Problem 56 - Algebra for College Students

Question

A mixture of acid and water is \(35 \%\) acid. If the mixture contains a total of \(40 \mathrm{~L},\) how many liters of pure acid are in the mixture? How many liters of pure water are in the mixture?

Step-by-Step Solution

Answer

14 liters of pure acid and 26 liters of pure water.

Step 1 - Identify given information

The mixture is 35% acid and the total volume of the mixture is 40 liters.

Step 2 - Calculate the volume of pure acid

To find the volume of pure acid, multiply the total volume of the mixture by the percentage of acid: Volume of pure acid = 35% of 40 liters = 0.35 * 40.

Step 3 - Perform the calculation for pure acid

Now multiply: 0.35 * 40 = 14 liters.So, there are 14 liters of pure acid in the mixture.

Step 4 - Calculate the volume of pure water

Since the total volume of the mixture is 40 liters and there are 14 liters of acid, the remaining volume must be water.Volume of pure water = Total volume - Volume of pure acid = 40 liters - 14 liters.

Step 5 - Perform the calculation for pure water

The remaining volume of water is: 40 liters - 14 liters = 26 liters.So, there are 26 liters of pure water in the mixture.

Key Concepts

percentage calculations

Understanding percentage calculations is essential in many real-world problems, like determining the concentration of a mixture. In our example, we have a mixture that is 35% acid and a total volume of 40 liters. To find out how much of that mixture is pure acid, you multiply the percentage by the total volume. Use the formula \( \text{Volume of pure acid} = \frac{\text{Percentage}}{100} \times \text{Total Volume} \). In the given exercise: \( \text{Volume of pure acid} = 0.35 \times 40 = 14 \text{ liters} \). This calculation shows that out of the 40 liters mixture, 14 liters is pure acid.

volume calculation

Volume calculation is determining how much space a substance occupies. For instance, knowing one component's volume helps us calculate the remaining mixture's volume. In our exercise, once we found there are 14 liters of pure acid in a 40-liter mixture, we can easily find the volume of pure water by subtracting the acid volume from the total mixture volume: \( \text{Volume of pure water} = \text{Total Volume} - \text{Volume of pure acid} \). Thus: \( 40 - 14 = 26 \text{ liters of pure water} \). Hence, 14 liters of acid and 26 liters of water make up the 40 liters mixture.

mixture problems in algebra

Mixture problems in algebra involve combining different quantities to form a mixture and often require solving for unknowns. You start by identifying known values: the percentage and total volume. You set up an equation reflecting these values. For example, if solution A has 35% acid in a total volume of 40 liters, you identify that 35% translates to 0.35 in decimal form. Then you multiply 0.35 by the volume to find the amount of pure substance, demonstrating the calculation clearly: \( 0.35 \times 40 = 14 \text{ liters} \). Subtract this from the total volume to find other components. With consistent practice, mixture problems become less intimidating and more intuitive.