight. So if each bag is $\frac{2}{3}$ lb, and there are $x$ bags, then $\frac{2}{3}x=6$. Wait, but looking at the options, option C (assuming the fraction is $\frac{2}{3}$) would be $\frac{2}{3}x = 6$. Let's check the options again. The options are:

A. $\frac{2}{3}(6)=x$

B. $x-\frac{2}{3}=6$

C. $\frac{2}{3}x = 6$

D. $\frac{3}{2}x = 6$

We know that:

Total weight of trail mix = weight per bag $\times$ number of bags.

Let $x$ be the number of bags. Weight per bag is $\frac{2}{3}$ lb, total weight is 6 lb. So, $\frac{2}{3}\times x=6$, which is $\frac{2}{3}x = 6$, which is option C.

Step2: Analyze each option

• Option A: $\frac{2}{3}(6)=x$ would be calculating the number of bags as $\frac{2}{3}$ of 6, which is incorrect. Because the total weight is 6, and we need to find the number of bags such that each bag is $\frac{2}{3}$ lb. So it's not $\frac{2}{3}$ of 6, but rather $\frac{2}{3}$ times the number of bags equals 6.
• Option B: $x-\frac{2}{3}=6$ is a subtraction equation, which has nothing to do with the relationship between the number of bags, weight per bag, and total weight.
• Option C: $\frac{2}{3}x = 6$: As we derived, weight per bag ($\frac{2}{3}$) times number of bags ($x$) equals total weight (6). This is correct.
• Option D: $\frac{3}{2}x = 6$: This would be if we had a different relationship, like if we were solving for $x$ in a different context, but it doesn't match the weight - per - bag and total - weight relationship here.

Answer:

C. $\frac{2}{3}x = 6$