Problem 1 - Technical Mathematics with Calculus

Question

Find the value of \(x.\) $$3: x=4: 6$$

Step-by-Step Solution

Answer

The value of \(x\) is 4.5 or \(4\frac{1}{2}\).

Step 1: Understand The Ratio Expression

The expression \(3 : x = 4 : 6\) represents two ratios that are equal. The first ratio compares the number 3 to the variable \(x\), and the second ratio compares the number 4 to the number 6.

Step 2: Set up the Proportion Equation

To find the value of \(x\), set up the proportion as a fraction equation: \(\frac{3}{x} = \frac{4}{6}\). This equation states that the two fractions are equal.

Step 3: Cross-Multiply

To solve the proportion, cross-multiply the numerators with the denominators across the equation to eliminate the fractions: \(3 \times 6 = 4 \times x\). This gives us \(18 = 4x\).

Step 4: Solve for x

Now, solve the equation for the variable \(x\). Divide both sides of the equation by 4: \(x = \frac{18}{4}\). Simplify the fraction to get the value of \(x\).

Step 5: Simplify the Result

After dividing \(18\) by \(4\), we find that \(x = \frac{18}{4} = 4.5\) or \(x = 4\frac{1}{2}\). Thus, the value of \(x\) is 4.5 or \(4\frac{1}{2}\).

Key Concepts

Cross-Multiplication

Cross-multiplication is a method used to solve proportions that involve two ratios or fractions. It consists of multiplying diagonally across the equality of the two ratios. When we have a proportion like \(\frac{a}{b} = \frac{c}{d}\), cross-multiplication involves multiplying \(a\) with \(d\) and \(b\) with \(c\), and setting the two products equal to each other, resulting in the equation \(ad = bc\). This method works because if two fractions are equal, their cross-products must also be equal.

For our exercise where we have the proportion \(3 : x = 4 : 6\), the cross-multiplication step involved multiplying 3 and 6 (the extremes) and then 4 and \(x\) (the means), leading to \(3 \times 6 = 4 \times x\) or \(18 = 4x\). By cross-multiplication, we remove the fractions, making it easier to solve for the unknown variable. This technique is particularly valuable for its simplicity and effectiveness in solving proportion problems.

Ratio and Proportion

Ratio and proportion are foundational concepts in mathematics that describe the relationship between two numbers or quantities. A ratio is a way to compare two quantities by using division, as in \(a:b\), which is the same as \(a/b\) or \(\frac{a}{b}\), where \(a\) and \(b\) are any two numbers. A proportion, on the other hand, states that two ratios are equal, as in \(\frac{a}{b} = \frac{c}{d}\), meaning the ratio of \(a\) to \(b\) is the same as the ratio of \(c\) to \(d\).

Understanding that a proportion is an equation that asserts the equality of two ratios allows us to use various algebraic methods to find an unknown quantity. In our exercise, the proportion \(3:x = 4:6\) indicates that the ratio of 3 to \(x\) is equivalent to the ratio of 4 to 6. To solve for \(x\), we translate these ratios into a fraction equation and then proceed to find the value of \(x\) that maintains the equality.

Fraction Equation

A fraction equation is an equation where the variables are included in fractions. In the context of solving proportions, we often convert ratios into fraction equations for easier manipulation. In a fraction equation like \(\frac{a}{b} = \frac{c}{d}\), we can solve for the unknown by finding a common denominator, cross-multiplying to eliminate fractions, or by using other algebraic operations.

In our original exercise, we transformed the proportion \(3 : x = 4 : 6\) into the fraction equation \(\frac{3}{x} = \frac{4}{6}\). This step simplifies the solving process because it provides a clear pathway to isolate the unknown variable \(x\). By cross-multiplying and then dividing to solve for \(x\), we apply these techniques to reach the final solution, highlighting the power of understanding and using fraction equations in solving mathematical problems.