Problem 35 - Prealgebra

Question

Find the missing term in each of the following proportions. Set up each problem like the examples in this section. Write your answers as fractions in lowest terms. $$\frac{0.5}{x}=\frac{1.4}{0.7}$$

Step-by-Step Solution

Answer

The missing term is \( x = \frac{1}{4} \).

Step 1: Identify the Proportion

We are given the proportion \( \frac{0.5}{x} = \frac{1.4}{0.7} \). We need to find the value of \( x \) that makes this proportion true.

Step 2: Calculate the Right Side

Simplify the right side of the equation by dividing 1.4 by 0.7: \[ \frac{1.4}{0.7} = 2. \] Now, the equation becomes \( \frac{0.5}{x} = 2 \).

Step 3: Solve for x

To find \( x \), we need to solve the equation \( \frac{0.5}{x} = 2 \). Multiply both sides by \( x \) to get \( 0.5 = 2x \).

Step 4: Isolate x

Divide both sides by 2 to isolate \( x \): \[ \frac{0.5}{2} = x. \] Simplifying gives \( x = 0.25 \).

Step 5: Write the Answer as a Fraction

Since the problem asks for the answer in fraction form, express \( x = 0.25 \) as a fraction: \[ x = \frac{1}{4}. \]

Key Concepts

Solving Proportions

Solving proportions is all about finding an unknown value in a fraction-based equation where two ratios are set to be equivalent. In simple terms, a proportion compares two ratios or fractions. For example, in a proportion like \( \frac{a}{b} = \frac{c}{d} \), the task is often to find one missing number that makes the equation true.

The goal is to ensure both sides of the equation balance. This is achieved by identifying one of the portions that need to be solved and uncovering the missing number through various mathematical operations. Solving proportions usually involves:

• Finding common denominators.
• Applying basic operations like multiplication and division.
• Ensuring that the two ratios are equated to each other.

For proportion \( \frac{0.5}{x} = \frac{1.4}{0.7} \), we need to isolate \( x \). By solving \( x \), we find \( x = 0.25 \) and further convert it into a fraction to get the solution \( \frac{1}{4} \).

Simplifying Fractions

Simplifying fractions means reducing them to their lowest terms to make them simpler. This involves dividing both the numerator (top number) and the denominator (bottom number) by their greatest common factor until we can’t anymore.

For example, when simplifying the right side of our given problem, the fraction \( \frac{1.4}{0.7} \) is divided by both the numerator and the denominator by \( 0.7 \):

• \( 1.4 \div 0.7 = 2 \)
• \( 0.7 \div 0.7 = 1 \)

The result is \( 2 \), which effectively simplifies the initial fraction to \( 2 \). This reduction makes the equation a lot cleaner and easier to work with. The same principle applies when converting decimals to fractions and simplifying them, as seen when \( x \) was simplified from \( 0.25 \) to \( \frac{1}{4} \). Always aim to convert and simplify fractions for clearer and more accurate results.

Cross-Multiplication

Cross-multiplication is a straightforward method used to solve proportions, especially when one term is unknown. It means multiplying diagonally across the equal sign to get an equation involving integers rather than fractions.

Consider the proportion \( \frac{0.5}{x} = \frac{2}{1} \). For cross-multiplication:

• Multiply the numerator of the first fraction by the denominator of the second: \( 0.5 \times 1 \).
• Multiply the denominator of the first fraction by the numerator of the second: \( x \times 2 \).

This gives the equation \( 0.5 = 2x \). By focusing on the equation \( 0.5 = 2x \) and solving it, we get \( x = 0.25 \). Cross-multiplication makes it clear which operations to perform and is essential for equations like finding \( x \) when proportions are at play. It quickly and easily shifts the problem from dealing with fractions to simple algebraic equations.