Problem 6 - Technical Mathematics with Calculus

Question

Find the value of \(x.\) $$x: 12=(x-12): 3$$

Step-by-Step Solution

Answer

x = 16

Step 1: Understand the Problem

Identify what the exercise is asking, which is to find the value of x that makes the two ratios equal. The given equation is a proportion, where `x:12` is equivalent to `(x-12):3`. This can be interpreted as `x/12 = (x-12)/3`.

Step 2: Cross-Multiply

To solve the proportion, cross-multiply the numerators with the denominators from the opposite sides of the equals sign. Multiply x with 3 and 12 with (x-12), setting the products equal to each other to form the equation `3x = 12(x - 12)`.

Step 3: Distribute the 12

Distribute the 12 across the `(x - 12)` in the equation `3x = 12(x - 12)` to get `3x = 12x - 144`.

Step 4: Collect like terms

Move the terms involving x to one side of the equation, by subtracting `12x` from both sides to get `3x - 12x = -144`.

Step 5: Simplify the equation

Combine like terms `3x - 12x` to get `-9x = -144`.

Step 6: Solve for x

Divide both sides of the equation `-9x = -144` by -9 to isolate x, resulting in `x = (-144) / (-9)`.

Step 7: Simplify the solution

Simplify `-144 / -9` to find the value of x, which is `x = 16`.

Key Concepts

Cross-Multiplication

Cross-multiplication is a technique often employed to solve proportions, which are equations where two ratios are set equal to each other. The process involves multiplying across the equal sign in a crisscross manner. For example, to solve the proportion \( \frac{x}{12} = \frac{x-12}{3} \) from the given exercise, we multiply the numerator of one fraction by the denominator of the other fraction and vice versa. This results in \( 3 \cdot x = 12 \cdot (x-12) \), effectively removing the fractions and allowing us to solve the equation more easily.

In solving proportions, cross-multiplication is critical because it simplifies the problem into a standard linear equation. It is particularly useful when dealing with variables in both ratios, as it quickly clears the denominators and moves us towards isolating the variable.

Proportion Word Problems

Proportion word problems can sometimes stump students, but they needn't be tricky. These problems involve real-world scenarios where two ratios are compared, and you are typically required to find a missing value that maintains the proportion. To effectively tackle them, identify the two ratios involved and set them equal to each other. For instance, if a recipe requires 3 cups of flour to make 12 cookies, and we want to know how much flour is needed for 8 cookies, we set up the proportion \( \frac{3}{12} = \frac{x}{8} \) and then solve for \(x\) using cross-multiplication.

To avoid common mistakes, always ensure that the units in both ratios are consistent and that corresponding parts are in the same position in both fractions. This positions students to solve the problems with greater accuracy.

Isolating Variables

Isolating the variable, or solving for the unknown, is a fundamental skill in algebra. Once you've cross-multiplied the proportion, you're often left with a linear equation with variables on one or both sides. The goal is to get the variable on one side of the equation by itself. In the case of our exercise, after cross-multiplication and distribution, we worked with the equation \(3x = 12x - 144\).

To isolate \(x\), subtract \(12x\) from both sides to get \(3x - 12x = -144\), which simplifies to \( -9x = -144\). From there, dividing both sides by -9 gives us \( x = 16 \). Remember, whatever operation you do to one side of the equation, you must do to the other side to maintain equality. This step is crucial for solving any equation and finding the correct value for the variable.

Equation Simplification

Simplification of an equation is an essential step in solving it efficiently. Simplifying an equation involves combining like terms and reducing expressions to their simplest form. After cross-multiplying and isolating the variable, you often need to simplify to determine the solution. In our exercise, once we moved all the \(x\)-terms to one side, we had to combine like terms \(3x - 12x\) to simplify the equation to \( -9x = -144\). The final simplification step is to divide both sides by -9 to solve for \(x\), yielding \( x = 16 \).

It is important to perform each operation carefully to avoid errors. Simplification can include distributing multiplication over addition and subtraction, combining like terms, and reducing fractions to their lowest terms. Mastering this process helps ensure that you reach the correct answer efficiently and with a clear understanding of the equation's structure.