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. 75% concentration
Answers
GPT-4o mini
To prepare a syrup with a specific concentration, you start with the amount of solute (in this case, sugar) and figure out how much solvent (water) to add in order to achieve the desired concentration.
1. **Determine the total weight of the syrup needed for a 75% sugar concentration**:
Let \( x \) be the total weight of the syrup (this includes both sugar and water). The formula for concentration is given by the ratio of the mass of solute (sugar) to the total mass of the solution:
\[
\text{concentration} = \frac{\text{mass of solute}}{\text{total mass}}
\]
In this case, we want a 75% sugar concentration, so:
\[
0.75 = \frac{0.5 \text{ lb}}{x}
\]
Rearranging the equation to solve for \( x \):
\[
x = \frac{0.5 \text{ lb}}{0.75}
\]
\[
x = \frac{0.5}{0.75} = \frac{0.5 \times 4}{3} = \frac{2}{3} \text{ lb} \approx 0.6667 \text{ lb}
\]
2. **Calculate the amount of water to add**:
The total weight of the syrup (solution) is approximately \( 0.6667 \text{ lb} \). Since Ella has 0.5 lb of sugar, the amount of water to add is:
\[
\text{water} = x - \text{mass of sugar}
\]
\[
\text{water} = 0.6667 \text{ lb} - 0.5 \text{ lb} = 0.1667 \text{ lb}
\]
Therefore, to make a syrup with a 75% sugar concentration, Ella should add approximately \( 0.1667 \text{ lb} \) of water. The total amount of syrup (sugar + water) will be about \( 0.6667 \text{ lb} \).
1. **Determine the total weight of the syrup needed for a 75% sugar concentration**:
Let \( x \) be the total weight of the syrup (this includes both sugar and water). The formula for concentration is given by the ratio of the mass of solute (sugar) to the total mass of the solution:
\[
\text{concentration} = \frac{\text{mass of solute}}{\text{total mass}}
\]
In this case, we want a 75% sugar concentration, so:
\[
0.75 = \frac{0.5 \text{ lb}}{x}
\]
Rearranging the equation to solve for \( x \):
\[
x = \frac{0.5 \text{ lb}}{0.75}
\]
\[
x = \frac{0.5}{0.75} = \frac{0.5 \times 4}{3} = \frac{2}{3} \text{ lb} \approx 0.6667 \text{ lb}
\]
2. **Calculate the amount of water to add**:
The total weight of the syrup (solution) is approximately \( 0.6667 \text{ lb} \). Since Ella has 0.5 lb of sugar, the amount of water to add is:
\[
\text{water} = x - \text{mass of sugar}
\]
\[
\text{water} = 0.6667 \text{ lb} - 0.5 \text{ lb} = 0.1667 \text{ lb}
\]
Therefore, to make a syrup with a 75% sugar concentration, Ella should add approximately \( 0.1667 \text{ lb} \) of water. The total amount of syrup (sugar + water) will be about \( 0.6667 \text{ lb} \).