Problem 28 - 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. $$\frac{x}{12}=\frac{12}{48}$$

Step-by-Step Solution

Answer

The missing term is 3.

Step 1: Identify the Problem Type

We are given a proportion where we need to find the missing term, denoted by \( x \), in the equation \( \frac{x}{12} = \frac{12}{48} \). A proportion means that two ratios are equivalent.

Step 2: Cross Multiply

To solve for \( x \) in the proportion \( \frac{x}{12} = \frac{12}{48} \), we can use cross-multiplication. This involves multiplying the numerator of one ratio by the denominator of the other ratio: \( x \times 48 = 12 \times 12 \).

Step 3: Compute the Products

Calculate the products from the cross-multiplication step: \( x \times 48 = 144 \). So, the equation becomes \( 48x = 144 \).

Step 4: Solve for the Unknown \( x \)

Divide both sides of the equation \( 48x = 144 \) by 48 to isolate \( x \): \( x = \frac{144}{48} \).

Step 5: Simplify the Fraction

Simplify \( \frac{144}{48} \) by finding the greatest common divisor (GCD) of 144 and 48, which is 48. Therefore, \( \frac{144}{48} = 3 \).

Step 6: Conclude the Solution

Thus, the missing term \( x \) in the original proportion is 3.

Key Concepts

Cross-Multiplication

Cross-multiplication is a powerful tool that allows us to solve for unknowns in proportions. It works by eliminating the fractions, which makes the equation easier to manage. When we have two fractions set equal to each other, like \( \frac{x}{12} = \frac{12}{48} \), cross-multiplication says to multiply the numerator of one fraction by the denominator of the other:

• Multiply \( x \) by 48, which gives us \( 48x \).
• Multiply 12 by 12, which provides 144.

This step transforms our equation into a more straightforward algebraic expression: \( 48x = 144 \). By removing the fractions, cross-multiplication simplifies the process, allowing us to focus on solving a simple equation.

Equivalent Ratios

When we say two ratios are equivalent, we mean they express the same relationship between numbers. In the problem \( \frac{x}{12} = \frac{12}{48} \), both ratios compare quantities in the same way.

• For a ratio \( \frac{a}{b} \), it is equivalent to another ratio \( \frac{c}{d} \) if \( a\times d = b \times c \).
• This means the cross-products are equal, confirming equivalence.

Understanding equivalent ratios helps in comparing different quantities by maintaining the same proportion. We're seeing here that, even though \( x \) wasn't initially obvious, it will uphold the equality when found. It guarantees that these two quantities bear the samerelation.

Simplifying Fractions

Simplifying fractions is all about making them as easy to work with as possible, by reducing them to their lowest terms. Once we found that \( x = \frac{144}{48} \), we further simplify this fraction by looking for the greatest common divisor (GCD) of the numerator and denominator. In this case:

• The GCD of 144 and 48 is 48.
• Divide both the numerator and denominator by 48.
• This simplifies the fraction to \( \frac{3}{1} \), or simply 3.

Simplifying fractions not only makes calculations easier but also ensures accuracy and clarity. It lets us see the essence of the fraction, providing a true representation of the ratio or relationship it describes.