Problem 61 - Fundamentals of Mathematics

Question

The ratio of chlorine to water in a solution is 2 to 7 . How many mL of water are in a solution that contains \(15 \mathrm{~mL}\) of chlorine?

Step-by-Step Solution

Answer

There are 52.5 mL of water in the solution.

Step 1: Understand the Problem

The problem gives us a ratio of chlorine to water as 2:7 and a specific amount of chlorine, 15 mL. We need to determine how many milliliters of water correspond to 15 mL of chlorine using this ratio.

Step 2: Set Up the Proportion

The given ratio can be written as \( \frac{2}{7} = \frac{15}{x} \), where \( x \) is the amount of water in mL. This equation represents the proportion of chlorine to water.

Step 3: Solve the Proportion

To solve for \( x \), we perform a cross multiplication. This gives us \( 2x = 7 \cdot 15 \). Simplifying the right-hand side, we have \( 2x = 105 \).

Step 4: Calculate the Amount of Water

Divide both sides of the equation by 2 to solve for \( x \). Thus, \( x = \frac{105}{2} = 52.5 \). Therefore, there are 52.5 mL of water in the solution.

Key Concepts

Understanding Ratios

Ratios are a way to compare two quantities. In our exercise, the ratio of chlorine to water is 2:7. This means for every 2 units of chlorine, there are 7 units of water. Ratios can provide a simple yet powerful method to understand the relationship between different quantities.
Think of them like a recipe. If a recipe calls for 2 cups of sugar and 7 cups of flour, the ratio of sugar to flour is 2:7. Whether you scale the amounts up or down, the ratio stays consistent.
Ratios can be represented in several forms, such as fractions (like \( \frac{2}{7} \)) or using a colon (2:7). This versatility makes them a useful tool in all sorts of mathematical and real-world contexts.

The Power of Cross Multiplication

Cross multiplication is a method to solve proportions, which are equations stating that two ratios are equal.
In our problem, the proportion is \( \frac{2}{7} = \frac{15}{x} \). By cross multiplying, we eliminate the fractions and make the equation easier to solve.
Here's how cross multiplication works: you multiply the denominator of one fraction by the numerator of the other fraction, and set that equal to the opposite product. So, for our equation, it results in \( 2x = 7 \times 15 \).
This simplifies to \( 2x = 105 \). Cross multiplication is a reliable way to clear proportions and find unknown values, especially when dealing with ratios in real-world problems.

Essentials of Mathematical Problem-Solving

Mathematical problem-solving requires a combination of strategies and logical thinking. In the context of ratios and proportions, it's important to approach problems step by step.
Start by fully understanding the problem and identifying what you know and what you need to find out. This sets a strong foundation for solving the problem effectively.
After understanding the problem, set up a proportion which represents the relationship described in the problem. Use known quantities and introduce variables for unknowns.
By using cross multiplication, you can solve the proportion to find the unknown. It's a technique that simplifies working with fractions and ratios, making the rest of the problem more straightforward.
Lastly, always double-check your solution to ensure accuracy and reasonableness. Rereading the problem helps verify that the solution fits the conditions described. Consistent practice with these steps improves proficiency in problem-solving across various mathematical topics.