Question

How many liters of water must be evaporated from 50 liters of a 3 percent sugar solution to get a 5 percent sugar solution? $$\begin{array}{l} 2 \\ 4 \\ 6 \\ 10 \\ 20 \end{array}$$

Short answer

20 liters must be evaporated.

Step-by-step solution

- Understand the Problem

You start with 50 liters of a 3% sugar solution. This means the solution contains a certain amount of sugar and mostly water. The goal is to increase the concentration of sugar to 5% by evaporating some of the water.

- Find the Amount of Sugar

Calculate the amount of sugar in the initial solution. Since the solution is 3% sugar, we use the formula: \(\text{Amount of Sugar} = \frac{3}{100} \times 50 = 1.5 \) liters of sugar.

- Set Up the New Concentration

Let \( x \) be the amount of water that needs to be evaporated. The new volume of the solution will be \( 50 - x \) liters. To get a 5% sugar concentration, the equation is: \(\text{Amount of Sugar} = 0.05 \times (50 - x) \)

- Solve the Equation

We have 1.5 liters of sugar, and the equation is: \( 1.5 = 0.05 \times (50 - x) \). Solving for \( x \): \( 1.5 = 2.5 - 0.05x \) \( 1.5 - 2.5 = -0.05x \) \( -1 = -0.05x \) \( x = \frac{-1}{-0.05} = 20 \).

- Validate and Verify

Double-check the values to ensure accuracy: Evaporating 20 liters of water from 50 liters leaves 30 liters. The sugar concentration would be \( \frac{1.5}{30} \times 100 = 5 \)%, confirming the solution is correct.

Key concepts

percentage concentration

Imagine you have a mixture of sugar and water. The percentage concentration tells you how much of the sugar there is compared to the entire mixture.

For example, in a 3% sugar solution, 3% of the mixture is sugar, and the rest is water.

This is important for understanding how much sugar is present originally. Remember, to calculate how much actual sugar is in the mixture, you multiply the volume of the solution by the percentage of sugar (expressed as a decimal).

In the exercise, starting with 50 liters of a 3% solution means you have 1.5 liters of sugar and the rest is water.

evaporation

Evaporation is the process where water turns into vapor and leaves the solution. When water evaporates, the amount of water decreases but the amount of sugar stays the same.

In the given problem, you are told to evaporate a certain amount of water to increase the sugar concentration from 3% to 5%.

Let's visualize this: by removing water, you make the solution more concentrated because the amount of sugar remains constant, but there's less water around it. Think of it like squeezing a sponge - the liquid inside the sponge becomes more concentrated as the sponge gets smaller.

It's like having a smaller bowl of soup with the same number of noodles; the soup will seem more 'noodley'.

algebraic equations

Algebraic equations help us solve for unknown quantities using known values.

In our problem, the unknown quantity is how much water needs to be evaporated. Let's understand the given equation:

Firstly, set up the equation based on the concentration formula: the amount of sugar divided by the new volume of the solution.
In the problem, we start with \ 1.5 \ liters of sugar and need to achieve a 5% concentration. Let \ x \ be the amount of water evaporated, and the new volume be \ 50 - x \ liters.
The equation becomes:

\( \ 1.5 = 0.05 \times (50 - x) \ \).
Solve for \ x \: Simplify this to:
\( \ 1.5 = 2.5 - 0.05x \ \)
Rearrange and solve:
\( \ -1 = -0.05x \ \)
Thus, \ x = 20 \.
This tells us that 20 liters of water need to evaporate.