Ella has 0.5 lbs of sugar. How much water should she add to make the following concentrations? Tell Ella how much syrup she will have in each case. 20%, 25% and 1.5% syrup?
The correct answer and explanation is:
To solve this problem, we need to calculate how much water Ella should add to her 0.5 lbs of sugar to create syrups with concentrations of 20%, 25%, and 1.5%. The concentration is defined as: Concentration=Amount of sugarTotal amount of syrup×100\text{Concentration} = \frac{\text{Amount of sugar}}{\text{Total amount of syrup}} \times 100
Let’s denote:
- Sugar = 0.5 lbs
- Water = xx lbs
- Total syrup = 0.5+x0.5 + x lbs
1. For 20% Syrup
0.50.5+x=0.20\frac{0.5}{0.5 + x} = 0.20
Solving: 0.5=0.20(0.5+x)0.5=0.1+0.20×0.4=0.20xx=20.5 = 0.20(0.5 + x) \\ 0.5 = 0.1 + 0.20x \\ 0.4 = 0.20x \\ x = 2
✅ Water to add: 2 lbs
✅ Total syrup: 0.5 + 2 = 2.5 lbs
2. For 25% Syrup
0.50.5+x=0.25\frac{0.5}{0.5 + x} = 0.25
Solving: 0.5=0.25(0.5+x)0.5=0.125+0.25×0.375=0.25xx=1.50.5 = 0.25(0.5 + x) \\ 0.5 = 0.125 + 0.25x \\ 0.375 = 0.25x \\ x = 1.5
✅ Water to add: 1.5 lbs
✅ Total syrup: 0.5 + 1.5 = 2 lbs
3. For 1.5% Syrup
0.50.5+x=0.015\frac{0.5}{0.5 + x} = 0.015
Solving: 0.5=0.015(0.5+x)0.5=0.0075+0.015×0.4925=0.015xx=32.83(approx.)0.5 = 0.015(0.5 + x) \\ 0.5 = 0.0075 + 0.015x \\ 0.4925 = 0.015x \\ x = 32.83 (approx.)
✅ Water to add: ~32.83 lbs
✅ Total syrup: 0.5 + 32.83 ≈ 33.33 lbs
📘 Explanation (300 words)
Ella wants to prepare syrups with specific sugar concentrations. Concentration in a solution is calculated as the ratio of the solute (sugar) to the total solution (sugar + water), expressed as a percentage. Given that she has 0.5 lbs of sugar, the challenge is to determine how much water to add so that this sugar constitutes 20%, 25%, and 1.5% of the total syrup weight.
We set up the relationship: Concentration=SugarSugar+Water×100\text{Concentration} = \frac{\text{Sugar}}{\text{Sugar} + \text{Water}} \times 100
Rewriting this in terms of xx, the unknown weight of water, allows us to isolate and solve for xx in each case.
For a 20% solution, the sugar must make up 20% of the total syrup. Solving the equation gives us that Ella needs to add 2 lbs of water, resulting in 2.5 lbs of syrup.
For a 25% syrup, less water is needed—only 1.5 lbs—giving 2 lbs of total syrup.
The most dilute solution, 1.5%, requires a very large amount of water—approximately 32.83 lbs—to dilute 0.5 lbs of sugar to that very low percentage. This results in about 33.33 lbs of total syrup.
This exercise shows how small changes in concentration demand significantly different amounts of added water, especially for very dilute mixtures. Understanding this relationship helps Ella control the sweetness and consistency of her syrup for cooking, baking, or preservation purposes.