Problem 10 - Technical Mathematics with Calculus

Question

Insert the missing quantity. $$\frac{?}{4 x}=\frac{7}{16 x}$$

Step-by-Step Solution

Answer

The missing quantity is 7/4.

Step 1: Cross Multiply

In a proportion, the cross products are equal. Multiply 4x by 7 and 16x by the missing quantity, which we'll represent with a variable, say 'a'.

Step 2: Set Up the Equation

Set up the equation from the cross multiplication. 16x * a = 4x * 7.

Step 3: Solve for the Missing Quantity

Divide both sides of the equation by 16x to isolate 'a'. Remember that when you divide both sides of an equation by the same nonzero quantity, the equality is maintained.

Step 4: Simplify the Equation

Simplify the right side of the equation by canceling out common factors. 4x divided by 16x reduces to 1/4, and the x's cancel out because they are the same in both the numerator and denominator.

Step 5: Find the Missing Quantity

Multiply 1/4 by 7 to get the value of 'a'.

Key Concepts

Understanding Proportions

Proportions are mathematical expressions that state the equivalence of two fractions. Let's consider the equation \( \frac{a}{b} = \frac{c}{d} \), where \(a, b, c,\) and \(d\) represent quantities, and \(b\) and \(d\) are non-zero. To solve for a missing quantity in a proportion, one common method used is cross multiplication.

Cross multiplication involves multiplying the numerator of one fraction by the denominator of the other fraction, and vice versa. The products obtained are also known as cross products, and they are equal to each other if the two fractions form a true proportion. In other words, \( a \times d = b \times c \). This property is very useful because it provides a simple way to solve for a variable or to check if two ratios are equivalent.

Solving Equations with Cross Multiplication

When solving equations, particularly those that involve proportions, cross multiplication is a valuable tool. The step-by-step solution to the given exercise demonstrates how cross multiplication can be used to solve for a variable in an algebraic equation. After cross multiplying, we get a new equation that no longer has the fraction, making it easier to isolate the variable and solve.

To better grasp this process, let's break down the steps. First, identify the known variables and replace the unknown with \(a\). Next, cross multiply and formulate the new equation, which in our case was \(16x \times a = 4x \times 7\). Then, isolate the variable \(a\) by dividing both sides by \(16x\), maintaining the equality. Finally, simplify and solve for \(a\). Understanding each step's purpose aids in recognizing patterns in different equations, sharpening problem-solving skills.

Working with Algebraic Fractions

Algebraic fractions are fractions where the numerator, the denominator, or both contain algebraic expressions. In the given exercise, the algebraic fractions involved variables with 'x'. When simplifying or solving equations with algebraic fractions, it's crucial to recognize common factors and cancel them accordingly, as seen in step 4 of our solution.

Simplification of algebraic fractions is much like simplifying numerical fractions: divide out the common factors in the numerator and denominator. In the exercise, we had a common factor of \(x\) in both, which canceled out. This concept is not only essential for solving proportions but is also widely applicable across various algebraic problems. These operations with algebraic fractions are the foundation for more advanced mathematics, including polynomial division and integration of rational functions in calculus.