Problem 46 - Fundamentals of Mathematics

Question

The ratio of acid to water in a solution is \(\frac{3}{8}\). How many \(\mathrm{mL}\) of acid are there in a solution that contain \(112 \mathrm{~mL}\) of water?

Step-by-Step Solution

Answer

There are 42 mL of acid in the solution.

Step 1: Understanding the Ratio

The given ratio of acid to water in the solution is \( \frac{3}{8} \). This means for every 3 parts of acid, there are 8 parts of water.

Step 2: Setting the Equation for Water

Let \( x \) be the amount of acid in the solution (in mL). The ratio equation can be expressed as \( \frac{x}{112} = \frac{3}{8} \), where 112 is the amount of water in mL.

Step 3: Solve for the Acid

Cross-multiply to solve for \( x \): \( 8x = 3 \times 112 \). This gives \( 8x = 336 \).

Step 4: Calculate the Amount of Acid

Divide both sides of the equation by 8 to solve for \( x \): \( x = \frac{336}{8} \). Thus, \( x = 42 \).

Step 5: Conclusion

There are 42 mL of acid in the solution.

Key Concepts

Understanding Acid-Water Solutions

An acid-water solution is a mixture where acid and water are combined in specific proportions. Here, we encounter a problem where the concentration of acid and water is given as a ratio. This ratio reveals how much acid is present relative to the amount of water. For our exercise, the ratio is \( \frac{3}{8} \). This means that for every 3 units of acid, there are 8 units of water.

Understanding the context of this ratio lets us determine how much acid is in any given volume of water. This is crucial when solving problems that require us to calculate and maintain the acidity of a solution. Knowing the acid-water ratio helps us maintain the desired chemical properties of the solution.

Cross-Multiplication Method

Cross-multiplication is a powerful technique used to solve proportions. When we have two ratios set equal to each other, like \( \frac{3}{8} = \frac{x}{112} \), cross-multiplication allows us to transform this equation into a more solvable form.

• Take the numerator of the first ratio (3) and multiply it by the denominator of the second ratio (112).
• Next, take the denominator of the first ratio (8) and multiply it by the numerator of the second ratio (\( x \)).
• The cross-multiplied equation is: \( 8x = 3 \times 112 \).

This technique helps to simplify and solve the equation for unknown variables systematically. It enables us to swiftly find the value of \( x \), representing the amount of acid in milliliters.

Solving Equations

Once we apply cross-multiplication in our equation, we move to solving it by isolating the unknown variable, typically represented as \( x \). Starting with the equation \( 8x = 336 \), our goal is to find \( x \).

• To isolate \( x \), divide both sides of the equation by 8.
• This simplifies the equation to: \( x = \frac{336}{8} \).

Performing the division gives us \( x = 42 \). Therefore, the calculated amount of acid is 42mL. The solution highlights the importance of understanding algebraic manipulation, a vital skill in mathematics that helps solve real-world problems efficiently.

Understanding Proportions

Proportions are central to many mathematical problems involving ratios and comparisons. They inform us how one quantity relates to another. In our specific case, the proportion \( \frac{3}{8} = \frac{x}{112} \) tells us that the relationship between acid and water must be maintained regardless of how much solution we have.

• The goal is to keep the ratio consistent throughout, which allows us to accurately determine quantities based on given information.
• By setting up the proportion correctly, we can apply mathematical techniques like cross-multiplication to find unknowns systematically.

Grasping the concept of proportions aids in understanding how different ratios operate, whether in chemistry, physics, cooking, or finance, where precise calculations play a critical role.