Problem 9 - 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. $$\frac{2}{5}=\frac{4}{x}$$

Step-by-Step Solution

Answer

The missing term is \( 10 \).

Step 1: Recognize the Proportion Structure

The given problem is a proportion, which is an equation that states two ratios are equal. In this case, the proportion is \( \frac{2}{5} = \frac{4}{x} \).

Step 2: Cross-Multiply to Eliminate the Fractions

To solve for \( x \), cross-multiply the terms of the proportion. This means multiplying the numerator of one fraction by the denominator of the other fraction, and equate the two products. This results in:\[ 2 \times x = 5 \times 4 \] which simplifies to \( 2x = 20 \).

Step 3: Solve the Equation for x

Now solve the linear equation \( 2x = 20 \) by dividing both sides by 2 to isolate \( x \). This gives us:\[ x = \frac{20}{2} \] which simplifies to \( x = 10 \).

Step 4: Express the Answer in Lowest Terms

Since \( x = 10 \) is already in its simplest form, there are no further steps needed to simplify the fraction.

Key Concepts

Cross-Multiplication

Cross-multiplication is a powerful and simple method used to find the unknown variable in a proportion. A proportion is essentially two fractions set equal to each other. In the given exercise, it takes the form \( \frac{a}{b} = \frac{c}{x} \). To cross-multiply, follow these steps:

• Multiply the numerator of the first fraction by the denominator of the second fraction.
• Do the same for the other diagonal: multiply the denominator of the first fraction by the numerator of the second.
• Set these two products equal to each other.

After cross-multiplying, proportion equations become easier to handle, as it removes the fractions from the equation. Cross-multiplication relies on the properties of equality and helps us find solutions efficiently.
In the problem \( \frac{2}{5} = \frac{4}{x} \), we obtained \( 2x = 20 \) by cross-multiplying, paving the way to solve for \( x \).

Solving Linear Equations

Solving linear equations is a fundamental skill in algebra. A linear equation is one in which the highest power of the variable is 1. Once you've used cross-multiplication to simplify your proportion, you'll typically be left with a linear equation to solve.
For example, from the cross-multiplication step, you get a simple equation like \( 2x = 20 \). Solving this involves a few straightforward steps:

• Isolate the variable by performing inverse operations. This means doing the opposite of what is currently being done to the variable.
• In the equation \( 2x = 20 \), the operation is multiplication. The inverse operation is division. So we divide both sides by 2.

Performing these steps gives us \( x = 10 \). Solving linear equations often requires practicing these steps until they become intuitive.

Fractions in Lowest Terms

Writing fractions in their lowest terms ensures they are presented in the simplest form. A fraction is in its simplest form when the numerator and the denominator have no common factors other than 1. Simplification can make working with fractions easier and results more clear.
To reduce a fraction, follow these steps:

• Find the greatest common factor (GCF) of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCF.
• The result is the fraction in its simplest form.

In this exercise, we found that \( x = 10 \), which can be expressed as \( \frac{10}{1} \). The only factor common to 10 and 1 is 1 itself, so \( x = 10 \) is already in its simplest form. Writing answers in lowest terms tidies up our results and ensures accuracy in mathematical communication.