Solving The Proportion 5/6 = (2x + 9)/9 A Step-by-Step Guide
Proportions, a fundamental concept in mathematics, play a crucial role in various fields, from basic arithmetic to advanced calculus. Understanding how to solve proportions is essential for tackling a wide range of mathematical problems. In this article, we will delve into the process of solving the proportion 5/6 = (2x + 9)/9, providing a step-by-step guide and explaining the underlying principles involved. By mastering the techniques discussed here, you will gain a solid foundation for solving more complex proportional equations in the future.
Solving proportions involves finding the value of an unknown variable that makes the two ratios equal. Proportions are mathematical statements that equate two ratios or fractions. A ratio represents the relationship between two quantities, expressed as a fraction. For example, the ratio 5/6 indicates that for every 5 units of one quantity, there are 6 units of another quantity. When two ratios are equal, they form a proportion. In the proportion 5/6 = (2x + 9)/9, we have two ratios set equal to each other. The goal is to find the value of 'x' that makes this equation true. Proportions are widely used in various real-world applications, including scaling recipes, calculating distances on maps, and determining the concentration of solutions in chemistry. The ability to solve proportions accurately is a valuable skill in both academic and practical settings.
Step 1: Understanding the Proportion
Before diving into the solution, let's break down the given proportion: 5/6 = (2x + 9)/9. This equation states that the ratio of 5 to 6 is equivalent to the ratio of (2x + 9) to 9. Our objective is to isolate 'x' and determine its value. To effectively solve this proportion, we need to understand the concept of equivalent fractions and how cross-multiplication can help us eliminate the fractions and simplify the equation. Equivalent fractions represent the same value even though they have different numerators and denominators. For example, 1/2 and 2/4 are equivalent fractions. In a proportion, the two ratios are equivalent fractions. Cross-multiplication is a technique used to solve proportions by multiplying the numerator of one fraction by the denominator of the other fraction and setting the results equal to each other. This method effectively eliminates the fractions and transforms the proportion into a linear equation, which is easier to solve. By understanding these fundamental concepts, we can confidently approach the solution of the proportion.
The Significance of Proportions in Mathematics
Proportions are not just abstract mathematical concepts; they are powerful tools that help us understand relationships between quantities in various real-world scenarios. Understanding the significance of proportions is crucial for appreciating their applications in diverse fields. Proportions allow us to scale quantities up or down while maintaining the same relative relationship. For instance, if a recipe calls for 2 cups of flour to make 12 cookies, we can use proportions to determine how much flour is needed to make 36 cookies. Similarly, proportions are used in mapmaking to represent large distances on a smaller scale. The scale of a map indicates the ratio between the distance on the map and the corresponding distance on the ground. By using proportions, we can accurately calculate real-world distances from map measurements. In scientific experiments, proportions are used to determine the concentration of solutions. For example, if we dissolve 10 grams of salt in 100 ml of water, we can use proportions to calculate the concentration of the solution in grams per liter. These examples highlight the practical relevance of proportions in everyday life and various professional fields.
Step 2: Applying Cross-Multiplication
To solve the proportion 5/6 = (2x + 9)/9, we'll employ the method of cross-multiplication. This technique involves multiplying the numerator of the first fraction by the denominator of the second fraction, and vice versa. This process eliminates the fractions, making the equation easier to solve. Cross-multiplication is a valid method for solving proportions because it is based on the fundamental property of equality. If two ratios are equal, then their cross-products are also equal. This property can be derived from the basic principles of fraction manipulation. When we cross-multiply, we are essentially multiplying both sides of the equation by the product of the denominators. This operation maintains the equality of the equation while eliminating the fractions. In our case, cross-multiplying 5/6 = (2x + 9)/9 gives us 5 * 9 = 6 * (2x + 9). This step transforms the proportion into a linear equation, which we can then solve for 'x'. Cross-multiplication is a powerful tool for solving proportions, and it is widely used in various mathematical and scientific contexts. Its simplicity and effectiveness make it an essential technique for anyone working with ratios and proportions.
Performing the Cross-Multiplication
By cross-multiplying, we get: 5 * 9 = 6 * (2x + 9). This step is crucial as it transforms the proportion into a linear equation, which is easier to manipulate and solve. The cross-multiplication process involves multiplying the numerator of the first fraction by the denominator of the second fraction and setting it equal to the product of the denominator of the first fraction and the numerator of the second fraction. This technique effectively eliminates the fractions, allowing us to work with a simpler equation. In this specific case, we multiply 5 (the numerator of the first fraction) by 9 (the denominator of the second fraction), resulting in 45. Similarly, we multiply 6 (the denominator of the first fraction) by (2x + 9) (the numerator of the second fraction), which gives us 6(2x + 9). Setting these two products equal to each other, we obtain the equation 45 = 6(2x + 9). This equation is now in a linear form, meaning that the variable 'x' is not raised to any power greater than 1. Linear equations are generally easier to solve than equations involving higher powers of the variable. The next step involves simplifying this equation by distributing the 6 on the right side and then isolating 'x' to find its value.
Step 3: Simplifying the Equation
Now, let's simplify the equation 45 = 6 * (2x + 9). We begin by distributing the 6 on the right side of the equation. This involves multiplying 6 by both terms inside the parentheses: 2x and 9. Distributing the 6, we get: 45 = 12x + 54. The distributive property is a fundamental concept in algebra that allows us to simplify expressions involving parentheses. It states that a(b + c) = ab + ac, where a, b, and c are any numbers or variables. In our case, we are applying the distributive property to multiply 6 by the binomial (2x + 9). This gives us 6 * 2x = 12x and 6 * 9 = 54. So, the equation becomes 45 = 12x + 54. The next step is to isolate the term with 'x' (12x) on one side of the equation. To do this, we subtract 54 from both sides of the equation. This operation maintains the equality of the equation because we are performing the same operation on both sides. Subtracting 54 from both sides is a common technique used to solve linear equations. It helps us to group the terms with 'x' on one side and the constant terms on the other side, making it easier to isolate 'x'.
The Importance of the Distributive Property
The distributive property is a cornerstone of algebraic manipulation, enabling us to simplify expressions and equations effectively. Understanding the distributive property is essential for solving a wide range of mathematical problems, from basic algebra to more advanced calculus. The distributive property states that multiplying a single term by a sum or difference inside parentheses is the same as multiplying the term by each individual term inside the parentheses and then adding or subtracting the results. Mathematically, this is expressed as a(b + c) = ab + ac and a(b - c) = ab - ac. In the context of our equation, 45 = 6(2x + 9), the distributive property allows us to multiply 6 by both 2x and 9, resulting in 12x + 54. Without the distributive property, we would not be able to simplify the equation and isolate 'x'. The distributive property is also crucial in various other mathematical operations, such as factoring polynomials and simplifying complex algebraic expressions. It is a fundamental tool that every mathematics student should master. By understanding and applying the distributive property correctly, we can simplify equations and solve problems more efficiently.
Step 4: Isolating the Variable
To isolate 'x', we need to subtract 54 from both sides of the equation: 45 - 54 = 12x + 54 - 54. This simplifies to -9 = 12x. Isolating the variable is a crucial step in solving any algebraic equation. The goal is to get the variable by itself on one side of the equation, so we can determine its value. To isolate 'x' in our equation, we need to eliminate the constant term (54) on the right side. We do this by subtracting 54 from both sides of the equation. Subtracting the same value from both sides maintains the equality of the equation, which is a fundamental principle in algebra. When we subtract 54 from both sides, the 54 on the right side cancels out, leaving us with 12x. On the left side, 45 - 54 equals -9. So, the equation becomes -9 = 12x. This equation is now simpler, with the 'x' term isolated on one side. The next step is to divide both sides of the equation by the coefficient of 'x' (which is 12) to solve for 'x'.
Maintaining Balance in Equations
The principle of maintaining balance in equations is fundamental to solving algebraic problems. This principle states that any operation performed on one side of an equation must also be performed on the other side to preserve the equality. Understanding and applying this principle is crucial for correctly solving equations and avoiding errors. In the context of our equation, 45 - 54 = 12x + 54 - 54, we subtracted 54 from both sides to isolate the term with 'x'. By subtracting the same value from both sides, we ensured that the equation remained balanced. If we had subtracted 54 from only one side, the equation would no longer be true. Similarly, in the next step, we will divide both sides of the equation by 12 to solve for 'x'. This operation also maintains the balance of the equation. The principle of maintaining balance applies to all algebraic operations, including addition, subtraction, multiplication, and division. It is a cornerstone of algebraic manipulation and is essential for solving equations correctly. By consistently applying this principle, we can confidently solve even complex equations.
Step 5: Solving for x
Finally, to solve for 'x', we divide both sides of the equation -9 = 12x by 12: (-9) / 12 = (12x) / 12. This simplifies to x = -9/12. Solving for 'x' is the ultimate goal of solving the proportion. To isolate 'x' completely, we need to divide both sides of the equation by its coefficient, which is 12 in this case. Dividing both sides by 12 maintains the balance of the equation, as we are performing the same operation on both sides. When we divide -9 by 12, we get -9/12. This fraction can be simplified further by dividing both the numerator and the denominator by their greatest common divisor, which is 3. Dividing both -9 and 12 by 3, we get -3/4. On the right side of the equation, dividing 12x by 12 cancels out the 12, leaving us with just 'x'. So, the solution to the equation is x = -3/4. This value of 'x' makes the original proportion 5/6 = (2x + 9)/9 true. We can verify this by substituting -3/4 for 'x' in the original equation and checking if both sides are equal. Solving for 'x' is the final step in the process, and it provides us with the value that satisfies the proportion.
Simplifying Fractions
In this step, we simplify the fraction -9/12 to its simplest form. Simplifying fractions is an essential skill in mathematics, as it allows us to express numbers in their most concise and understandable form. A fraction is in its simplest form when the numerator and denominator have no common factors other than 1. To simplify a fraction, we find the greatest common divisor (GCD) of the numerator and denominator and divide both by the GCD. The greatest common divisor is the largest number that divides both the numerator and the denominator without leaving a remainder. In the case of -9/12, the GCD of 9 and 12 is 3. Dividing both the numerator (-9) and the denominator (12) by 3, we get -3/4. This fraction is in its simplest form because 3 and 4 have no common factors other than 1. Simplifying fractions makes it easier to compare and work with numbers. It also helps to avoid confusion and ensures that our answers are expressed in the most standard form. Mastering the skill of simplifying fractions is crucial for success in various mathematical topics.
Solution: x = -3/4
Therefore, the solution to the proportion 5/6 = (2x + 9)/9 is x = -3/4. This value of 'x' satisfies the original equation, making the two ratios equal. To verify our solution, we can substitute x = -3/4 back into the original proportion and check if both sides are equal. Substituting x = -3/4 into the equation 5/6 = (2x + 9)/9, we get:
5/6 = (2*(-3/4) + 9)/9
Simplifying the right side:
5/6 = (-3/2 + 9)/9
5/6 = (-3/2 + 18/2)/9
5/6 = (15/2)/9
5/6 = 15/18
Simplifying the fraction 15/18 by dividing both numerator and denominator by their greatest common divisor (3), we get:
5/6 = 5/6
Since both sides of the equation are equal, our solution x = -3/4 is correct. Verifying the solution is an important step in solving any equation. It helps to ensure that we have not made any errors in our calculations and that our answer is accurate. By substituting the solution back into the original equation, we can confirm that it satisfies the equation and makes it true. This gives us confidence in our answer and demonstrates our understanding of the problem-solving process.
Importance of Verification
Verifying the solution is a critical step in the problem-solving process, ensuring the accuracy and validity of the answer. It is a practice that should be adopted in all mathematical problem-solving scenarios. Verification involves substituting the obtained solution back into the original equation or problem statement to check if it satisfies the given conditions. This process helps to identify any errors that may have occurred during the solution process, such as mistakes in calculations or incorrect application of formulas. In the context of our proportion, we substituted x = -3/4 back into the original equation 5/6 = (2x + 9)/9 and verified that both sides were equal. This confirmed that our solution was correct. Verification is particularly important in complex problems where there are multiple steps involved, as errors can easily occur at any stage. By verifying the solution, we can catch these errors and correct them. Additionally, verification helps to build confidence in our problem-solving abilities. When we verify our solution and find that it is correct, we gain assurance in our understanding of the concepts and the problem-solving process. Therefore, verification should always be considered an integral part of solving mathematical problems.
Conclusion
In conclusion, we have successfully solved the proportion 5/6 = (2x + 9)/9 by applying the principles of cross-multiplication, simplification, and isolating the variable. The solution we found is x = -3/4. Throughout this process, we emphasized the importance of understanding the underlying mathematical concepts, such as equivalent fractions, the distributive property, and maintaining balance in equations. Mastering these concepts is crucial for solving not only proportions but also a wide range of algebraic problems. We also highlighted the significance of verification as a critical step in the problem-solving process. By substituting the solution back into the original equation, we confirmed its accuracy and gained confidence in our answer. Solving proportions is a fundamental skill in mathematics with numerous practical applications in various fields, including science, engineering, and finance. By mastering the techniques discussed in this article, you will be well-equipped to tackle more complex proportional equations and apply these skills in real-world scenarios. Continuous practice and a solid understanding of the underlying principles are key to success in mathematics.
The Broader Significance of Proportional Reasoning
Proportional reasoning is a fundamental skill that extends beyond solving mathematical equations. It is a crucial cognitive ability that enables us to understand relationships between quantities and make informed decisions in various aspects of life. Proportional reasoning involves understanding ratios, rates, and proportions, and applying these concepts to solve problems in diverse contexts. In everyday life, we use proportional reasoning when scaling recipes, calculating fuel efficiency, and comparing prices. For example, when doubling a recipe, we need to adjust the quantities of all the ingredients proportionally. Similarly, when comparing the prices of two products, we consider the price per unit to determine which is the better deal. Proportional reasoning is also essential in various professional fields, such as science, engineering, and finance. Scientists use proportions to analyze experimental data and make predictions. Engineers use proportions to design structures and systems. Financial analysts use proportions to assess investment opportunities and manage risk. Developing strong proportional reasoning skills can enhance our ability to make sound judgments and solve problems effectively in both personal and professional settings. It is a skill that is highly valued in today's data-driven world.
