Problem 22 - 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. $$\frac{\frac{2}{3}}{y}=\frac{\frac{1}{3}}{5}$$

Step-by-Step Solution

Answer

The missing term is \( y = 10 \).

Step 1: Identify the Proportion Formula

The given exercise is a proportion where the ratio \( \frac{2}{3} \) to \( y \) is equal to the ratio \( \frac{1}{3} \) to \( 5 \). The equation resembles the proportion formula: \( \frac{a}{b} = \frac{c}{d} \). For our problem: \( a = \frac{2}{3} \), \( b = y \), \( c = \frac{1}{3} \), and \( d = 5 \).

Step 2: Cross-Multiply

In order to solve for \( y \), we employ cross-multiplication. Multiply the numerator of each ratio by the denominator of the other ratio. This gives: \[ \frac{2}{3} \times 5 = \frac{1}{3} \times y \].

Step 3: Simplify the Equation

Simplify the equation obtained from cross-multiplication: \[ \frac{10}{3} = \frac{y}{3} \]. Our goal is to solve for \( y \).

Step 4: Eliminate the Fraction

To eliminate the fraction, multiply both sides of the equation by 3: \[ 10 = y \]. This gives us the value of \( y \).

Step 5: Write the Solution

The missing term in the proportion is \( y = 10 \). Since it is already in its simplest form, this is the final answer.

Key Concepts

Understanding Cross-Multiplication

Cross-multiplication is a powerful technique used to solve equations that involve proportions. In simple terms, if you have a proportion like \( \frac{a}{b} = \frac{c}{d} \), cross-multiplication involves multiplying the numerator of one fraction by the denominator of the other. For this technique:

• Multiply \( a \) by \( d \) to get one product.
• Multiply \( b \) by \( c \) to get the other product.
• Set these two products equal to one another, resulting in the equation \( a \times d = b \times c \).

Cross-multiplication is effective because it transforms a proportion into an equation without fractions, making it easier to work with. By applying this method, you maintain the equality of the original proportion while simplifying your work.

Getting Comfortable with Fractions

Fractions are everywhere in math, and being at ease with them makes solving equations and understanding proportions much simpler. A fraction represents a part of a whole and is expressed as \( \frac{numerator}{denominator} \). In dealing with fractions, keep these tips in mind:

• The numerator is the top number and represents how many parts you have.
• The denominator is the bottom number and indicates the total number of equal parts the whole is divided into.
• Fractions can be equivalent: \( \frac{1}{2} = \frac{2}{4} \), because they represent the same part of a whole.
• Always simplify fractions when possible by dividing the numerator and the denominator by their greatest common divisor (GCD). For example, the fraction \( \frac{4}{8} \) simplifies to \( \frac{1}{2} \).

Fractions are integral to solving many types of equations and problems, such as those involving proportions. Recognizing developments in fraction manipulation will significantly enhance your mathematical skills and ease transitioning from fractions to whole numbers in proportion problems.

Solving Equations Involving Proportions

Solving equations, especially those involving proportions, requires patience and a step-by-step approach. When faced with an equation like a proportion, the goal is to isolate the variable you're solving for. This sometimes includes the process of simplifying expressions and ensuring simplification of fractions. Here's how to tackle such equations:

• Identify the proportion you are working with, just like \( \frac{\frac{2}{3}}{y} = \frac{\frac{1}{3}}{5} \).
• Apply cross-multiplication: Multiply across the equals sign \( \frac{2}{3} \times 5 \) and \( \frac{1}{3} \times y \).
• Simplify the resulting equation: You might end up with something like \( \frac{10}{3} = \frac{y}{3} \).
• Solve for the unknown: After eliminating fractions by multiplication or other means, solve for the variable. In this example, multiplying both sides by 3 gives \( y = 10 \).

By following these steps faithfully, you can solve not only simple but also complex equations, leading to a correct solution. This method ensures all steps are taken carefully and logically, reinforcing understanding of both the method and the solution.