Problem 7 - Technical Mathematics with Calculus

Question

Find the value of \(x.\) $$x: 6=(x+6): 10 \frac{1}{2}$$

Step-by-Step Solution

Answer

\(x = 8\)

Step 1 - Express The Proportion

First, express the given proportion as a fraction. That means, we write down \( \frac{x}{6} = \frac{x + 6}{10 \frac{1}{2}} \). Now, solve for \(x\).

Step 2 - Convert Mixed Number to Improper Fraction

Before we solve the proportion, we need to convert the mixed number \(10 \frac{1}{2}\) into an improper fraction. To do this, multiply the whole number \(10\) by the denominator \(2\), add the numerator \(1\) to this product, making the improper fraction \(\frac{21}{2}\). So the equation becomes \( \frac{x}{6} = \frac{x + 6}{\frac{21}{2}} \)

Step 3 - Cross Multiply

Cross multiplying the two fractions, we get \( x \cdot \frac{21}{2} = 6(x + 6) \)

Step 4 - Distribute and Simplify

Distributing on the right side of the equation, we get \( \frac{21}{2}x = 6x + 36 \)

Step 5 - Clear The Fraction

Multiply both sides of the equation by \(2\) to clear the fraction, resulting in \( 21x = 12x + 72 \)

Step 6 - Isolate \(x\)

Subtract \(12x\) from both sides to isolate \(x\) on one side: \((21x - 12x = 12x - 12x + 72) \Rightarrow (9x = 72)\)

Step 7 - Solve For \(x\)

Divide both sides by \(9\) to find the value of \(x\): \(x = \frac{72}{9} = 8\)

Key Concepts

Cross Multiplication

Cross multiplication is a method used to solve proportions, which are equations that set two fractions equal to each other. It involves multiplying the numerator of each fraction by the denominator of the other fraction. The process creates a straightforward means of finding unknown quantities in proportional relationships.

For example, when you have a proportion such as \( \frac{a}{b} = \frac{c}{d} \) and you want to find the value of one of the variables, cross multiplication results in the equation \( a \cdot d = b \cdot c \). This step eliminates the fractions and gives you a simpler equation to work with. From here, standard algebraic techniques can be applied to find the variable in question.

In the context of our original problem, the proportion \( \frac{x}{6} = \frac{x + 6}{10 \frac{1}{2}} \) was simplified using cross multiplication, which lead to the equation \( x \cdot \frac{21}{2} = 6(x + 6) \) after the conversion of the mixed number to an improper fraction.

Converting Mixed Numbers to Improper Fractions

When solving equations that include mixed numbers, it's essential to convert them to improper fractions to simplify the calculations. An improper fraction is a fraction where the numerator is greater than or equal to the denominator.

To convert a mixed number to an improper fraction, follow these steps:

• Multiply the whole number by the denominator of the fractional part.
• Add the result to the numerator of the fractional part.
• Write the sum over the original denominator to form the improper fraction.

For instance, take the mixed number \(10 \frac{1}{2}\). To convert it, multiply \(10\) by \(2\) to get \(20\), then add \(1\), the numerator of the fractional part, resulting in \(21\). The improper fraction is \(\frac{21}{2}\), which replaces the mixed number in the equation. This step is vital to simplifying the proportion and allowing the use of cross multiplication.

Isolating Variables

Isolating variables is a fundamental practice in solving algebraic equations. The goal is to manipulate the equation to get the unknown variable by itself on one side of the equation. Isolating the variable is a multi-step process that may involve simplifying expressions, distributing multiplication over addition, combining like terms, and using inverse operations to cancel out other terms.

Let's illustrate this with the equation obtained from our cross multiplication step: \( \frac{21}{2}x = 6x + 36 \). To isolate \(x\), we clear the fraction by multiplying all terms by \(2\), yielding \(21x = 12x + 72\). Next, we combine like terms by subtracting \(12x\) from both sides, which gives us \(9x = 72\). We are now close to isolating \(x\), we simply divide both sides by \(9\) to get \(x = 8\). By isolating \(x\), we've effectively solved the proportion for the unknown variable.