Which scenario models the equation 2x=125−12x

Question

2
x
=
125
−
1
2
x
? Answer 1 or 2.
After two hours of driving at the same rate of miles per hour, the Garcia family is one-half hour short of reaching their vacation spot, which is 125 miles away.
A construction company has three barrels of cement. The two full barrels together weigh 125 pounds more than the third one, which is half full.

Answer 1

To solve the equation \(2x = 125 - \frac{1}{2}x\), we can analyze both scenarios to determine which one fits the equation better:

1. Scenario 1: After two hours of driving at the same rate of miles per hour, the Garcia family is one-half hour short of reaching their vacation spot, which is 125 miles away.

   In this case, let \(x\) represent the speed in miles per hour. The time they've already driven is 2 hours, and they are half an hour short, meaning they need a total of \(2.5\) hours for the trip. Therefore, the relationship can be expressed as:

   \[ 2x = 125 - x \times \frac{1}{2} \] Rearranging gives us: \[ 2x = 125 - 0.5x \] This simplifies to \(2.5x = 125\), which does not match the original equation.

2. Scenario 2: A construction company has three barrels of cement. The two full barrels together weigh 125 pounds more than the third one, which is half full.

   Let \(x\) be the weight of a full barrel of cement. Therefore, the weight of the two full barrels is \(2x\) and the weight of the half-full barrel is \(\frac{1}{2}x\). Thus, the equation representing this scenario is:

   \[ 2x = \frac{1}{2}x + 125 \] Rearranging gives us: \[ 2x - \frac{1}{2}x = 125 \] Which simplifies to: \[ \frac{3}{2}x = 125 \] Again, this does not exactly match the original equation but incorporates the right elements.

To determine which scenario better models the equation \(2x = 125 - \frac{1}{2}x\) and fits neatly without contradictions, we can reanalyze:

Scenario 1 combines distance, time, and rate. Scenario 2 combines weight and operational constraints on the barrels.

The closest matching roots and transformations lead us back to Scenario 2, as it begins similarly with the comparison between weights. Thus:

Answer: 2.

Answer 2

Its one

Answer 3

Thank you for your clarification. Let's break down the first scenario again to understand how it directly relates to the equation \(2x = 125 - \frac{1}{2}x\) properly:

Scenario 1: After two hours of driving at the same rate of miles per hour, the Garcia family is one-half hour short of reaching their vacation spot, which is 125 miles away.

In this scenario:

1. Let \(x\) represent the speed in miles per hour.
2. They drove for 2 hours and still have \(0.5\) hours left to reach their destination.
3. The total time needed to cover 125 miles is \(125 / x\).

The total time they have already driven plus the time left must equal the total required time to reach their destination, so we can set up the equation as follows:

\[ 2 + 0.5 = \frac{125}{x} \] This simplifies to: \[ 2.5 = \frac{125}{x} \] Multiplying both sides by \(x\) yields: \[ 2.5x = 125 \]

Now, we can isolate \(x\): \[ x = \frac{125}{2.5} = 50 \]

Now reevaluating \(2x = 125 - \frac{1}{2}x\), we can also manipulate the original equation in a check:

Given: \[ 2x + \frac{1}{2}x = 125 \]

Combining the \(x\) terms gives: \[ \frac{5}{2}x = 125 \] Therefore, \[ x = 50 \]

From this review, it indeed confirms that Scenario 1 models the relationship expressed in the equation \(2x = 125 - \frac{1}{2}x\) correctly.

So, you are correct! The answer is 1. Thank you for your patience!

Answer 4

Thank you as well

Answer 5

You're welcome! If you have any more questions or need further clarification on anything else, feel free to ask. I'm here to help!