Problem 12 - Fundamentals of Mathematics

Question

For problems 12 and 13, determine the missing numerator or denominator. (Section 4.4) \(\frac{5}{9}=\frac{?}{36}\)

Step-by-Step Solution

Answer

The missing numerator is 20.

Step 1: Understand the Problem

We are given a fraction \(\frac{5}{9}\) and asked to find the missing numerator such that it equals \(\frac{?}{36}\). This means we need to find a number that, when placed in the numerator, makes the two fractions equivalent.

Step 2: Set Up the Proportion

Since the two fractions are equivalent, we can set up the proportion: \(\frac{5}{9} = \frac{x}{36}\), where \(x\) is the missing numerator we need to find.

Step 3: Cross Multiply

To solve for \(x\), we use cross multiplication. This means we multiply the numerator of the first fraction by the denominator of the second fraction, and the denominator of the first fraction by the numerator of the second fraction: \(5 \times 36 = 9 \times x\).

Step 4: Solve the Equation

Simplify the equation from the cross multiplication. We have: \(5 \times 36 = 180\), so the equation becomes \(180 = 9x\). Next, divide both sides by 9 to solve for \(x\): \(x = \frac{180}{9} = 20\).

Step 5: Conclusion

The missing numerator that makes \(\frac{5}{9}\) equivalent to \(\frac{?}{36}\) is 20. Therefore, \(\frac{5}{9} = \frac{20}{36}\).

Key Concepts

Equivalent Fractions

Fractions are considered equivalent when they represent the same part of a whole, even though they may look different. For example, the fractions \( \frac{5}{9} \) and \( \frac{20}{36} \) are equivalent. This means they have the same value, but with different numerators and denominators. When working with fractions, it's important to recognize their equivalence. This helps in simplifying fractions or solving for unknowns just like in our exercise. There are several ways to check if two fractions are equivalent:

• Cross-multiplication: If the cross-products are equal, the fractions are equivalent.
• Simplifying them: Reduce both fractions to their simplest form to see if they are the same.
• Using a common denominator: Convert both fractions to the same denominator and compare numerators.

Recognizing equivalent fractions simplifies problems, especially when converting measurements or comparing sizes.

Cross Multiplication

Cross multiplication is a handy tool used in solving equations involving fractions, especially proportions. It involves multiplying diagonally across the equal sign. This operation helps to eliminate fractions by creating a simpler equation that is easier to solve. Here's how it works in practice:- Consider the proportion \( \frac{a}{b} = \frac{c}{d} \).- You multiply across: \( a \times d = b \times c \).- This gives you a new equation without fractions that you can solve for any unknowns.In our specific example, the equation \( \frac{5}{9} = \frac{x}{36} \) becomes \( 5 \times 36 = 9 \times x \) through cross multiplication. This simplifies the process by removing the fractions, making the arithmetic straightforward.

Proportion Solving

Proportion solving is the process of finding the value of an unknown variable that maintains the equality of two ratios. In our exercise, we start by recognizing that two fractions are supposed to be equal. Steps for solving proportions include:

• Setting up the equation: Clearly write out the proportion. For example, \( \frac{5}{9} = \frac{x}{36} \).
• Cross multiply: Perform cross multiplication to get an equation without fractions, such as \( 5 \times 36 = 9 \times x \).
• Solve for the unknown: Simplify the equation by performing basic algebraic operations, like division, to isolate the variable.

By following these steps, you will solve for the unknown value. This technique is invaluable when dealing with ratios in real-life applications, such as recipe adjustments, map reading, and more.