Question
Ella has 0.5 lb of sugar. How much water should she add to make the following concentrations? Tell Ella how much syrup she will have in each case.
Answers
GPT-4o mini
To determine how much water Ella should add to achieve different concentrations of syrup, we first need to understand the definition of concentration in a mixture. The concentration is usually expressed as a percentage, with the formula:
\[
\text{Concentration} (\%) = \left( \frac{\text{mass of solute}}{\text{mass of solution}} \right) \times 100
\]
Where:
- The mass of the solute is the mass of sugar (0.5 lb in this case).
- The mass of the solution is the sum of the mass of the solute and mass of the solvent (water).
Let's calculate how much water she needs to add for various concentrations, and then find out how much syrup (total solution) she will have in each case.
### Case 1: 5% Concentration
Set up the equation:
\[
5 = \left( \frac{0.5}{0.5 + w} \right) \times 100
\]
Where \( w \) is the weight of water in pounds. Rearranging gives:
\[
0.05(0.5 + w) = 0.5
\]
\[
0.025 + 0.05w = 0.5
\]
\[
0.05w = 0.5 - 0.025
\]
\[
0.05w = 0.475
\]
\[
w = \frac{0.475}{0.05} = 9.5 \text{ lb of water}
\]
Total mass of the syrup:
\[
0.5 + 9.5 = 10 \text{ lb of syrup}
\]
### Case 2: 10% Concentration
Set up the equation:
\[
10 = \left( \frac{0.5}{0.5 + w} \right) \times 100
\]
Rearranging gives:
\[
0.10(0.5 + w) = 0.5
\]
\[
0.05 + 0.10w = 0.5
\]
\[
0.10w = 0.5 - 0.05
\]
\[
0.10w = 0.45
\]
\[
w = \frac{0.45}{0.10} = 4.5 \text{ lb of water}
\]
Total mass of the syrup:
\[
0.5 + 4.5 = 5 \text{ lb of syrup}
\]
### Case 3: 15% Concentration
Set up the equation:
\[
15 = \left( \frac{0.5}{0.5 + w} \right) \times 100
\]
Rearranging gives:
\[
0.15(0.5 + w) = 0.5
\]
\[
0.075 + 0.15w = 0.5
\]
\[
0.15w = 0.5 - 0.075
\]
\[
0.15w = 0.425
\]
\[
w = \frac{0.425}{0.15} \approx 2.833 \text{ lb of water}
\]
Total mass of the syrup:
\[
0.5 + 2.833 \approx 3.333 \text{ lb of syrup}
\]
### Summary
- For **5% concentration**: Add **9.5 lb of water** → Total syrup: **10 lb**
- For **10% concentration**: Add **4.5 lb of water** → Total syrup: **5 lb**
- For **15% concentration**: Add **approximately 2.833 lb of water** → Total syrup: **approximately 3.333 lb**
Let me know if you need calculations for different concentrations or any further assistance!
\[
\text{Concentration} (\%) = \left( \frac{\text{mass of solute}}{\text{mass of solution}} \right) \times 100
\]
Where:
- The mass of the solute is the mass of sugar (0.5 lb in this case).
- The mass of the solution is the sum of the mass of the solute and mass of the solvent (water).
Let's calculate how much water she needs to add for various concentrations, and then find out how much syrup (total solution) she will have in each case.
### Case 1: 5% Concentration
Set up the equation:
\[
5 = \left( \frac{0.5}{0.5 + w} \right) \times 100
\]
Where \( w \) is the weight of water in pounds. Rearranging gives:
\[
0.05(0.5 + w) = 0.5
\]
\[
0.025 + 0.05w = 0.5
\]
\[
0.05w = 0.5 - 0.025
\]
\[
0.05w = 0.475
\]
\[
w = \frac{0.475}{0.05} = 9.5 \text{ lb of water}
\]
Total mass of the syrup:
\[
0.5 + 9.5 = 10 \text{ lb of syrup}
\]
### Case 2: 10% Concentration
Set up the equation:
\[
10 = \left( \frac{0.5}{0.5 + w} \right) \times 100
\]
Rearranging gives:
\[
0.10(0.5 + w) = 0.5
\]
\[
0.05 + 0.10w = 0.5
\]
\[
0.10w = 0.5 - 0.05
\]
\[
0.10w = 0.45
\]
\[
w = \frac{0.45}{0.10} = 4.5 \text{ lb of water}
\]
Total mass of the syrup:
\[
0.5 + 4.5 = 5 \text{ lb of syrup}
\]
### Case 3: 15% Concentration
Set up the equation:
\[
15 = \left( \frac{0.5}{0.5 + w} \right) \times 100
\]
Rearranging gives:
\[
0.15(0.5 + w) = 0.5
\]
\[
0.075 + 0.15w = 0.5
\]
\[
0.15w = 0.5 - 0.075
\]
\[
0.15w = 0.425
\]
\[
w = \frac{0.425}{0.15} \approx 2.833 \text{ lb of water}
\]
Total mass of the syrup:
\[
0.5 + 2.833 \approx 3.333 \text{ lb of syrup}
\]
### Summary
- For **5% concentration**: Add **9.5 lb of water** → Total syrup: **10 lb**
- For **10% concentration**: Add **4.5 lb of water** → Total syrup: **5 lb**
- For **15% concentration**: Add **approximately 2.833 lb of water** → Total syrup: **approximately 3.333 lb**
Let me know if you need calculations for different concentrations or any further assistance!