Calculate Ruth's Water Intake from a Proportional Equation Involving 5x-1 Liters

It is recommended that people drink a certain number of liters of water per day.

Ruth drinks half of the recommended amount.

Gabriel drinks 3 times the recommended amount.

Leah drinks 7 liters daily.

If the three of them drink 5 times the recommended amount minus 1 liter in total, then how many liters per day does Ruth drink?

To solve this problem, we'll follow these steps:

• Step 1: Define variables based on given information.
• Step 2: Formulate an equation for the total water consumption.
• Step 3: Solve the equation for the recommended amount ($x$).
• Step 4: Calculate Ruth's daily water intake.

Now, let's work through each step:
Step 1: Define the variables:
- Let $x$ be the recommended daily amount of water (in liters).
- Ruth drinks $\frac{x}{2}$ liters, Gabriel drinks $3x$ liters, and Leah drinks 7 liters.

Step 2: Set up the equation for total intake:
- According to the problem, the total amount of water they drink together is $5x - 1$ liters.

Step 3: Form the equation:
- $\frac{x}{2} + 3x + 7 = 5x - 1$

Solve the equation:
- Combine like terms:
$\frac{x}{2} + 3x = \frac{x}{2} + 6x$
Thus, $\frac{x}{2} + 6x + 7 = 5x - 1$

Multiply the whole equation by 2 to eliminate the fraction:
- $x + 12x + 14 = 10x - 2$
- Rearrange the equation:
$3x + 14 = -2$

Solving for $x$:
- Subtract $6x$ from both sides:
$1 - x + 7 = 0$
Thus, $x = 3$

Step 4: Calculate Ruth's water consumption:
- Ruth drinks $\frac{x}{2} = \frac{3}{2} = 1.5$ liters.

Therefore, Ruth drinks 1.5 liters of water daily.

The correct answer is 1.5 liters, which corresponds to choice ($3$).