Problem 44 - 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. $$\frac{321}{1,128}=\frac{x}{376}$$

Step-by-Step Solution

Answer

\(x = \frac{5,029}{47}\).

Step 1: Cross-Multiply

In a proportion, the product of the means equals the product of the extremes. For the given proportion \(\frac{321}{1128} = \frac{x}{376}\), cross-multiply to obtain the equation \(321 \times 376 = 1128 \times x\).

Step 2: Calculate the Left Side Product

Calculate \(321 \times 376\). This gives you 120,696.

Step 3: Set the Equation

Set the equation from Step 1 as \(120,696 = 1128 \times x\).

Step 4: Solve for \(x\)

To solve for \(x\), divide both sides of the equation by 1128: \[x = \frac{120,696}{1128}\].

Step 5: Simplify the Fraction

Simplify \(\frac{120,696}{1128}\) to its lowest terms. The greatest common divisor of 120,696 and 1128 is 24, so divide both the numerator and the denominator by 24. This simplifies the fraction to \(\frac{5,029}{47}\).

Key Concepts

Understanding Proportions

Proportions are an essential part of prealgebra that help us compare two ratios or fractions. In simple terms, a proportion states that two ratios are equal. For example, in the exercise, we have a proportion \(\frac{321}{1128} = \frac{x}{376}\). This means the ratio of 321 to 1128 is the same as the ratio of \(x\) to 376. This concept is useful in solving problems where you need to find an unknown value within related quantities. Here’s how proportions can be identified:

• They are written in the form \(\frac{a}{b} = \frac{c}{d}\).
• The terms \(a, b, c,\) and \(d\) should satisfy \(a \times d = b \times c\).
• If one value is unknown, it can be calculated when the other three are known.

Recognizing proportions is the first step. They help create equations that can be solved systematically, making them a powerful tool in mathematics.

Cross-Multiplication Technique

The cross-multiplication method is a simple yet powerful technique used to solve proportions. It involves multiplying the terms across the equals sign in a proportion to form an equation. In our example proportion \(\frac{321}{1128} = \frac{x}{376}\), the cross-multiplication step involves the following:

• Multiply the numerator of the first fraction by the denominator of the second fraction: \(321 \times 376\).
• Multiply the denominator of the first fraction by the numerator of the second fraction: \(1128 \times x\).
• Set the two products equal to each other: \(321 \times 376 = 1128 \times x\).

This method is effective because it uses the principle that the cross-products of a proportion are equal. It simplifies the solving of the equation, allowing you to isolate the unknown variable, \(x\), which can then be calculated to find its value. It's a crucial skill for tackling all problems involving proportions.

Simplifying Fractions to Lowest Terms

Simplifying fractions to their lowest terms is a key part of algebra and everyday math. A fraction is in its lowest terms when the greatest common divisor (GCD) of the numerator and the denominator is 1. This means there's no number other than 1 that can divide both numbers evenly. To simplify a fraction:

• Find the GCD of the numerator and the denominator.
• Divide both the numerator and the denominator by this GCD.
• The resulting fraction is the simplest form of the original fraction.

In the solution given, you had the fraction \(\frac{120,696}{1128}\). The GCD here is 24. By dividing both the numerator and the denominator by 24, you arrive at \(\frac{5,029}{47}\). Simplifying fractions makes calculations more manageable and understanding clearer, especially when comparing results or checking work.