Problem 26 - Elementary Algebra

Question

For the following problems, determine the missing numerator or denomin ator. \(\frac{1}{3}=\frac{?}{12}\)

Step-by-Step Solution

Answer

}{12}\)? Answer: The missing numerator is 4.

Step 1: Set up the proportion

To find the missing numerator, set up a proportion with the fractions: \(\frac{1}{3}=\frac{?}{12} \)

Step 2: Cross-multiply

Cross-multiply to solve for the missing numerator. \( 1 \times 12 = 3\times? \)

Step 3: Solve for the missing numerator

Simplify the equation by multiplying the numbers: \( 12 = 3\times? \) Divide both sides by 3 to solve for the missing numerator (?): \( \frac{12}{3}= ? \) \( ?=4 \)

Step 4: Write the complete fraction

Now that we have determined the missing numerator, write the complete fraction: \(\frac{1}{3}=\frac{4}{12}\)

Key Concepts

Proportion

A proportion is a statement that shows two ratios are equal. Think of a ratio like a comparison between two numbers or quantities. When you set two of these equal to each other, you form a proportion.In the exercise, the equation \( \frac{1}{3} = \frac{?}{12} \) forms a proportion. You can think about this visually as two fractions showing that if you multiply the terms of one fraction by the same number, you should get the terms of the other fraction. Key points about proportions:

• They compare two quantities and show their relationship in terms of relative size or quantity.
• They are used to solve for unknown values in problems such as the missing numerator or denominator in fractions.

Understanding proportions helps in solving many mathematical problems, especially those involving comparisons and scale.

Cross-Multiplication

Cross-multiplication is a handy technique for solving equations involving proportions. When you have two fractions set equal to each other, you can cross-multiply to solve for the unknown. Let’s break it down:

• Take the numerator of the first fraction and multiply it by the denominator of the second fraction.
• Do the same for the numerator of the second fraction and the denominator of the first one, essentially forming a cross.

In our example \( \frac{1}{3} = \frac{?}{12} \), cross-multiplying gives us the equation \( 1 \times 12 = 3 \times ? \). This is a crucial step because it transforms a proportion into a simple multiplication equation, making it much easier to solve.With cross-multiplication, you can quickly discover unknown values and understand relationships between different quantities.

Fractions

Fractions are a fundamental concept in mathematics, representing parts of a whole. The fraction \( \frac{1}{3} \) means that 1 is a part of the total value 3. Understanding Fractions:

• The top number (numerator) represents how many parts we are talking about.
• The bottom number (denominator) shows how many parts make up a whole.

In the given problem, the fraction \( \frac{?}{12} \) is looked at. We needed to find what the missing part, or numerator, is when related to the whole of 12.Fractions are seen everywhere, from measuring ingredients in a recipe to examining probabilities. Comprehending how to manipulate and solve problems with fractions, like finding missing numerators or denominators, is essential for building a solid foundation in math.