Problem 56 - Fundamentals of Mathematics

Question

For the following problems, determine the missing numerator or denominator. $$\frac{9}{13}=\frac{?}{286}$$

Step-by-Step Solution

Answer

The missing numerator is 198.

Step 1: Understand the Problem

We are given the fraction \( \frac{9}{13} \) and need to find the missing numerator such that the two fractions \( \frac{9}{13} \) and \( \frac{?}{286} \) are equivalent.

Step 2: Set Up Proportion Equation

Since the two fractions are equivalent, we can write the equation: \( \frac{9}{13} = \frac{x}{286} \), where \( x \) is the missing numerator. The basic principle here is that the cross-products of equivalent fractions are equal.

Step 3: Cross-Multiply

To solve for \( x \), we cross-multiply the fractions. This gives us the equation: \( 9 \times 286 = 13 \times x \).

Step 4: Calculate the Cross-Product

Calculate \( 9 \times 286 \,\), which gives us 2574, so the equation becomes \( 2574 = 13 \times x \).

Step 5: Solve for x

To find \( x \), divide both sides of the equation by 13: \( x = \frac{2574}{13} \).

Step 6: Compute the Division

Perform the division: \( \frac{2574}{13} = 198 \). Thus, the value of \( x \) is 198.

Step 7: Verify the Solution

Verify your solution by ensuring that \( \frac{9}{13} = \frac{198}{286} \). Simplify \( \frac{198}{286} \), which reduces to \( \frac{9}{13} \), confirming that our solution is correct.

Key Concepts

Numerator

In a fraction, the numerator is the top number which indicates how many parts of a whole are being considered. For example, in the fraction \( \frac{9}{13} \), 9 is the numerator. It represents the part of the whole we are focusing on or comparing— in this case, 9 parts of a possible 13. When looking for a missing numerator, as seen in the problem \( \frac{9}{13} = \frac{x}{286} \), the unknown numerator \( x \) represents the number corresponding proportionally to 9 in the given fraction.

• The numerator can be found by solving a proportion equation.
• A change in the numerator affects the size of the fraction value, assuming the denominator is constant.

Denominator

The denominator in a fraction is the bottom number which shows the number of equal parts that make up a whole. In \( \frac{9}{13} \), the denominator is 13, indicating that the whole is split into 13 equal parts. In problems involving fraction equivalence, understanding the role of the denominator is essential because it affects the size of parts each unit of the numerator represents. As you analyze \( \frac{9}{13} = \frac{x}{286} \), note that the denominator 286 in the second fraction tells us how many parts the whole is assumed to have in this proportion.

• The denominator provides a scale or basis for comparison across fractions.
• Changing the denominator alters the fraction's division, affecting its equivalency.

Cross-multiplication

Cross-multiplication is a technique used to compare two fractions by multiplying the numerator of each fraction by the denominator of the other. It is most useful for solving equations involving equivalent fractions. In this exercise, for instance, solving \( \frac{9}{13} = \frac{x}{286} \) requires cross-multiplying: \( 9 \times 286 = 13 \times x \). This fundamental step allows us to eliminate the fractions and compare the whole numbers directly.

• Enables easy manipulation and resolution of a fraction equation.
• Clarifies the relationship between the numerators and denominators.

Cross-multiplication is an invaluable skill for anyone working with fractions. It provides a pathway directly to the solution without needing to convert to a common denominator.

Proportion Equation

A proportion equation shows that two fractions are equal to each other. In the context of this problem, \( \frac{9}{13} = \frac{x}{286} \) is our proportion equation. It provides a framework to find an unknown value, either a numerator or denominator, by using relationships between other known values. Solving this involves the methods of cross-multiplication and will eventually help identify the missing piece of the fraction.

• Proportion equations are valuable in many fields such as science, engineering, and everyday problem-solving.
• The equality shows a balance, where altering one element requires a corresponding change elsewhere to maintain equivalency.

Understanding these equations is also an important foundation for studying ratios in deeper mathematical contexts.