Questions: How much water should be added to 77 gallons of a 37% salt solution to get a 30% salt solution?

Transcript text: How much water should be added to 77 gallons of a $37 \%$ salt solution to get a $30 \%$ salt solution?

Solution

Solution Steps

To solve this problem, we need to determine the amount of water to add to dilute the salt solution from 37% to 30%. We can use the concept of concentration and the equation of mass balance. The initial amount of salt in the solution is 37% of 77 gallons. After adding water, the total volume will be 77 gallons plus the amount of water added, and the concentration should be 30%. We set up an equation to solve for the amount of water added.

Step 1: Calculate Initial Amount of Salt

The initial amount of salt in the solution can be calculated as follows: \[ \text{Initial Salt} = \text{Initial Volume} \times \text{Initial Concentration} = 77 \, \text{gallons} \times 0.37 = 28.49 \, \text{gallons of salt} \]

Step 2: Set Up the Equation for Final Concentration

After adding $ x $ gallons of water, the total volume becomes $ 77 + x $ gallons. The final concentration of salt should be 30%, which gives us the equation: \[ \text{Initial Salt} = (77 + x) \times 0.30 \]

Step 3: Solve for $ x $

Substituting the value of the initial salt into the equation: \[ 28.49 = (77 + x) \times 0.30 \] Rearranging the equation to solve for $ x $: \[ x = \frac{28.49}{0.30} - 77 \] Calculating this gives: \[ x \approx 17.9667 \]