Problem 27 - 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. $$\frac{x}{10}=\frac{10}{2}$$

Step-by-Step Solution

Answer

The missing term is \( \frac{50}{1} \).

Step 1: Cross-Multiply to Set Up Equation

We start by using the property of proportions: if \( \frac{a}{b} = \frac{c}{d} \), then \( a \times d = b \times c \). Here, \( a = x \), \( b = 10 \), \( c = 10 \), and \( d = 2 \). Thus, cross-multiplying gives us: \( x \times 2 = 10 \times 10 \).

Step 2: Simplify the Equation

Now, let's simplify the equation obtained from the cross multiplication: \( 2x = 100 \). This sets us up to solve for \( x \).

Step 3: Solve for x by Isolating x

To isolate \( x \), divide both sides of the equation \( 2x = 100 \) by 2: \( x = \frac{100}{2} \). This simplifies to \( x = 50 \).

Step 4: Write the Answer as a Simplified Fraction

Since the question asks to write the answer as a fraction in lowest terms, note that \( x \) in this problem is a whole number. Therefore, the fraction representation of \( x = 50 \) in lowest terms is \( \frac{50}{1} \).

Key Concepts

Cross-Multiplying

Cross-multiplying is a useful mathematical technique often applied when solving proportions. It involves multiplying the numerator of one fraction by the denominator of the other fraction, and vice versa. By doing so, we effectively eliminate the fractions and are left with a simple equation.

• This is particularly handy when trying to determine an unknown value in a proportion. For example, in the given exercise, we have the proportion \( \frac{x}{10} = \frac{10}{2} \).
• The process of cross-multiplying here means calculating \( x \times 2 = 10 \times 10 \), as if we are creating an equivalent scenario without fractions.

The resulting expression from cross-multiplying in this case is \( 2x = 100 \).
This serves as the foundation for solving for the unknown variable, making it an indispensable step in proportion problems.

Isolating Variables

Once the equation \( 2x = 100 \) is derived through cross-multiplying, the next goal is to isolate the variable \( x \). Isolating the variable refers to getting the variable by itself on one side of the equation.

• To achieve this with the equation \( 2x = 100 \), we can divide both sides of the equation by 2.
• Performing this division simplifies the equation to \( x = \frac{100}{2} \).

The reason we divide by 2 is because \( x \) is being multiplied by 2, and division is the inverse operation of multiplication.
Ultimately, this results in \( x = 50 \), with \( x \) isolated and clearly visible on its own. This step is crucial as it provides the solution for \( x \).

Simplifying Fractions

In mathematics, simplifying fractions is an essential skill that allows us to express a fraction in its simplest form. It involves reducing the fraction to its lowest terms where the numerator and denominator have no common factors other than 1.

• In situations like the exercise provided, where \( x \) is a whole number, simplifying might involve expressing this as a fraction over 1.
• Here, our calculation revealed \( x = 50 \), which can be written as \( \frac{50}{1} \).

Since 50 and 1 have no common factors other than 1, this fraction is already in its simplest form.
Simplifying makes fractions more comprehensible and is an excellent practice for ensuring clarity and precision in mathematical solutions.