Problem 41 - Fundamentals of Mathematics

Question

For the following problems, determine the missing numerator or denominator. $$\frac{1}{5}=\frac{?}{30}$$

Step-by-Step Solution

Answer

The missing numerator is 6.

Step 1: Understand the Proportion

The problem provides a proportion: \( \frac{1}{5} = \frac{?}{30} \). Our task is to find the unknown number in the numerator of the second fraction. This means we need to determine what number in the numerator of \( \frac{?}{30} \) will make the two fractions equal.

Step 2: Set Up the Equation

To solve for the missing numerator, we set up an equation based on the equivalence of two fractions. We have:\[ \frac{1}{5} = \frac{x}{30} \]Here, \(x\) represents the unknown numerator.

Step 3: Cross Multiply

To solve for \(x\), we use cross-multiplication. According to the rule of cross multiplication for proportions: \( a \cdot d = b \cdot c \), where \( \frac{a}{b} = \frac{c}{d} \). Applying this to \( \frac{1}{5} = \frac{x}{30} \) gives:\[1 \cdot 30 = 5 \cdot x\]Which simplifies to:\[30 = 5x\]

Step 4: Solve for x

To find \(x\), divide both sides of the equation by 5:\[ x = \frac{30}{5} \]Simplifying gives:\[ x = 6 \]

Step 5: Conclusion

The number that completes the proportion is \(6\). Thus, \( \frac{1}{5} = \frac{6}{30} \). This confirms that both fractions are equivalent, as multiplying the numerator and denominator of \(\frac{1}{5}\) by 6 keeps the two sides of the equation equal.

Key Concepts

Cross Multiplication

Cross multiplication is a technique used to solve proportions, which are equations that express that two ratios or fractions are equivalent. It can help us find an unknown variable quickly. By relating the terms diagonally across the equals sign, we create a simple equation to solve. In the proportion \( \frac{1}{5} = \frac{x}{30} \), we perform cross multiplication by multiplying the numerator of the first fraction by the denominator of the second fraction and vice versa.

Let's look at how this works:

• Multiply the numerator of the first fraction by the denominator of the second fraction: \( 1 \times 30 \).
• Multiply the denominator of the first fraction by the numerator of the second fraction: \( 5 \times x \).

Set the two products equal, leading to the equation \( 30 = 5x \). This allows us to solve for \( x \) by isolating it on one side of the equation. Cross multiplication streamlines our calculations and is a reliable method for solving proportions quickly, making it especially valuable in algebraic operations.

Numerators and Denominators

In fractions, the numerator and denominator play crucial roles. The numerator is the top number and the denominator is the bottom number of a fraction. In the context of proportions, understanding these roles helps us keep balance between two equal fractions.

For example, in the exercise \( \frac{1}{5} = \frac{x}{30} \), \(1\) and \(x\) are numerators, and \(5\) and \(30\) are denominators. The numerator counts how many parts you have, while the denominator shows into how many parts the whole is divided.

To maintain equality between two fractions, the same factors must scale both the numerators and denominators. This principle helps us find the missing piece in a proportion. By solving \( 5x = 30 \) and finding \( x = 6 \), we ensure that the fractions keep the same ratio, thus proving their equivalence.

Equivalent Fractions

Equivalent fractions represent the same part of a whole, even if their numerators and denominators differ. They are a key concept when dealing with proportions, as seen in the exercise where \( \frac{1}{5}\) became \( \frac{6}{30} \) after solving the proportion.

There are several ways to verify equivalent fractions:

• Simplification – simplify both fractions to see if they become identical. Here, \( \frac{6}{30} \) simplifies down by dividing both numbers by their greatest common divisor (6), resulting in \( \frac{1}{5} \).
• Cross multiplication – a quick check for equivalence, as seen when \( 1 \times 30 = 5 \times 6 \).

Recognizing equivalent fractions is essential for understanding proportions and simplifying complex mathematical expressions. It also aids in comparing different sets of fractions without confusion.