Problem 28 - Elementary Algebra

Question

For the following problems, determine the missing numerator or denomin ator. \(\frac{3}{3}=\frac{?}{9}\)

Step-by-Step Solution

Answer

}{9}\)? Answer: The missing numerator is 9.

Step 1: Set up a proportion between the two fractions

Since the two fractions are equal, we can set up a proportion like this: \(\frac{3}{3} = \frac{?}{9}\)

Step 2: Cross-multiply

To find the missing value, cross-multiply both sides of the proportion: \(3 \times 9 = 3 \times ?\)

Step 3: Simplify and solve

Simplify the equation by multiplying the numbers, and then solve for the unknown value: \(27 = 3 \times ?\) Divide both sides by 3 to solve for the missing value: \(27 \div 3 = ?\)

Step 4: Determine the missing numerator

After dividing, we find the missing numerator: \(9 = ?\) So, the missing numerator is 9, and the fraction is \(\frac{9}{9}\).

Key Concepts

Cross-Multiplication

Cross-multiplication is a reliable technique used to solve equations that involve two fractions set equal to each other, known as proportions. This method involves multiplying the numerator of one fraction by the denominator of the other fraction and setting these products equal to each other. It's crucial for students to understand that cross-multiplication only works when the fractions represent an equality, meaning they are proportional.

For example, let's examine the given exercise. We have two fractions, \frac{3}{3}\text{ and }\frac{?}{9}\. We can use cross-multiplication here by multiplying across the equal sign: \(3 \times 9\) on one side and \(3 \times ?\) on the other. This results in \(27 = 3 \times ?\), which simplifies our problem to a basic algebraic equation that we can solve for the unknown value.

Solving Proportions

Solving proportions is a foundational skill in algebra, important for understanding relationships between quantities. When two ratios are set equal, we say they are in proportion. To solve for an unknown in a proportion, one common strategy is to cross-multiply as illustrated in our exercise. Another strategy could involve scaling up or down if the proportional relationship is immediately apparent.

Once the equation is set up, as in \(27 = 3 \times ?\), we proceed by isolating the variable. We do this by performing the inverse operation. In this case, since the variable is being multiplied by 3, we divide both sides by 3, giving us \(9 = ?\), thereby finding the value of our missing numerator. It's important for students to practice these steps to gain fluency in solving various proportional problems.

Fractions Equivalence

Fraction equivalence is built on the idea that different fractions can represent the same value. Understanding this is crucial when working with proportions because it allows us to manipulate the terms of a fraction without changing its value. For instance, \(\frac{3}{3}\) and \(\frac{9}{9}\) are equivalent fractions because they both simplify to 1.

In the context of the exercise, the proportion \(\frac{3}{3} = \frac{?}{9}\) highlights this concept. By finding the missing numerator, we confirmed that the two fractions are equivalent, as \(\frac{9}{9}\) also simplifies to 1. It's helpful for students to visualize fractional equivalence by thinking of pie charts or other divided shapes, which can represent fractions visually and aid in their understanding of how different fractions can be equivalent.