Problem 36 - 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. $$\frac{0.3}{x}=\frac{2.4}{0.8}$$

Step-by-Step Solution

Answer

The missing term \( x \) is \( \frac{1}{10} \).

Step 1: Setting Up the Proportion Equation

We have the proportion \( \frac{0.3}{x} = \frac{2.4}{0.8} \). The goal is to find the value of \( x \) that makes this equation true.

Step 2: Cross Multiply to Eliminate Fractions

Cross-multiply to solve for \( x \). Multiply \( 0.3 \) by \( 0.8 \) and \( x \) by \( 2.4 \):\[ 0.3 \times 0.8 = x \times 2.4 \]

Step 3: Compute the Left Side Product

Calculate the product on the left side: \[ 0.3 \times 0.8 = 0.24 \] So our equation now is:\[ 0.24 = x \times 2.4 \]

Step 4: Isolate the Variable

To solve for \( x \), divide both sides by \( 2.4 \):\[ x = \frac{0.24}{2.4} \]

Step 5: Simplify the Division Fraction

Divide the numbers to simplify:\[ x = 0.1 \] Convert 0.1 to a fraction: \[ x = \frac{1}{10} \].

Step 6: Express in Lowest Terms

The fraction \( \frac{1}{10} \) is already in its lowest terms, so the solution is complete.

Key Concepts

Understanding Cross-Multiplication

Cross-multiplication is an essential technique used when solving proportions. A proportion is an equation that states two ratios are equal. To solve a proportion like \( \frac{0.3}{x} = \frac{2.4}{0.8} \), we use cross-multiplication to eliminate the fractions.

Here's how cross-multiplication works:

• Multiply the numerator of the first fraction by the denominator of the second fraction.
• Then multiply the numerator of the second fraction by the denominator of the first fraction.
• Set the two products equal to each other.

In our example, we cross-multiply to get:\[ 0.3 \times 0.8 = x \times 2.4 \]
By equalizing these products, we transform the fraction into a simple equation that can be solved for the unknown variable, \( x \). This removes the complication of dealing with fractions, making the equation much easier to handle.

Simplifying Fractions Effectively

Simplifying fractions involves reducing them to their lowest terms. This means expressing the fraction in its most basic form, where the numerator and denominator are as small as possible.

In the context of our equation, once we've solved for \( x \) using cross-multiplication and reached \( x = \frac{0.24}{2.4} \), we need to simplify this fraction. Reducing \( \frac{0.24}{2.4} \) involves carrying out the division:
\[ x = 0.1 \]
To convert 0.1 into a fraction, you express it with factors of 10 (since our decimal system is base-10):
\[ x = \frac{1}{10} \]
Now, \( \frac{1}{10} \) is already simplified because 1 and 10 have no common factors other than 1. Simplifying fractions ensures our answer is clear and easy to understand.

Solving Proportion Equations

The process of solving equations, especially those involving proportions, can be straightforward. Once the cross-multiplication step has transformed the fractions into a solvable equation, the next goal is to isolate the variable.

For the equation \( 0.24 = x \times 2.4 \), isolating \( x \) involves dividing both sides by 2.4. This helps us find the exact value of \( x \):
\[ x = \frac{0.24}{2.4} \]
Once simplified, we found \( x = 0.1 \), which we then expressed as a fraction \( \frac{1}{10} \) in its lowest terms. Isolating the variable step reveals the solution to the original proportion question.

• First, simplify any fractions if not already done.
• Use arithmetic operations to isolate the variable on one side of the equation.
• Finally, always check if the solution is simplified and accurate.

This holistic approach to solving equations ensures you always reach the most efficient and clear solution.