Short Answer 
The process of determining the amount of water needed for different syrup concentrations involves understanding the concept of concentration, where the percentage indicates the sugar-to-mixture ratio. For example, a 50% syrup requires 0.5 lbs of water, while a 5% syrup necessitates 9.5 lbs of water, with corresponding calculations provided for various concentrations.
Step 1: Understanding Syrup Concentration
To determine how much water Ella needs for various syrup concentrations, we need to first grasp the concept of *concentration*. The percentage indicates the amount of sugar in relation to the total mixture. For instance, in a 50% syrup, half is sugar, so we require an equal amount of water, which is 0.5 lbs.
Step 2: Calculating Water for Different Concentrations
Next, we will calculate the required water for each concentration. The formula used involves the total mass of the mixture and the given sugar weight. For each concentration:
- 50% syrup: 0.5 lbs of water
- 5% syrup: 9.5 lbs of water
- 75% syrup: 0.167 lbs of water
- 1.5% syrup: 32.83 lbs of water
Step 3: Applying the Calculation Method
To find the mass of water needed, apply the formula: *M + sugar weight = total mass*. For example, with the 5% syrup, the equation leads to solving for the mass of water (M) needed, which amounts to 9.5 lbs. Repeat this for each syrup concentration to find the necessary water amounts.