Problem 5 - Technical Mathematics with Calculus

Question

Find the value of \(x.\) $$x:(14-x)=4: 3$$

Step-by-Step Solution

Answer

The value of \(x\) is 8.

Step 1: Set up the proportion

First, we need to set up the proportion as given in the question. The colon ':' represents the ratio, so we can rewrite it using fractions. The given proportion is \(x:(14-x) = 4:3\), which can be written as \(\frac{x}{14-x} = \frac{4}{3}\).

Step 2: Cross-multiply

Next, we cross multiply to remove the fractions and solve for \(x\). This gives us \(3x = 4(14 - x)\).

Step 3: Distribute and solve for x

Now, distribute the 4 on the right side of the equation: \(3x = 56 - 4x\). Then, add \(4x\) to both sides of the equation to get the \(x's\) on one side: \(3x + 4x = 56\). This simplifies to \(7x = 56\). Now, divide both sides by 7 to solve for \(x\): \(x = \frac{56}{7} = 8\).

Key Concepts

Cross-Multiplication

Cross-multiplication is a pivotal technique used to solve proportions, which are statements that two ratios are equivalent. To elucidate this concept, imagine you are holding a cross; on one side you have a fraction and on the other side, another fraction. The cross-multiplication method involves multiplying the numerator of one fraction by the denominator of the other fraction and doing the same with the remaining pair.

For instance, in the proportion \(\frac{a}{b} = \frac{c}{d}\) you would cross-multiply to get \(a \times d = b \times c\). Applying cross-multiplication makes proportions easier to handle because it eliminates the fractions, allowing you to solve for the unknown variable using basic algebra.

Solving Proportions

Solving proportions involves finding the value of a variable within two ratios that are set equal to each other. It's crucial to comprehend that a proportion denotes that two ratios are balanced, just as two sides of a scale. When a proportion is disrupted by an unknown variable, the goal is to isolate this variable to maintain the equilibrium.

The process usually begins by setting up two equivalent fractions, followed by cross-multiplication. Once the variable is isolated through algebraic manipulation, such as collecting like terms and isolating the variable on one side, the solution to the proportion can be readily determined. This process is not only about finding a numerical answer but also about understanding how to manipulate and balance ratios.

Algebraic Fractions

Algebraic fractions, akin to traditional fractions, comprise a numerator and a denominator, but with algebraic expressions. Mastering algebraic fractions is a cornerstone of algebra, especially when it comes to solving equations involving ratios.

Essential techniques for working with algebraic fractions include finding a common denominator when adding or subtracting them, simplifying them by canceling common factors, and cross-multiplication when dealing with proportions. It's important to keep in mind that the principles of regular fractions apply to algebraic fractions as well; for example, you cannot divide by zero, and the process of simplifying can often make complex problems much more manageable.